```julia
using Distributions
using Random
using LinearAlgebra
using ProgressMeter
E = LinearAlgebra.I
eye(n) = Matrix(1.0E,n,n) #identity matrix

#
#import Fontconfig, Cairo
using Gadfly
```

    ┌ Info: Precompiling Distributions [31c24e10-a181-5473-b8eb-7969acd0382f]
    └ @ Base loading.jl:1273
    ┌ Info: Precompiling ProgressMeter [92933f4c-e287-5a05-a399-4b506db050ca]
    └ @ Base loading.jl:1273
    ┌ Info: Precompiling Gadfly [c91e804a-d5a3-530f-b6f0-dfbca275c004]
    └ @ Base loading.jl:1273

# Numerical Calculation of Real Log Canonical Threshold in LDA using Gibbs Sampler

## True, Learner, and Prior

Let $x \in \mathrm{Onehot}(M)$, $y\in \mathrm{Onehot}(H)$ and $z\in \mathrm{Onehot}(N)$ be the word, the topic, and the document, respectively. The true or optimal topic is $y^0 \in \mathrm{Onehot}(H_0)$.

The true (data generating) distribution is
$$q(x,y^0|z) = \prod_{j=1}^N \left[\prod_{k=1}^{H_0} \left(b^0_{kj} \prod_{i=1}^M (a^0_{ik})^{x_{i}}\right)^{y^0_k}\right]^{z_j}.$$

The learning machine (statistical or likelihood model) is
$$p(x,y|z,A,B) = \prod_{j=1}^N \left[\prod_{k=1}^{H} \left(b_{kj} \prod_{i=1}^M (a_{ik})^{x_{i}}\right)^{y_k}\right]^{z_j}.$$

The parameters $A=(a_{ik})$ and $B=(b_{kj})$ are stochastic matrices;
$\sum_{i=1}^M a_{ik}=1$ and $\sum_{k=1}^H b_{kj}=1$ hold.
Thus, the distributions $p$ and $q$ are *mixtures of categorical mixture models*.
The element $a_{ik}$ means the probability that the word is $i$ when the topic is $k$.
Same as, the element $b_{kj}$ means the probability that the topic is $k$ when the document is $j$.

The prior distribution is a product of (symmetry) Dirichlet distributions since the parameters are in simplexes;
$$\varphi(A) = \prod_{k=1}^H\mathrm{Dir}(a_k|\alpha_k) \propto \prod_{k=1}^H \prod_{i=1}^M a_{ik}^{\alpha_k -1 },$$
$$\varphi(B) = \prod_{j=1}^N\mathrm{Dir}(b_j|\beta_j)\propto \prod_{j=1}^N \prod_{k=1}^H b_{kj}^{\beta_j -1 }.$$

## Deriving Gibbs Sampler

### Claim: Gibbs Sampler

The posterior is a product of Dirichlet distributions;
$$\psi(A,B|X^n,Y^n,Z^n)=\prod_{k=1}^H \mathrm{Dir}(a_k|\hat{\alpha}_k)\prod_{j=1}^N \mathrm{Dir}(b_j|\hat{\beta}_j),$$
where
$$\hat{\alpha}_{k} =(\hat{\alpha}_{ik}),  \hat{\alpha}_{ik}=\sum_{l=1}^n y(l)_k  x(l)_i + \alpha_k,$$
$$\hat{\beta}_{j} =(\hat{\beta}_{kj}),  \hat{\beta}_{kj}=\sum_{l=1}^n z(l)_j  y(l)_k + \beta_k.$$

The topic distribution is a product of categorical distributions;
$$Pdf(Y^n|X^n,Z^n,A,B) = \prod_{l=1}^n \mathrm{Cat}(y(l) | \eta_l),$$
where
$$\eta_l = (\eta_{lk}),$$
$$\eta_{lk} \propto \exp \left( \sum_{j=1}^N z(l)_j \left( \log b_{kj}
+\sum_{i=1}^M x(l)_i \log a_{ik}\right) \right),$$
$$\sum_{k=1}^H \eta_{lk}=1.$$

### Proof Focus

We derive the GS algorithm for alternately sampling from the conditional distribution of the parameters given the topics and the one of the topics given the parameters. Our goal posterior distribution is
$$\psi(A,B|X^n,Y^n,Z^n) \propto Pdf(X^n,Y^n,Z^n,A,B) = P(X^n, Y^n|Z^n,A,B)\varphi(A)\varphi(B),$$
where $$P(X^n, Y^n|Z^n,A,B) = \prod_{l=1}^n p(x(l), y(l)|z(l),A,B)$$
is the likelihood.

On the other hand, the probability density function of the topic distribution given the parameter and the data is
$$Pdf(Y^n|X^n,Z^n,A,B) \propto Pdf(X^n,Y^n,Z^n,A,B) = P(X^n, Y^n|Z^n,A,B)\varphi(A)\varphi(B).$$

Both densities are in proportion to the joint distribution. For this reason, the simultaneous densities are transformed to be regarded as respective densities.

### Proof

The log of the prior:
$$\log \varphi(A) = \sum_{k=1}^H \sum_{i=1}^M (\alpha_k - 1)\log a_{ik}+const. ,$$
$$\log \varphi(B) = \sum_{j=1}^N \sum_{k=1}^H (\beta_k - 1)\log b_{kj}+const.$$

The log of the likelihood:
\begin{align}
\log P(X^n, Y^n|Z^n,A,B) &= \sum_{l=1}^n \log p(x(l), y(l)|z(l),A,B) \\
&= \sum_{l=1}^n\sum_{j=1}^N z(l)_j \sum_{k=1}^{H} y(l)_k \left( \log b_{kj}+ \sum_{i=1}^M x(l)_i \log a_{ik} \right).
\end{align}

The log of the posterior is
\begin{align}
\log \psi(A,B|X^n,Y^n,Z^n) &= \log P(X^n, Y^n|Z^n,A,B)+\log\varphi(A)+\log\varphi(B)+const. \\
&= \sum_{k=1}^{H}\sum_{j=1}^N\sum_{l=1}^n z(l)_j  y(l)_k \left( \log b_{kj}+ \sum_{i=1}^M x(l)_i \log a_{ik} \right)
+\sum_{k=1}^H \sum_{i=1}^M (\alpha_k - 1)\log a_{ik}
+\sum_{j=1}^N \sum_{k=1}^H (\beta_k - 1)\log b_{kj}+const. \\
&= \sum_{k=1}^{H}\sum_{j=1}^N\sum_{l=1}^n z(l)_j  y(l)_k \log b_{kj}
+\sum_{i=1}^M\sum_{k=1}^{H}\sum_{j=1}^N\sum_{l=1}^n z(l)_j  y(l)_k  x(l)_i \log a_{ik}
+\sum_{i=1}^M \sum_{k=1}^H (\alpha_k - 1)\log a_{ik}
+\sum_{k=1}^H \sum_{j=1}^N (\beta_k - 1)\log b_{kj}+const. \\
&= \sum_{k=1}^H \sum_{j=1}^N \left( \sum_{l=1}^n z(l)_j  y(l)_k + \beta_k - 1\right)\log b_{kj}
+ \sum_{i=1}^M \sum_{k=1}^H \left( \sum_{j=1}^N\sum_{l=1}^n z(l)_j  y(l)_k  x(l)_i + \alpha_k - 1 \right)\log a_{ik} + const.
\end{align}
Therefore, the posterior is a product of Dirichlet distributions;
$$\psi(A,B|X^n,Y^n,Z^n)=\prod_{k=1}^H \mathrm{Dir}(a_k|\hat{\alpha}_k)\prod_{j=1}^N \mathrm{Dir}(b_j|\hat{\beta}_j),$$
where
$$\hat{\alpha}_{k} =(\hat{\alpha}_{ik}),  \hat{\alpha}_{ik}=\sum_{l=1}^n y(l)_k  x(l)_i + \alpha_k,$$
$$\hat{\beta}_{j} =(\hat{\beta}_{kj}),  \hat{\beta}_{kj}=\sum_{l=1}^n z(l)_j  y(l)_k + \beta_k.$$

On the other hand, the log of the topic distribution is
\begin{align}
\log Pdf(Y^n|X^n,Z^n,A,B) &= \log P(X^n, Y^n|Z^n,A,B)+\log\varphi(A)+\log\varphi(B)+const. \\
&= \sum_{l=1}^n\sum_{k=1}^{H}\sum_{j=1}^N z(l)_j  y(l)_k \log b_{kj}
+\sum_{l=1}^n\sum_{k=1}^{H}\sum_{i=1}^M\sum_{j=1}^N z(l)_j  y(l)_k  x(l)_i \log a_{ik}
+\sum_{k=1}^H\sum_{i=1}^M  (\alpha_k - 1)\log a_{ik}
+\sum_{k=1}^H \sum_{j=1}^N (\beta_k - 1)\log b_{kj}+const. \\
&= \sum_{l=1}^n\sum_{k=1}^{H} y(l)_k\left( \sum_{j=1}^N z(l)_j   \log b_{kj}
+\sum_{i=1}^M\sum_{j=1}^N z(l)_j x(l)_i \log a_{ik}\right)
+\sum_{k=1}^H \left( \sum_{i=1}^M(\alpha_k - 1)\log a_{ik}+\sum_{j=1}^N(\beta_k - 1)\log b_{kj}\right)+const. \\
&= \sum_{l=1}^n\sum_{k=1}^{H} y(l)_k\left[ \sum_{j=1}^N z(l)_j \left( \log b_{kj}
+\sum_{i=1}^M x(l)_i \log a_{ik}\right)\right]
+const. \\
\end{align}
Therefore, the topic distribution is a product of categorical distributions;
$$Pdf(Y^n|X^n,Z^n,A,B) = \prod_{l=1}^n \mathrm{Cat}(y(l) | \eta_l),$$
where
$$\eta_l = (\eta_{lk}),$$
$$\eta_{lk} \propto \exp \left( \sum_{j=1}^N z(l)_j \left( \log b_{kj}
+\sum_{i=1}^M x(l)_i \log a_{ik}\right) \right),$$
$$\sum_{k=1}^H \eta_{lk}=1.$$

```julia
@doc """
    rlct_R(M,N,H,r)

Calculate the real log canonical threshold of reduced rank regression
when the input dimmension is M, the output one is N,
the learner's rank is H, and the true rank is r.

The arguments must be positive integers and H>=r must hold.

# Example
```julia-repl
julia> rlct_R(4,3,2,1)
4.0
```
"""
function rlct_R(M,N,H,r)
    H_0 = r
    @assert (M>=1)&(N>=1)&(H>=H_0)&(H_0>=0)
    if (N+H_0<=M+H) & (M+H_0<=N+H) & (H+H_0<=M+N)
        if (M+N+H+H_0)%2 == 0
            return (2*(H+H_0)*(M+N)-(M-N)^2-(H+H_0)^2)/8
        else
            return (2*(H+H_0)*(M+N)-(M-N)^2-(H+H_0)^2 +1)/8
        end
    elseif M+H<N+H_0
        return (M*H+N*H_0-H*H_0)/2
    elseif N+H<M+H_0
        return (N*H+M*H_0-H*H_0)/2
    elseif M+N<H+H_0
        return M*N/2
    else #must not occur!
        return -1
    end
end

@doc """
    rlct_LDA(M,N,H,r)

Calculate the real log canonical threshold of latent Dirichlet allocation
when the input dimmension is M, the output one is N,
the learner's rank is H, and the rank of the transformed matrix U_0V_0 is r.
The rigorous definition of r is stated in the paper.

M>=2 and N>=2 and H>=r+1>=1 must hold.

# Example
```julia-repl
julia> rlct_LDA(4,3,2,1)
3.5
```
"""
function rlct_LDA(M,N,H,r)
    @assert (M>=2)&(N>=2)&(H>=r+1)&(r+1>=1)
    @assert (rlct_R(M,N,H,r+1)-N/2)==(rlct_R(M-1,N-1,H-1,r)+M/2-1/2)
    return rlct_R(M,N,H,r+1)-N/2
end

@doc """
    calc_r(A0, B0)

Calculate an intrinsic value of the true parameter as (A0, B0).
Transform from A0 and B0 to U0 and V0 and compute the rank of U_0V_0.
"""
function calc_r(A0, B0)
    M, H0 = size(A0)
    H02, N = size(B0)
    @assert H0==H02
    tildeA = A0[1:(M-1), 1:(H0-1)]
    a = A0[1:(M-1), H0]
    tildea = reduce(hcat, [a for _ in 1:(H0-1)])
    tildeB = B0[1:(H0-1), 1:(N-1)]
    b = B0[1:(H0-1), N]
    tildeb = reduce(hcat, [b for _ in 1:(N-1)])
    U0 = tildeA - tildea
    V0 = tildeB - tildeb
    return rank(U0*V0)
end

```

    calc_r

## Experiment

```julia
# fundamaental setting
## calc. RLCT
SIM_ITERS = 100
## MCMC
MCMC_K = 2000
BURNIN = 10000
THIN = 20
MCMC_ITER = BURNIN + THIN*MCMC_K
## hyperparameter in prior
ALPHA = 1.0
BETA = 1.0
## true and model
M = 4
N = 4
H_0 = 2
Random.seed!(1)
A_0 = hcat([rand(Dirichlet(ALPHA*ones(M))) for k in 1:H_0]...)
B_0 = hcat([rand(Dirichlet(BETA*ones(H_0))) for j in 1:N]...)
#### control variable (number of learner's topics)
H = 4
r = calc_r(A_0, B_0)

## data
SEED = 2020 #don't tune
L = 500 #number of words in each document
SAMPLESIZE = N*L
TESTSIZE = 200*SAMPLESIZE
## save path
LOG_FILE_PATH = joinpath("log","YYYYMMDD-LDA-H=$H-n=$SAMPLESIZE-T=$TESTSIZE.csv")
println(A_0)
println(B_0)
```

    [0.3472736053441858 0.06429606435264494; 0.3692397045645508 0.14468121592345295; 0.27001287227749055 0.41156519234939487; 0.01347381781377272 0.37945752737450733]
    [0.542803694828061 0.4266223529710219 0.19671787990538073 0.6861112342805497; 0.457196305171939 0.5733776470289782 0.8032821200946193 0.31388876571945046]

```julia
rlct_LDA(M,N,H,H_0)
```

    5.5

```julia
@doc """
    generate_data(A::Array{Float64,2}, B::Array{Float64,2}; n::Int=SAMPLESIZE, seed=1)

    Generate artificiall data from the given LDA model like the following.
    for l=1:n
        generate words from Cat(x|AB[:,j]) for j=1,...,N
        as two format: docs array (N, n//N, M) and words array (n, M)
    end

    # Argments

    - A::Array{Float64,2}: stochastic matrix A= (the probability that the word is i when the topic is k)
      whose shape is (M,H).
    - B::Array{Float64,2}: stochastic matrix B= (the probability that the topic is k when the doc is j)
      whose shape is (H,N).
    - n::Int=SAMPLESIZE: the number of all words in all document. sample size.
    - seed::Int=1: the random seed.

    # Return
    - return: generated words (onehot(M) vectors) as two format
    - rtype: tuple(Array; shape is (N, n//N, M), Array; shape is (n, M))
"""
function generate_data(A::Array{Float64,2}, B::Array{Float64,2}; n::Int=SAMPLESIZE, seed=1)
    Random.seed!(seed);
    C = A*B
    M = size(C)[1]
    N = size(C)[2]
    L = div(n, N)
    @assert n%N==0
    words_arr = zeros(Bool,M,n)
    for j in 1:N
        words = rand(Multinomial(1, C[:,j]), L)
        words_arr[:,(1+(j-1)*L):(j*L)] = words
    end
    docs_arr = reshape(words_arr, (M,L,N))
    return (docs_arr, words_arr)
end
```

    generate_data

```julia
function make_onehot(hot_ind::Int, dim::Int; binary_type=Bool)
    @assert hot_ind <= dim
    @assert 1 <= hot_ind
    v = zeros(binary_type, dim);
    v[hot_ind] = 1;
    return v
end

function get_doc_onehot(docs_arr::Array{Bool,3}; n::Int=SAMPLESIZE)
    #===
    :return: Array of onehot vectors which mean each document id
    :rtype: Array{Float64,2}; shape is (N, n). each column means the document id corresponding word.
    ===#
    N = size(docs_arr)[3]
    L = div(n, N)
    doc_onehot = hcat([hcat([make_onehot(j, N) for l in 1:L]...) for j in 1:N]...)
    return doc_onehot
end
```

    get_doc_onehot (generic function with 1 method)

```julia
# each sampling step for Gibbs sampler
@doc """
    topic_step(docs_arr::Array{Bool,3}, A::Array{Float64,2}, B::Array{Float64,2}; n=SAMPLESIZE)

Generate a hidden variable (i.e. topic) in Gibbs sampler.

# Arguments
- docs_arr::Array{Bool,3}: data as documents form whose shape is (M, L==n//N, N).
- A::Array{Float64,2}: parameter matrix which represents word appearance probability for each topic.
- B::Array{Float64,2}: parameter matrix which represents topic propotion for each document.
- n::Int=SAMPLESIZE: sample size i.e. number of words.

# Return
- return: Bool topic matrix Y whose shape is (H, n).
"""
function topic_step(docs_arr::Array{Bool,3}, A::Array{Float64,2}, B::Array{Float64,2}; n=SAMPLESIZE)
    H = size(A)[2]
    N = size(B)[2]
    L = div(n, N)
    tmp = hcat([vcat(reshape(sum(docs_arr.*log.(A[:,k]),dims=1), (L,N)) .+ log.(B[k,:])'...) for k in 1:H]...)
    tmp = exp.(tmp)
    eta = tmp ./ sum(tmp, dims=2)
    Y = Array{Bool}(hcat([rand(Multinomial(1,eta[l,:])) for l in 1:n]...))
    return Y
end

@doc """
    A_step(words_arr::Array{Bool,2}, Y::Array{Bool,2}, alpha; n=SAMPLESIZE)

Generate a stochastic matrix in Gibbs sampler.

# Arguments
- words_arr::Array{Bool,2}: data as words form whose shape is (M, n).
- Y::Array{Bool,2}: topic matrix whose shape is (H, n).
- alpha: hyperparameter of Dirichlet prior for A.
- n::Int=SAMPLESIZE: sample size i.e. number of words.

# Return
- return: stochastic matrix which represents word appearance probability for each topic.
"""
function A_step(words_arr::Array{Bool,2}, Y::Array{Bool,2}, alpha; n=SAMPLESIZE)
    H = size(Y)[1]
    M = size(words_arr)[1]
    #use_words = reshape(words_arr, (n, M))
    hat_alpha = (Y*words_arr')' .+ alpha
    A = hcat([rand(Dirichlet(hat_alpha[:,k])) for k in 1:H]...)
    return A
end

@doc """
    B_step(doc_onehot::Array{Bool,2}, Y::Array{Bool,2}, beta; n=SAMPLESIZE)

Generate a stochastic matrix in Gibbs sampler.

# Arguments
- docs_arr::Array{Bool,3}: data as documents form whose shape is (M, L==n//N, N).
- Y::Array{Bool,2}: topic matrix whose shape is (H, n).
- beta: hyperparameter of Dirichlet prior for B.
- n::Int=SAMPLESIZE: sample size i.e. number of words.

# Return
- return: stochastic matrix which topic propotion for each document.
"""
function B_step(doc_onehot::Array{Bool,2}, Y::Array{Bool,2}, beta; n=SAMPLESIZE)
    #===
    (topic x num_of_docs) stochastic matrix sampling step.

    :param docs_arr: data as documents form; shape is (M, L=n//N, N)
    :param Y: topic matrix; shape is (H, n)
    :param beta: hyperparameter of Dirichlet prior for B
    :type beta: T(which means scalar) or Array{T,1}(which means 1-d vector); length is N. Where T is Int or Float.
    :return: param matrix B
    ===#
    H = size(Y)[1]
    N = size(doc_onehot)[1]
    #useZ = get_doc_onehot(docs_arr)'
    hat_beta = Y*(doc_onehot') .+ beta
    B = hcat([rand(Dirichlet(hat_beta[:,j])) for j in 1:N]...)
    return B
end
```

    B_step

The pmf of model is
$$p(x|z,A,B) = \prod_{j=1}^N \left(\sum_{k=1}^{H} b_{kj} \prod_{i=1}^M (a_{ik})^{x_{i}}\right)^{z_j}. $$
If $z_j=1$, then the pmf is
$$p(x|z_j=1,A,B) = \sum_{k=1}^{H} b_{kj} \prod_{i=1}^M (a_{ik})^{x_{i}}.$$
And if $x_i=1$, then the pmf is
$$p(x_i=1|z_j=1, A,B) = \sum_{k=1}^{H} b_{kj} a_{ik} = (AB)_{ij}.$$

Intuitively, this is why Latent Dirichlet Allocation is equivalent to Stochastic Matrix Factorization.

```julia
@doc """
    model_pmf(x::Array{Bool,1}, z::Array{Bool,1}, A::Array{Float64,2}, B::Array{Float64,2})

Calculate a probability mass for given data and parameter.

# Arguments
- x::Array{Bool,1}: M-dim onehot vector which means a word.
- z::Array{Bool,1}: N-dim onehot vector which means a document.
- A::Array{Float64,2}: (M,H) stochastic matrix
  whose (i,k) element means a probability that the word is i when the topic is k.
- B::Array{Float64,2}: (H,N) stochastic matrix
  whose (k,j) element means a probability that the topic is k when the document is j.
"""
function model_pmf(x::Array{Bool,1}, z::Array{Bool,1}, A::Array{Float64,2}, B::Array{Float64,2})
    #===
    probability mass function of model p(x|z,A,B).
    See also the above markdown cell.

    :param x: an M-dim onehot vector which means a word.
    :param x: an N-dim onehot vector which means a document.
    :param A: an (M,H) stochastic matrix whose (i,k) element means a probability that the word is i when the topic is k.
    :param B: an (H,N) stochastic matrix whose (k,j) element means a probability that the topic is k when the document is j.
    ===#
    @assert sum(z) == 1
    @assert sum(x) == 1
    @assert size(A)[2] == size(B)[1]
    j = argmax(z) #z_j=1, z_{j'}=0 (j' != j)
    i = argmax(x) #x_i=1, x_{i'}=0 (i' != i)
    AB = A*B
    return AB[i,j]
end
```

    model_pmf

```julia
# Gibbs sampler
function run_all_sampling!(
        docs_arr::Array{Bool,3}, words_arr::Array{Bool,2}, doc_onehot::Array{Bool,2}, init_Y::Array{Bool,2},
        allYs::Array{Bool,3}, allAs::Array{Float64,3}, allBs::Array{Float64,3})
    iters = size(allYs)[3];
    alpha = ALPHA;
    beta = BETA;
    sampling_progress = Progress(iters);
    Y = init_Y;
    @time for k in 1:iters
        ## parameter matrix sampling and saving
        A = A_step(words_arr, Y, alpha);
        B = B_step(doc_onehot, Y, beta);
        allAs[:,:,k] = A;
        allBs[:,:,k] = B;
        ## hidden variable (topic indicator) sampling and saving
        Y = topic_step(docs_arr, A, B);
        allYs[:,:,k] = Y;
        ## Progress update
        next!(sampling_progress);
    end
    #return (allYs, allAs, allBs)
end

#===
function calc_likelihoodMat!(
        k::Int, words_arr::Array{Bool,2}, doc_onehot::Array{Bool,2}, A::Array{Float64,2}, B::Array{Float64,2},
        likelihoodMat::Array{Float64,2}; n::Int=SAMPLESIZE)
    #===
    ===#
    ## generated quantity: likelihood matrix
    for l in 1:n
        x = words_arr[:,l];
        z = doc_onehot[:,l];
        likelihoodMat[l,k] = model_pmf(x,z,A,B);
    end
    return likelihoodMat
end
===#

function run_thining!(
        words_arr::Array{Bool,2}, doc_onehot::Array{Bool,2},
        allYs::Array{Bool,3}, allAs::Array{Float64,3}, allBs::Array{Float64,3},
        gsYs::Array{Bool,3}, gsAs::Array{Float64,3}, gsBs::Array{Float64,3}, likelihoodMat::Array{Float64,2};
        K::Int=MCMC_K, burn::Int=BURNIN, th::Int=THIN, n::Int=SAMPLESIZE)
    ## burnin and thining
    K = size(gsYs)[3]
    thining_progress = Progress(K);
    @time for k in 1:K
        A = allAs[:,:,burn+k*th];
        gsAs[:,:,k] = A;
        B = allBs[:,:,burn+k*th];
        gsBs[:,:,k] = B;
        gsYs[:,:,k] = allYs[:,:,burn+k*th];
        ## generated quantity: likelihood matrix
        #likelihoodMat = calc_likelihoodMat!(k, words_arr, doc_onehot, A, B, likelihoodMat)
        for l in 1:n
            x = words_arr[:,l];
            z = doc_onehot[:,l];
            likelihoodMat[l,k] = model_pmf(x,z,A,B);
        end
        next!(thining_progress);
    end
    #return (gsYs, gsAs, gsBs, likelihoodMat)
end

function run_Gibbs_sampler_cored(
    docs_arr::Array{Bool,3}, words_arr::Array{Bool,2}; K::Int=MCMC_K, burn::Int=BURNIN, th::Int=THIN, n::Int=SAMPLESIZE)
    #===
    Version: separating core functions
    ===#
    ## initial value of A and B is sampled from the prior distributuion
    #### Assume that prior is SYMMETRIC Dirichlet distribution. Hyparam is scalar.
    init_A = hcat([rand(Dirichlet(M, ALPHA)) for k in 1:H]...);
    init_B = hcat([rand(Dirichlet(H, BETA)) for j in 1:N]...);
    init_Y = topic_step(docs_arr, init_A, init_B);
    ## sampling iteration
    #### allocate tensors for MCMC sample
    iters = burn + K*th;
    allAs = zeros(Float64, M, H, iters);
    gsAs = zeros(Float64, M, H, K);
    allBs = zeros(Float64, H, N, iters);
    gsBs = zeros(Float64, H, N, K);
    allYs = zeros(Bool, H, n, iters);
    gsYs = zeros(Bool, H, n, K);
    #### all sampling
    doc_onehot = get_doc_onehot(docs_arr);
    println("Start $iters iteration for GS")
    run_all_sampling!(docs_arr, words_arr, doc_onehot, init_Y, allYs, allAs, allBs)
    ## burnin and thining
    likelihoodMat = zeros(Float64, n, K)
    println("Start burn-in and thining from $iters to $K")
    run_thining!(words_arr, doc_onehot, allYs, allAs, allBs, gsYs, gsAs, gsBs, likelihoodMat)
    return (gsYs, gsAs, gsBs, likelihoodMat)
end

@doc """
    run_Gibbs_sampler(docs_arr, words_arr, alpha, beta; K=MCMC_K, burn=BURNIN, th=THIN, n=SAMPLESIZE, seed_MCMC=2)

Run Gibbs sampler for LDA.
Global variable `M` , `H` , and `N` are used
as vocab. size, num. of topics, and num. of documents, respectively.
Note that rtype is tuple of tensor(Array{T,3}), not tuple of Array{Array{T,2}}.
"""
function run_Gibbs_sampler(docs_arr, words_arr, alpha, beta; K=MCMC_K, burn=BURNIN, th=THIN, n=SAMPLESIZE, seed_MCMC=2)
    #===
    Run Gibbs sampler for LDA.
    Global variable `M` , `H` , and `N` are used
    as vocab. size, num. of topics, and num. of documents, respectively.
    Note that rtype is tuple of tensor(Array{T,3}), not tuple of Array{Array{T,2}}.

    :param docs_arr: data as document format
    :param words_arr: data as word format
    :param alpha: hyperparameter of Dirichlet prior for A
    :param beta: hyperparameter of Dirichlet prior for B
    :return: sample of Y(topic indicator variable),A and B from GS and likelihood matrix
    :rtype: tuple(Array{Bool,3};shape(H,n,K), Array{Float64,3};shape(M,H,K), Array{Float64,3};shape(H,N,K), Array{Float64,2};shape(n,K))
    ===#
    Random.seed!(seed_MCMC);
    ## initial value of A and B is sampled from the prior distributuion
    #### Assume that prior is SYMMETRIC Dirichlet distribution. Hyparam is scalar.
    init_A = hcat([rand(Dirichlet(M, alpha)) for k in 1:H]...);
    init_B = hcat([rand(Dirichlet(H, beta)) for j in 1:N]...);
    Y = topic_step(docs_arr, init_A, init_B);
    ## sampling iteration
    #### allocate tensors for MCMC sample
    iters = burn + K*th;
    allAs = zeros(Float64, M, H, iters);
    gsAs = zeros(Float64, M, H, K);
    allBs = zeros(Float64, H, N, iters);
    gsBs = zeros(Float64, H, N, K);
    allYs = zeros(Bool, H, n, iters);
    gsYs = zeros(Bool, H, n, K);
    #### all sampling
    doc_onehot = get_doc_onehot(docs_arr);
    println("Start $iters iteration for GS")
    sampling_progress = Progress(iters);
    @time for k in 1:iters
        ## parameter matrix sampling and saving
        A = A_step(words_arr, Y, alpha);
        B = B_step(doc_onehot, Y, beta);
        allAs[:,:,k] = A;
        allBs[:,:,k] = B;
        ## hidden variable (topic indicator) sampling and saving
        Y = topic_step(docs_arr, A, B);
        allYs[:,:,k] = Y;
        ## Progress update
        next!(sampling_progress);
    end
    ## burnin and thining
    likelihoodMat = zeros(Float64, n, K)
    println("Start burn-in and thining from $iters to $K")
    thining_progress = Progress(K);
    @time for k in 1:K
        A = allAs[:,:,burn+k*th];
        gsAs[:,:,k] = A;
        B = allBs[:,:,burn+k*th];
        gsBs[:,:,k] = B;
        gsYs[:,:,k] = allYs[:,:,burn+k*th];
        ## generated quantity: likelihood matrix
        for l in 1:n
            x = words_arr[:,l];
            z = doc_onehot[:,l];
            likelihoodMat[l,k] = model_pmf(x,z,A,B);
        end
        next!(thining_progress);
    end
    return (gsYs, gsAs, gsBs, likelihoodMat)
end
```

    run_Gibbs_sampler

```julia
function run_prior_sampling(docs_arr, words_arr, alpha, beta; K=MCMC_K, burn=BURNIN, th=THIN, n=SAMPLESIZE, seed_MCMC=2)
    #===
    Run prior sampling K times.

    :return: sample of Y(topic indicator variable),A and B from GS and likelihood matrix
    :rtype: tuple(Array{Bool,3};shape(H,n,K), Array{Float64,3};shape(M,H,K), Array{Float64,3};shape(H,N,K), Array{Float64,2};shape(n,K))
    ===#
    Random.seed!(seed_MCMC);
    ## sampling iteration
    #### allocate tensors for MCMC sample
    gsAs = zeros(Float64, M, H, K);
    gsBs = zeros(Float64, H, N, K);
    gsYs = zeros(Bool, H, n, K);
    #### all sampling
    doc_onehot = get_doc_onehot(docs_arr);
    println("Start $K iteration for prior sampling (no thining)")
    sampling_progress = Progress(K);
    @time for k in 1:K
        ## parameter matrix sampling and saving
        A = hcat([rand(Dirichlet(M, alpha)) for k in 1:H]...);
        B = hcat([rand(Dirichlet(H, beta)) for j in 1:N]...);
        gsAs[:,:,k] = A;
        gsBs[:,:,k] = B;
        ## hidden variable (topic indicator) sampling and saving
        Y = topic_step(docs_arr, A, B);
        gsYs[:,:,k] = Y;
        ## Progress update
        next!(sampling_progress);
    end
    ## calc likelihood matrix
    likelihoodMat = zeros(Float64, n, K)
    println("Start calculating likelihood matrix")
    thining_progress = Progress(K);
    @time for k in 1:K
        A = gsAs[:,:,k];
        B = gsBs[:,:,k];
        ## generated quantity: likelihood matrix
        for l in 1:n
            x = words_arr[:,l];
            z = doc_onehot[:,l];
            likelihoodMat[l,k] = model_pmf(x,z,A,B);
        end
        next!(thining_progress);
    end
    return (gsYs, gsAs, gsBs, likelihoodMat)
end
```

    run_prior_sampling (generic function with 1 method)

### Theoretical Property of Generalization Error and WAIC

The numerical WAIC is defined by
$$W_n = T_n + V_n / n,$$
where
$$T_n = -\frac{1}{n}\sum_{l=1}^n \log p^*(x_l|z_l),$$
$$p^*(x_l | z_l) = \frac{1}{K}\sum_{k=1}^K p(x_l|z_l,w_k),$$
and
$$V_n = \sum_{l=1}^n \left[ \frac{1}{K}\sum_{k=1}^K(\log p(x_l|z_l,w_k))^2 - \left\{ \frac{1}{K}\sum_{k=1}^K \log p(x_l|z_l,w_k) \right\}^2 \right].$$
The normalized WAIC is defined by
$$W_n - S_n = T_n - S_n + \frac{V_n}{n} = \frac{1}{n}\sum_{l=1}^n \log \frac{q(x_l|z_l)}{p^*(x_l|z_l)} + \frac{V_n}{n}.$$

On the other hand, the numerical generalization error is defined by
$$G_n = \frac{1}{T}\sum_{t=1}^T \log \frac{q(x_t|z_t)}{p^*(x_t|x_l)}.$$

In the singular learning theory ($K \rightarrow \infty$, i.e. theoretical posterior mean is considered),
the following property is known and useful for calculating the RLCT $\lambda$:
$$G_n + W_n - S_n = \frac{2\lambda}{n} + o_p\left( \frac{1}{n} \right).$$
This equation means that $G_n$ and $W_n - S_n$ asymptotically has same variance.

```julia
# define functions for calculating RLCT
@doc """
    calc_functional_var(loglikeMat::Array{Float64,2})

Calculate a functional variance from given loglikelihood matrix.
"""
function calc_functional_var(loglikeMat::Array{Float64,2})
    #===
    Calculate functional variance from loglike matrix.

    :param loglikeMat: a matrix whose (l,k) element is log p(x_l|z_l,A_k,B_k)
    ===#
    n = size(loglikeMat)[1]
    K = size(loglikeMat)[2]
    first_term = reshape(mean(loglikeMat.^2, dims=2),n)
    second_term = reshape(mean(loglikeMat, dims=2).^2, n)
    func_var = sum(first_term - second_term)
    return func_var
end

@doc """
    calc_predict_dist(x, z, As, Bs)

Calculate predictive distribution from given MCMC sample.
"""
function calc_predict_dist(x, z, As, Bs)
    #===
    Calculate pmf of predictive distribution.

    :param x: an M-dim onehot vector which means a word.
    :param z: an N-dim onehot vector which means a document.
    :param As: an Array whose [:,:,k] element means an MCMC sample of A.
    :param Bs: an Array whose [:,:,k] element means an MCMC sample of B.
    ===#
    @assert size(As)[3]==size(Bs)[3]
    K = size(As)[3]
    mass = 0.0
    for k in 1:K
        mass += model_pmf(x, z, As[:,:,k], Bs[:,:,k])
    end
    return mass/K
end

@doc """
    calc_normalized_WAIC(words_arr, doc_onehots, true_A, true_B, likelihoodMat)

Calculate normalized WAIC, i.e. emperical loss - WAIC.
"""
function calc_normalized_WAIC(words_arr, doc_onehots, true_A, true_B, likelihoodMat)
    #===
    Calculating normalized WAIC.
    ===#
    n = size(words_arr)[2]
    emp_loss = -mean(log.(mean(likelihoodMat, dims=2)))
    emp_entropy = -mean([log(model_pmf(words_arr[:,l], doc_onehots[:,l], true_A, true_B)) for l in 1:n])
    func_var = calc_functional_var(log.(likelihoodMat))
    #println(typeof(emp_entropy))
    normalized_WAIC = emp_loss - emp_entropy + func_var/n
    return normalized_WAIC
end

@doc """
    calc_generalization_error(true_A, true_B, As, Bs; nT=TESTSIZE, seed_T=3)

Calculate generalization error, i.e. KL divergence from the true distribution to the predictive one
using many test data generated by the true distribution.
"""
function calc_generalization_error(true_A, true_B, As, Bs; nT=TESTSIZE, seed_T=3)
    #===
    Calculating generalization error using test data generated by true distribution.
    ===#
    test_doc, test_words = generate_data(true_A, true_B, n=nT, seed=seed_T)
    test_doc_onehot = get_doc_onehot(test_doc, n=nT)
    ge = 0.0
    calc_gerr_progress = Progress(nT)
    println(nT)
    @time for t in 1:nT
        q = model_pmf(test_words[:,t], test_doc_onehot[:,t], true_A, true_B)
        pred = calc_predict_dist(test_words[:,t], test_doc_onehot[:,t], As, Bs)
        ge += log(q) - log(pred) #log(q/pred), but q and pred is small number.
        next!(calc_gerr_progress)
    end
    ge /= nT
    return ge
end

```

    calc_generalization_error

```julia
@doc """
    run_single_inference(true_A, true_B, seed)

Carry out an inference.
- Generate sample.
- Run Gibbs sampler and make the posterior distribution.
- Calculate (generalization error, normalized WAIC) (this is the return).
"""
function run_single_inference(true_A, true_B, seed)
    #===
    Run single inference.

    Random seed of each module is determined by the following rule:
    generate train data: seed + 1
    run Gibbs sampler: seed + 2
    generate test data: seed + 3

    Macro seed (SEED) must be disjoint from 3 and larger than 10.
    ===#
    Random.seed!(seed);
    train_X_docs, train_X_words = generate_data(true_A, true_B, n=SAMPLESIZE, seed=seed+1)
    train_Z = get_doc_onehot(train_X_docs, n=SAMPLESIZE)
    println("Gibbs Sampling")
    topic_Ys, param_As, param_Bs, likelihoodMat = run_Gibbs_sampler(train_X_docs, train_X_words, ALPHA, BETA, seed_MCMC=seed+2)
    #topic_Ys, param_As, param_Bs, likelihoodMat = run_Gibbs_sampler_cored(train_X_docs, train_X_words)
    println("Calculation Normalized WAIC")
    normalized_WAIC = calc_normalized_WAIC(train_X_words, train_Z, true_A, true_B, likelihoodMat)
    println("Calculaton Generalization Error")
    ge = calc_generalization_error(true_A, true_B, param_As, param_Bs, seed_T=seed+3)
    return (ge, normalized_WAIC)
end
```

    run_single_inference

```julia
function run_single_inference_debug(true_A, true_B, seed)
    #===
    Run single inference for debug mode.
    ===#
    Random.seed!(seed);
    train_X_docs, train_X_words = generate_data(true_A, true_B, n=SAMPLESIZE, seed=seed+1)
    train_Z = get_doc_onehot(train_X_docs, n=SAMPLESIZE)
    println("Gibbs Sampling")
    topic_Ys, param_As, param_Bs, likelihoodMat = run_Gibbs_sampler(train_X_docs, train_X_words, ALPHA, BETA, seed_MCMC=seed+2)
    #topic_Ys, param_As, param_Bs, likelihoodMat = run_Gibbs_sampler_cored(train_X_docs, train_X_words)
    println("Calculation Normalized WAIC")
    normalized_WAIC = calc_normalized_WAIC(train_X_words, train_Z, true_A, true_B, likelihoodMat)
    println("Calculaton Generalization Error")
    ge = calc_generalization_error(true_A, true_B, param_As, param_Bs, seed_T=seed+3)
    return (topic_Ys, param_As, param_Bs, likelihoodMat, ge, normalized_WAIC)
end
```

    run_single_inference_debug (generic function with 1 method)

```julia
function run_single_inference_PS(true_A, true_B, seed)
    #===
    Run single inference using prior sampling (worst benchmark).
    ===#
    Random.seed!(seed);
    train_X_docs, train_X_words = generate_data(true_A, true_B, n=SAMPLESIZE, seed=seed+1)
    train_Z = get_doc_onehot(train_X_docs, n=SAMPLESIZE)
    println("Prior Sampling")
    topic_Ys, param_As, param_Bs, likelihoodMat = run_prior_sampling(train_X_docs, train_X_words, ALPHA, BETA, seed_MCMC=seed+2)
    #topic_Ys, param_As, param_Bs, likelihoodMat = run_Gibbs_sampler_cored(train_X_docs, train_X_words)
    println("Calculation Normalized WAIC")
    normalized_WAIC = calc_normalized_WAIC(train_X_words, train_Z, true_A, true_B, likelihoodMat)
    println("Calculaton Generalization Error")
    ge = calc_generalization_error(true_A, true_B, param_As, param_Bs, seed_T=seed+3)
    return (topic_Ys, param_As, param_Bs, likelihoodMat, ge, normalized_WAIC)
end
```

    run_single_inference_PS (generic function with 1 method)

```julia
@doc """
    run_multi_inference(true_A, true_B, sim_iters, log_file_path)

Independently repeat inferences.
"""
function run_multi_inference(true_A, true_B, sim_iters, log_file_path)
    #===
    for it in 1:sim_iters
        run_single_inference
    end
    and calculate RLCT from above $sim_iters simulation results.

    :param true_A: true parameter stochastic matrix A
    :param true_B: true parameter stochastic matrix B
    :param sim_iters: number of simulations
    :param log_file_path: path of the file to write the experimental log
    :return: generalization errors and normalized WAICs in the simulations
    :rtype: tuple(Array{Float64,1}, Array{Float64,1})
    ===#
    seeds = SEED .+ SEED * (1:sim_iters)
    gerrors = zeros(Float64, sim_iters)
    normalized_WAICs = zeros(Float64, sim_iters)
    open(log_file_path, "a") do fp
        println(fp, "## Simulation Setting")
        println(fp, "M,N,H,H_0,L,MCMC_K,BURNIN,THIN,TESTSIZE,SIM_ITERS,SEED")
        println(fp, "$M,$N,$H,$H_0,$L,$MCMC_K,$BURNIN,$THIN,$TESTSIZE,$sim_iters,$SEED")
        println(fp, "## Simulation Log")
        println(fp, "iter,gerror,normalized_WAIC,RLCT,RLCT_SEM")
    end
    simulation_progress = Progress(sim_iters)
    for it in 1:sim_iters
        println("# start $it th simulation")
        ge, norm_W = run_single_inference(true_A, true_B, seeds[it])
        gerrors[it] = ge
        normalized_WAICs[it] = norm_W
        now_ges = gerrors[1:it]
        now_nwaics = normalized_WAICs[1:it]
        rlct = (SAMPLESIZE/2)*mean(now_ges + now_nwaics)
        rlct_sem = (SAMPLESIZE/2)*std(now_ges + now_nwaics)/sqrt(it)
        open(log_file_path, "a") do fp
            println(fp,"$it,$ge,$norm_W,$rlct,$rlct_sem")
            println("$it,$ge,$norm_W,$rlct,$rlct_sem")
        end
        next!(simulation_progress)
        #@assert ge>=0
        #@assert norm_W>=0
    end
    return (gerrors, normalized_WAICs)
end
```

    run_multi_inference

```julia
# main process
##===
gerrors, normalized_WAICs = @time run_multi_inference(A_0, B_0, SIM_ITERS, LOG_FILE_PATH)
ge_and_nW = gerrors + normalized_WAICs
each_lams = (SAMPLESIZE/2) .* ge_and_nW
lam = (SAMPLESIZE/2)*mean(ge_and_nW)
lam_sd = (SAMPLESIZE/2)*std(ge_and_nW)/sqrt(SIM_ITERS)
println("The numerical RLCT is $lam ± $lam_sd")
#===#
```

    # start 1 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress:   0%|█                                        |  ETA: 1:25:50

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:09

     69.630321 seconds (608.12 M allocations: 67.212 GiB, 6.56% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress:   4%|██                                       |  ETA: 0:00:03

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.112392 seconds (21.97 M allocations: 1.655 GiB, 8.74% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress:   0%|█                                        |  ETA: 0:12:34

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:11

    551.111497 seconds (2.41 G allocations: 465.249 GiB, 15.90% gc time)
    1,0.0024217591440052963,0.002570125241062803,4.991884385068099,NaN

    Progress:   1%|█                                        |  ETA: 17:12:28

    # start 2 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress:   0%|█                                        |  ETA: 0:01:00

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:16

     76.983160 seconds (607.99 M allocations: 67.206 GiB, 8.62% gc time)

    Progress:   4%|██                                       |  ETA: 0:00:03

    Start burn-in and thining from 50000 to 2000

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.866925 seconds (21.97 M allocations: 1.655 GiB, 7.74% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress:   0%|█                                        |  ETA: 0:08:45

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:14

    554.355026 seconds (2.41

    Progress:   2%|█                                        |  ETA: 17:10:08

     G allocations: 465.246 GiB, 15.73% gc time)
    2,0.0012821823387435664,0.00363825000633795,4.956158365074808,0.03572601999329156
    # start 3 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress:   0%|█                                        |  ETA: 0:00:57

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:02

     62.895320 seconds (607.97 M allocations: 67.204 GiB, 6.43% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.202384 seconds (21.97 M allocations: 1.655 GiB, 7.20% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:17
    Progress:   3%|██                                       |  ETA: 16:55:53

    557.217675 seconds (2.41 G allocations: 465.249 GiB, 16.11% gc time)
    3,0.0017286486266155888,0.0034147049450427925,5.018556767269332,0.06571917602914687
    # start 4 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress:   0%|█                                        |  ETA: 0:01:10

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:15

     75.196374 seconds (607.99 M allocations: 67.205 GiB, 8.57% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress:   4%|██                                       |  ETA: 0:00:03

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.242235 seconds (21.97 M allocations: 1.655 GiB, 8.21% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress:   0%|█                                        |  ETA: 0:08:14

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:11

    551.813425 seconds (2.41 G allocations: 465.246 GiB, 15.72% gc time)
    4,0.0016956014296069524,0.0026940956550819206,4.861341846624216,0.16393912383084383

    Progress:   4%|██                                       |  ETA: 16:46:20

    # start 5 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:02

     62.340286 seconds (607.97 M allocations: 67.204 GiB, 6.46% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress:   4%|██                                       |  ETA: 0:00:02

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.083401 seconds (21.97 M allocations: 1.655 GiB, 7.51% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress:   0%|█                                        |  ETA: 0:08:02

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:14

    554.893264 seconds (2.41 G allocations: 465.248 GiB, 16.13% gc time)
    5,0.0006690875163942158,0.004657626888851093,4.9544163583484355,0.1574435979205141

    Progress:   5%|███                                      |  ETA: 16:33:17

    # start 6 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:26

     86.373759 seconds (608.01 M allocations: 67.207 GiB, 8.30% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.723436 seconds (21.97 M allocations: 1.655 GiB, 7.86% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress:   0%|█                                        |  ETA: 0:08:43

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:53
    Progress:   6%|███                                      |  ETA: 16:37:41

    593.692084 seconds (2.41 G allocations: 465.248 GiB, 15.68% gc time)
    6,0.003061328681552959,0.001485333945664225,4.886457403159894,0.14541003154673543
    # start 7 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress:   0%|█                                        |  ETA: 0:01:09

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:03

     63.028650 seconds (607.97 M allocations: 67.204 GiB, 6.52% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress:   4%|██                                       |  ETA: 0:00:02

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.100765 seconds (21.97 M allocations: 1.655 GiB, 7.68% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:13
    Progress:   7%|███                                      |  ETA: 16:23:18

    553.176293 seconds (2.41 G allocations: 465.248 GiB, 16.01% gc time)
    7,0.002410659444316768,0.002231186821168756,4.851512954920699,0.12776551491546964
    # start 8 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:23

     83.154517 seconds (608.01 M allocations: 67.206 GiB, 8.57% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.655434 seconds (21.97 M allocations: 1.655 GiB, 8.10% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:49
    Progress:   8%|████                                     |  ETA: 16:20:47

    589.242585 seconds (2.41 G allocations: 465.248 GiB, 15.60% gc time)
    8,0.0021211410894848267,0.0032695799129853053,4.918913960864377,0.1295604712226349
    # start 9 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:05

     65.014501 seconds (607.97 M allocations: 67.204 GiB, 6.43% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress:   4%|██                                       |  ETA: 0:00:03

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.096980 seconds (21.97 M allocations: 1.655 GiB, 7.23% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress:   0%|█                                        |  ETA: 0:07:05

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:41
    Progress:   9%|████                                     |  ETA: 16:11:58

    581.955764 seconds (2.41 G allocations: 465.251 GiB, 15.85% gc time)
    9,0.0031385190938971763,0.0016176898308097399,4.900835623513548,0.11568292222774375
    # start 10 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress:   0%|█                                        |  ETA: 0:01:05

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:22

     82.428417 seconds (608.00 M allocations: 67.206 GiB, 8.55% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress:   3%|██                                       |  ETA: 0:00:03

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.397091 seconds (21.97 M allocations: 1.655 GiB, 8.65% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:44
    Progress:  10%|█████                                    |  ETA: 16:05:48

    584.697548 seconds (2.41 G allocations: 465.247 GiB, 15.64% gc time)
    10,0.0007134641651682472,0.004144506349333991,4.8965491126124165,0.10355870302309547
    # start 11 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress:   0%|█                                        |  ETA: 0:01:11

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:10

     70.404462 seconds (607.98 M allocations: 67.205 GiB, 6.32% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.450893 seconds (21.97 M allocations: 1.655 GiB, 7.51% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress:   0%|█                                        |  ETA: 0:08:39

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:47

    587.306956 seconds (2.41 G allocations: 465.252 GiB, 15.87% gc time)

    Progress:  11%|█████                                    |  ETA: 15:57:29

    11,0.0010926648661800044,0.0036072940871877375,4.87867727995381,0.09536202368523705
    # start 12 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress:   0%|█                                        |  ETA: 0:01:16

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:22

     82.585079 seconds (608.00 M allocations: 67.206 GiB, 8.37% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress:   3%|██                                       |  ETA: 0:00:03

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.580847 seconds (21.97 M allocations: 1.655 GiB, 8.34% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress:   0%|█                                        |  ETA: 0:09:23

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:40
    Progress:  12%|█████                                    |  ETA: 15:49:20

    580.182166 seconds (2.41 G allocations: 465.247 GiB, 15.65% gc time)
    12,0.002895620319178235,0.001445179525879958,4.8338541603791745,0.09791514191476433
    # start 13 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress:   0%|█                                        |  ETA: 0:01:03

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:03

     63.206825 seconds (607.97 M allocations: 67.204 GiB, 6.57% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress:   4%|██                                       |  ETA: 0:00:03

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.184057 seconds (21.97 M allocations: 1.655 GiB, 7.23% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:09
    Progress:  13%|██████                                   |  ETA: 15:35:09

    549.948331 seconds (2.41 G allocations: 465.247 GiB, 16.02% gc time)
    13,0.0027125786927423155,0.002047540306757108,4.828182224926887,0.09024724425843746
    # start 14 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress:   0%|█                                        |  ETA: 0:00:55

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:14

     74.691291 seconds (607.99 M allocations: 67.205 GiB, 8.66% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress:   3%|██                                       |  ETA: 0:00:03

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.358145 seconds (21.97 M allocations: 1.655 GiB, 8.68% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress:   0%|█                                        |  ETA: 0:07:24

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:10

    550.646979 seconds (2.41 G allocations: 465.246 GiB, 15.69% gc time)
    14,0.0031874875189593136,0.0015975760610265916,4.825102321716816,0.0836094587984058

    Progress:  14%|██████                                   |  ETA: 15:22:46

    # start 15 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:04

     64.190529 seconds (607.97 M allocations: 67.204 GiB, 6.48% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.171297 seconds (21.97 M allocations: 1.655 GiB, 7.48% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error

    Progress:   0%|█                                        |  ETA: 0:07:50

    400000

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:06
    Progress:  15%|███████                                  |  ETA: 15:09:13

    546.054653 seconds (2.41 G allocations: 465.247 GiB, 16.12% gc time)
    15,0.0009757647246791552,0.003966728240523068,4.832928364615842,0.07822861582957619
    # start 16 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress:   0%|█                                        |  ETA: 0:01:03

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:14

     74.671345 seconds (607.99 M allocations: 67.205 GiB, 8.60% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.213858 seconds (21.97 M allocations: 1.655 GiB, 8.29% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress: 100%|█████████████████████████████████████████| Time: 0:08:56
    Progress:  16%|███████                                  |  ETA: 14:56:12

    536.984175 seconds (2.41 G allocations: 465.245 GiB, 15.83% gc time)
    16,0.0036384287242274133,0.0005887144250938852,4.795066788659933,0.08239084127893138
    # start 17 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress:   0%|█                                        |  ETA: 0:00:59

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:03

     63.511434 seconds (607.97 M allocations: 67.204 GiB, 6.63% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.111181 seconds (21.97 M allocations: 1.655 GiB, 7.64% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress:   0%|█                                        |  ETA: 0:09:54

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:12
    Progress:  17%|███████                                  |  ETA: 14:43:50

    552.178569 seconds (2.41 G allocations: 465.248 GiB, 16.05% gc time)
    17,0.0025253013660822784,0.0021657875068309034,4.78895044067483,0.07763403059727012
    # start 18 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress:   0%|█                                        |  ETA: 0:01:02

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:16

     76.572994 seconds (608.00 M allocations: 67.206 GiB, 8.51% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress:   4%|██                                       |  ETA: 0:00:03

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.395699 seconds (21.97 M allocations: 1.655 GiB, 8.68% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress:   0%|█                                        |  ETA: 0:06:56

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:12
    Progress:  18%|████████                                 |  ETA: 14:32:42

    552.242019 seconds (2.41 G allocations: 465.246 GiB, 15.72% gc time)
    18,0.002575582887855905,0.0027071294359516324,4.816381656404427,0.07816548401511522
    # start 19 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress:   0%|█                                        |  ETA: 0:00:58

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:04

     64.956989 seconds (607.97 M allocations: 67.204 GiB, 6.51% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.282746 seconds (21.97 M allocations: 1.655 GiB, 8.06% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:09

    549.677849 seconds (2.41 G allocations: 465.248 GiB, 16.14% gc time)
    19,0.0008100636270389622,0.004202633310993887,4.826714039648027,0.07465560714259259

    Progress:  19%|████████                                 |  ETA: 14:20:38

    # start 20 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:16

     76.667947 seconds (608.00 M allocations: 67.206 GiB, 8.52% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.274685 seconds (21.97 M allocations: 1.655 GiB, 8.02% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:09
    Progress:  20%|█████████                                |  ETA: 14:09:31

    549.795198 seconds (2.41 G allocations: 465.246 GiB, 15.76% gc time)
    20,0.001181611844776133,0.0035331054084337997,4.8211142003261225,0.07104556221522466
    # start 21 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:05

     65.252509 seconds (607.97 M allocations: 67.204 GiB, 6.42% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress:   3%|██                                       |  ETA: 0:00:03

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.267757 seconds (21.97 M allocations: 1.655 GiB, 7.63% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress:   0%|█                                        |  ETA: 0:07:56

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:03
    Progress:  21%|█████████                                |  ETA: 13:57:23

    543.979995 seconds (2.41 G allocations: 465.247 GiB, 16.09% gc time)
    21,0.001826801454799901,0.002938164067998402,4.818440453777178,0.06763068253805882
    # start 22 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:14

     74.269268 seconds (607.99 M allocations: 67.205 GiB, 8.74% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.256302 seconds (21.97 M allocations: 1.655 GiB, 8.39% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress:   0%|█                                        |  ETA: 0:07:05

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:13
    Progress:  22%|██████████                               |  ETA: 13:46:31

    553.508520 seconds (2.41 G allocations: 465.246 GiB, 15.86% gc time)
    22,0.004756496328551393,4.825246264352715e-5,4.817818105477985,0.06448632885214546
    # start 23 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:08

     68.307476 seconds (607.98 M allocations: 67.205 GiB, 6.30% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress:   3%|██                                       |  ETA: 0:00:03

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.398644 seconds (21.97 M allocations: 1.655 GiB, 7.16% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress:   0%|█                                        |  ETA: 0:07:50

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:06
    Progress:  23%|██████████                               |  ETA: 13:34:57

    546.366655 seconds (2.41 G allocations: 465.247 GiB, 16.22% gc time)
    23,0.0030040075917577196,0.00218877185704268,4.8341207725789594,0.06373896792567241
    # start 24 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress:   0%|█                                        |  ETA: 0:01:08

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:15

     75.444177 seconds (607.99 M allocations: 67.206 GiB, 8.61% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress:   4%|██                                       |  ETA: 0:00:03

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.310463 seconds (21.97 M allocations: 1.655 GiB, 8.25% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress:   0%|█                                        |  ETA: 0:06:57

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:12
    Progress:  24%|██████████                               |  ETA: 13:24:13

    552.764234 seconds (2.41 G allocations: 465.246 GiB, 15.85% gc time)
    24,0.002010732914652271,0.002657391550313539,4.827204259761745,0.06141611769257396
    # start 25 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:09

     69.364505 seconds (607.98 M allocations: 67.205 GiB, 6.23% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress:   4%|██                                       |  ETA: 0:00:02

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.104940 seconds (21.97 M allocations: 1.655 GiB, 7.59% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress:   0%|█                                        |  ETA: 0:08:25

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:08

    548.800337 seconds (2.41 G allocations: 465.248 GiB,

    Progress:  25%|███████████                              |  ETA: 13:12:58

    16.14% gc time)
    25,0.001448817168920329,0.0035251499231203276,4.833074773052902,0.05920006146732868
    # start 26 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:14

     74.136308 seconds (607.99 M allocations: 67.205 GiB, 8.66% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress:   4%|██                                       |  ETA: 0:00:02

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.247561 seconds (21.97 M allocations: 1.655 GiB, 9.07% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress:   0%|█                                        |  ETA: 0:07:01

    Progress: 100%|█████████████████████████████████████████| Time: 0:08:58

    538.467741 seconds (2.41 G allocations: 465.245 GiB, 16.14% gc time)
    26,0.0006053898072771067,0.004319002132828639,4.836586971785703,0.05698591550147009

    Progress:  26%|███████████                              |  ETA: 13:01:32

    # start 27 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:02

     62.215011 seconds (607.97 M allocations: 67.204 GiB, 6.45% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress:   4%|██                                       |  ETA: 0:00:02

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.111951 seconds (21.97 M allocations: 1.655 GiB, 7.57% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error

    Progress:   0%|█                                        |  ETA: 0:06:52

    400000

    Progress: 100%|█████████████████████████████████████████| Time: 0:08:57
    Progress:  27%|████████████                             |  ETA: 12:49:38

    537.866819 seconds (2.41 G allocations: 465.246 GiB, 16.24% gc time)
    27,0.0028864109830302265,0.0016811408754120565,4.826622708328541,0.05573269566590917
    # start 28 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:14

     74.680780 seconds (607.99 M allocations: 67.205 GiB, 8.73% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.242358 seconds (21.97 M allocations: 1.655 GiB, 8.23% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress:   0%|█                                        |  ETA: 0:07:24

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:01
    Progress:  28%|████████████                             |  ETA: 12:38:32

    541.379332 seconds (2.41 G allocations: 465.246 GiB, 16.05% gc time)
    28,0.0027653295593963374,0.0021436866770783124,4.829565334333759,0.05378592515688418
    # start 29 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:02

     62.495503 seconds (607.97 M allocations: 67.204 GiB, 6.57% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.153094 seconds (21.97 M allocations: 1.655 GiB, 7.55% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress:   0%|█                                        |  ETA: 0:07:55

    Progress: 100%|█████████████████████████████████████████| Time: 0:08:55

    535.250932 seconds (2.41 G allocations: 465.246 GiB, 16.26% gc time)
    29,0.0014991627060163416,0.003182814103045103,4.82447607484161,0.05214704389336745

    Progress:  29%|████████████                             |  ETA: 12:26:45

    # start 30 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress:   0%|█                                        |  ETA: 0:01:01

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:14

     74.162043 seconds (607.99 M allocations: 67.205 GiB, 8.79% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.263947 seconds (21.97 M allocations: 1.655 GiB, 8.58% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress: 100%|█████████████████████████████████████████| Time: 0:08:51
    Progress:  30%|█████████████                            |  ETA: 12:15:24

    531.816473 seconds (2.41 G allocations: 465.245 GiB, 16.11% gc time)
    30,0.003626624400750939,0.0019212613109657788,4.848589729404113,0.05585243871776853
    # start 31 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress:   0%|█                                        |  ETA: 0:00:54

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:02

     62.168531 seconds (607.97 M allocations: 67.204 GiB, 6.54% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.138599 seconds (21.97 M allocations: 1.655 GiB, 7.50% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress:   0%|█                                        |  ETA: 0:06:52

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:02
    Progress:  31%|█████████████                            |  ETA: 12:04:03

    542.148961 seconds (2.41 G allocations: 465.247 GiB, 16.18% gc time)
    31,0.0024471342718231743,0.002215714539518655,4.842598086885975,0.054351972867704004
    # start 32 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:19

     79.735690 seconds (608.00 M allocations: 67.206 GiB, 8.50% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.353886 seconds (21.97 M allocations: 1.655 GiB, 8.20% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:13
    Progress:  32%|██████████████                           |  ETA: 11:53:50

    553.882744 seconds (2.41 G allocations: 465.246 GiB, 15.97% gc time)
    32,0.0034765878046164557,0.00098945739780373,4.830830809246419,0.05392561745585114
    # start 33 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress:   0%|█                                        |  ETA: 0:01:01

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:04

     64.883007 seconds (607.97 M allocations: 67.204 GiB, 6.46% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress:   4%|██                                       |  ETA: 0:00:03

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.075633 seconds (21.97 M allocations: 1.655 GiB, 7.37% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:06
    Progress:  33%|██████████████                           |  ETA: 11:42:49

    546.029167 seconds (2.41 G allocations: 465.247 GiB, 16.28% gc time)
    33,0.0027857739320823922,0.0019201276185343147,4.827045074136429,0.052402893697135515
    # start 34 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:18

     78.696397 seconds (608.00 M allocations: 67.206 GiB, 8.57% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.397115 seconds (21.97 M allocations: 1.655 GiB, 9.10% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress:   0%|█                                        |  ETA: 0:08:08

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:24
    Progress:  34%|██████████████                           |  ETA: 11:32:54

    564.525310 seconds (2.41 G allocations: 465.246 GiB, 15.86% gc time)
    34,0.00306956331359231,0.002378803374880125,4.845319239263957,0.05402291407856065
    # start 35 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress:   0%|█                                        |  ETA: 0:01:17

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:07

     67.088286 seconds (607.98 M allocations: 67.205 GiB, 6.38% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress:   4%|██                                       |  ETA: 0:00:03

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.326626 seconds (21.97 M allocations: 1.655 GiB, 7.40% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:07
    Progress:  35%|███████████████                          |  ETA: 11:22:03

    547.778740 seconds (2.41 G allocations: 465.248 GiB, 16.30% gc time)
    35,0.0014191794908529165,0.0036669680380708794,4.852200047539953,0.05290605592973003
    # start 36 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:14

     74.472873 seconds (607.99 M allocations: 67.205 GiB, 8.67% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress:   4%|██                                       |  ETA: 0:00:03

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.270395 seconds (21.97 M allocations: 1.655 GiB, 9.04% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:15
    Progress:  36%|███████████████                          |  ETA: 11:11:41

    555.639677 seconds (2.41 G allocations: 465.246 GiB, 15.98% gc time)
    36,0.0012306151210446073,0.0035011438773812206,4.848854462842338,0.05152417754161401
    # start 37 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress:   0%|█                                        |  ETA: 0:01:04

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:05

     65.347271 seconds (607.97 M allocations: 67.204 GiB, 6.32% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.207712 seconds (21.97 M allocations: 1.655 GiB, 7.15% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress:   0%|█                                        |  ETA: 0:06:57

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:08
    Progress:  37%|████████████████                         |  ETA: 11:00:51

    548.915024 seconds (2.41 G allocations: 465.248 GiB, 16.26% gc time)
    37,0.0032114713277053715,0.001757512246707225,4.852101195587481,0.05021735395557231
    # start 38 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress:   0%|█                                        |  ETA: 0:00:58

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:17

     77.115792 seconds (608.00 M allocations: 67.206 GiB, 8.58% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress:   4%|██                                       |  ETA: 0:00:03

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.222063 seconds (21.97 M allocations: 1.655 GiB, 8.72% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress: 100%|█████████████████████████████████████████| Time: 0:08:58
    Progress:  38%|████████████████                         |  ETA: 10:50:06

    538.755117 seconds (2.41 G allocations: 465.245 GiB, 16.08% gc time)
    38,0.004285548550370712,0.0005460484667820951,4.8515616119444624,0.0488809613938934
    # start 39 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:03

     63.777460 seconds (607.97 M allocations: 67.204 GiB, 6.40% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.272807 seconds (21.97 M allocations: 1.655 GiB, 7.27% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress:   0%|█                                        |  ETA: 0:07:39

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:01

    541.752286 seconds (2.41 G allocations: 465.247 GiB, 16.22% gc time)
    39,0.003926276211556378,0.0009596477422044803,4.852442697632062,0.047619260937965764

    Progress:  39%|████████████████                         |  ETA: 10:39:06

    # start 40 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress:   0%|█                                        |  ETA: 0:01:05

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:14

     74.993623 seconds (607.99 M allocations: 67.206 GiB, 8.68% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.230889 seconds (21.97 M allocations: 1.655 GiB, 8.41% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress:   0%|█                                        |  ETA: 0:07:39

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:03

    543.191806 seconds (2.41 G allocations: 465.245 GiB, 16.09% gc time)
    40,0.002653939684026488,0.0025787059460206105,4.861947770942439,0.04737679527257821

    Progress:  40%|█████████████████                        |  ETA: 10:28:27

    # start 41 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:03

     63.503760 seconds (607.97 M allocations: 67.204 GiB, 6.64% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.051348 seconds (21.97 M allocations: 1.655 GiB, 7.99% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:02
    Progress:  41%|█████████████████                        |  ETA: 10:17:31

    542.364342 seconds (2.41 G allocations: 465.247 GiB, 16.36% gc time)
    41,0.0012592633970719897,0.003403391097936889,4.857086959334303,0.04646178486087118
    # start 42 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:15

     75.336041 seconds (607.99 M allocations: 67.206 GiB, 8.58% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress:   4%|██                                       |  ETA: 0:00:03

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.332566 seconds (21.97 M allocations: 1.655 GiB, 8.67% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:01
    Progress:  42%|██████████████████                       |  ETA: 10:06:52

    541.403151 seconds (2.41 G allocations: 465.245 GiB, 16.16% gc time)
    42,0.0012232142587574118,0.003567189216009216,4.855499257320787,0.045369848284492774
    # start 43 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress:   0%|█                                        |  ETA: 0:00:56

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:03

     63.516293 seconds (607.97 M allocations: 67.204 GiB, 6.54% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.117031 seconds (21.97 M allocations: 1.655 GiB, 7.37% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress:   0%|█                                        |  ETA: 0:07:34

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:05
    Progress:  43%|██████████████████                       |  ETA: 9:56:04

    545.319380 seconds (2.41 G allocations: 465.247 GiB, 16.36% gc time)
    43,0.004440462359676357,0.000853544584980508,4.86569711051465,0.04546073836034347
    # start 44 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress:   0%|█                                        |  ETA: 0:00:57

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:14

     74.499393 seconds (607.99 M allocations: 67.205 GiB, 8.70% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.336587 seconds (21.97 M allocations: 1.655 GiB, 8.79% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:00

    540.656279 seconds (2.41 G allocations: 465.245 GiB, 16.19% gc time)
    44,0.0051576401596996335,0.0006800093232592289,4.88778693716111,0.04960543550017913

    Progress:  44%|███████████████████                      |  ETA: 9:45:26

    # start 45 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:03

     63.632995 seconds (607.97 M allocations: 67.204 GiB, 6.51% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.108609 seconds (21.97 M allocations: 1.655 GiB, 7.23% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:04

    544.574306 seconds (2.41 G allocations: 465.247 GiB, 16.29% gc time)
    45,0.004740890496135313,-0.00022041484268667776,4.879624464189721,0.04917276467616185

    Progress:  45%|███████████████████                      |  ETA: 9:34:40

    # start 46 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:15

     75.369971 seconds (607.99 M allocations: 67.206 GiB, 8.63% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress:   4%|██                                       |  ETA: 0:00:03

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.324246 seconds (21.97 M allocations: 1.655 GiB, 8.27% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:01
    Progress:  46%|███████████████████                      |  ETA: 9:24:06

    541.820036 seconds (2.41 G allocations: 465.245 GiB, 16.13% gc time)
    46,0.002724260884878366,0.0019508040955273243,4.875177518890068,0.04829707422374235
    # start 47 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:03

     63.823924 seconds (607.97 M allocations: 67.204 GiB, 6.60% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress:   4%|██                                       |  ETA: 0:00:03

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.134358 seconds (21.97 M allocations: 1.655 GiB, 7.46% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress:   0%|█                                        |  ETA: 0:07:50

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:03
    Progress:  47%|████████████████████                     |  ETA: 9:13:21

    543.719980 seconds (2.41 G allocations: 465.248 GiB, 16.30% gc time)
    47,0.0030743416441549273,0.0023555472150249843,4.886979887832405,0.04870978754598277
    # start 48 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress:   0%|█                                        |  ETA: 0:01:01

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:15

     75.402922 seconds (607.99 M allocations: 67.206 GiB, 8.67% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.204670 seconds (21.97 M allocations: 1.655 GiB, 8.87% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress:   0%|█                                        |  ETA: 0:07:01

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:00
    Progress:  48%|████████████████████                     |  ETA: 9:02:47

    540.494385 seconds (2.41 G allocations: 465.245 GiB, 16.15% gc time)
    48,0.003071248302626933,0.0019847339994728272,4.890500771462975,0.04781401345351079
    # start 49 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:03

     63.868763 seconds (607.97 M allocations: 67.204 GiB, 6.55% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.153934 seconds (21.97 M allocations: 1.655 GiB, 7.71% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:04
    Progress:  49%|█████████████████████                    |  ETA: 8:52:06

    544.102277 seconds (2.41 G allocations: 465.247 GiB, 16.33% gc time)
    49,0.00234359753183401,0.002666999544497759,4.8929517164602965,0.04689214805614312
    # start 50 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:14

     74.372830 seconds (607.99 M allocations: 67.205 GiB, 8.70% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.206643 seconds (21.97 M allocations: 1.655 GiB, 8.94% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:09

    549.920193 seconds (2.41 G allocations: 465.246 GiB, 16.02% gc time)
    50,0.0022612246562657024,0.0024477801523693473,4.889272778303791,0.046091790953551204
    # start 51 th simulation
    Gibbs Sampling

    Progress:  50%|█████████████████████                    |  ETA: 8:41:42

    Start 50000 iteration for GS

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:03

     63.796755 seconds (607.97 M allocations: 67.204 GiB, 6.61% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress:   4%|██                                       |  ETA: 0:00:03

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.250632 seconds (21.97 M allocations: 1.655 GiB, 7.31% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error

    400000

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:03
    Progress:  51%|█████████████████████                    |  ETA: 8:31:02

    543.837572 seconds (2.41 G allocations: 465.247 GiB, 16.20% gc time)
    51,0.0017168806017927995,0.0030749623407270712,4.887362389366851,0.045219364096177675
    # start 52 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:14

     74.802360 seconds (607.99 M allocations: 67.206 GiB, 8.67% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress:   3%|██                                       |  ETA: 0:00:03

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.318369 seconds (21.97 M allocations: 1.655 GiB, 8.29% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:02
    Progress:  52%|██████████████████████                   |  ETA: 8:20:31

    542.155106 seconds (2.41 G allocations: 465.245 GiB, 16.16% gc time)
    52,0.0023998871044473833,0.0031185170057261916,4.899497807074674,0.04597187671580107
    # start 53 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress:   0%|█                                        |  ETA: 0:01:15

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:03

     63.597957 seconds (607.97 M allocations: 67.204 GiB, 6.54% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.174808 seconds (21.97 M allocations: 1.655 GiB, 7.34% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:04

    544.848464 seconds (2.41 G allocations: 465.247 GiB, 16.33% gc time)
    53,0.003084805035732281,0.0022002809313678446,4.906773055377041,0.04567922105002251

    Progress:  53%|██████████████████████                   |  ETA: 8:09:54

    # start 54 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:15

     75.042894 seconds (607.99 M allocations: 67.205 GiB, 8.66% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress:   4%|██                                       |  ETA: 0:00:03

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.252346 seconds (21.97 M allocations: 1.655 GiB, 8.36% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress:   0%|█                                        |  ETA: 0:10:12

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:05
    Progress:  54%|███████████████████████                  |  ETA: 7:59:27

    545.544938 seconds (2.41 G allocations: 465.245 GiB, 16.09% gc time)
    54,0.003383785300196111,0.0018214771391898378,4.912300636562391,0.045164856167359976
    # start 55 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress:   0%|█                                        |  ETA: 0:00:52

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:04

     64.763944 seconds (607.97 M allocations: 67.204 GiB, 6.47% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress:   4%|██                                       |  ETA: 0:00:03

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.196897 seconds (21.97 M allocations: 1.655 GiB, 7.43% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:05
    Progress:  55%|███████████████████████                  |  ETA: 7:48:52

    545.396116 seconds (2.41 G allocations: 465.247 GiB, 16.27% gc time)
    55,0.002758106256233777,0.0018177504806599796,4.906183474750234,0.04475608360958659
    # start 56 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:15

     75.473661 seconds (607.99 M allocations: 67.206 GiB, 8.66% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress:   3%|██                                       |  ETA: 0:00:03

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.323842 seconds (21.97 M allocations: 1.655 GiB, 8.49% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:04
    Progress:  56%|███████████████████████                  |  ETA: 7:38:26

    544.381843 seconds (2.41 G allocations: 465.245 GiB, 16.14% gc time)
    56,0.0010887977247245142,0.003831193641646814,4.906430044243468,0.043950293288581
    # start 57 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress:   0%|█                                        |  ETA: 0:00:59

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:04

     64.074217 seconds (607.97 M allocations: 67.204 GiB, 6.61% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.136804 seconds (21.97 M allocations: 1.655 GiB, 7.66% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:05
    Progress:  57%|████████████████████████                 |  ETA: 7:27:52

    545.299402 seconds (2.41 G allocations: 465.247 GiB, 16.34% gc time)
    57,0.0037508378124633795,0.00030490477666544457,4.891505702925667,0.045679183585523586
    # start 58 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:15

     75.299897 seconds (607.99 M allocations: 67.206 GiB, 8.76% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.361069 seconds (21.97 M allocations: 1.655 GiB, 8.44% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress:   0%|█                                        |  ETA: 0:08:02

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:02

    542.646207 seconds (2.41 G allocations: 465.245 GiB, 16.10% gc time)
    58,0.000999010732478565,0.0035627112657094246,4.885819776981915,0.045243411265949884
    # start 59 th simulation
    Gibbs Sampling

    Progress:  58%|████████████████████████                 |  ETA: 7:17:24

    Start 50000 iteration for GS

    Progress:   0%|█                                        |  ETA: 0:00:57

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:03

     63.752114 seconds (607.97 M allocations: 67.204 GiB, 6.47% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress:   4%|██                                       |  ETA: 0:00:02

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.098781 seconds (21.97 M allocations: 1.655 GiB, 7.42% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:06

    546.888012 seconds (2.41 G allocations: 465.248 GiB, 16.30% gc time)
    59,0.002729571210341925,0.0023003544201193847,4.888262249074785,0.04453698739276896

    Progress:  59%|█████████████████████████                |  ETA: 7:06:52

    # start 60 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:14

     74.834468 seconds (607.99 M allocations: 67.205 GiB, 8.67% gc time)

    Progress:   4%|██                                       |  ETA: 0:00:03

    Start burn-in and thining from 50000 to 2000

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.284657 seconds (21.97 M allocations: 1.655 GiB, 8.67% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:05
    Progress:  60%|█████████████████████████                |  ETA: 6:56:27

    545.011730 seconds (2.41 G allocations: 465.245 GiB, 16.08% gc time)
    60,0.001551892537627175,0.003278005416927959,4.887289510832791,0.04379921640292564
    # start 61 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress:   0%|█                                        |  ETA: 0:00:57

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:04

     64.462672 seconds (607.97 M allocations: 67.204 GiB, 6.52% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress:   4%|██                                       |  ETA: 0:00:02

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.168472 seconds (21.97 M allocations: 1.655 GiB, 7.24% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress:   0%|█                                        |  ETA: 0:07:29

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:04
    Progress:  61%|██████████████████████████               |  ETA: 6:45:54

    544.800237 seconds (2.41 G allocations: 465.247 GiB, 16.26% gc time)
    61,0.004273178601487479,0.0003429040550334897,4.882843496827679,0.04330405265660763
    # start 62 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:14

     74.701344 seconds (607.99 M allocations: 67.205 GiB, 8.72% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress:   4%|██                                       |  ETA: 0:00:03

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.211907 seconds (21.97 M allocations: 1.655 GiB, 8.42% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress:   0%|█                                        |  ETA: 0:08:50

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:04
    Progress:  62%|██████████████████████████               |  ETA: 6:35:29

    544.107129 seconds (2.41 G allocations: 465.245 GiB, 16.13% gc time)
    62,0.0008505514570360685,0.0041920351193836,4.885419998111421,0.042677718905168624
    # start 63 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress:   0%|█                                        |  ETA: 0:00:55

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:04

     64.472097 seconds (607.97 M allocations: 67.204 GiB, 6.49% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress:   4%|██                                       |  ETA: 0:00:03

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.110318 seconds (21.97 M allocations: 1.655 GiB, 7.41% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress:   0%|█                                        |  ETA: 0:08:38

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:17
    Progress:  63%|██████████████████████████               |  ETA: 6:25:05

    557.833884 seconds (2.41 G allocations: 465.249 GiB, 16.24% gc time)
    63,0.00485302604392376,-6.476100708763602e-5,4.883877855868955,0.042023137224461195
    # start 64 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress:   0%|█                                        |  ETA: 0:01:16

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:20

     80.569916 seconds (608.00 M allocations: 67.206 GiB, 8.50% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.442239 seconds (21.97 M allocations: 1.655 GiB, 8.09% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress:   0%|█                                        |  ETA: 0:08:07

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:21
    Progress:  64%|███████████████████████████              |  ETA: 6:14:53

    561.224422 seconds (2.41 G allocations: 465.246 GiB, 15.96% gc time)
    64,0.0016232550555929756,0.0035668137498870538,4.888662089456628,0.04163708925061894
    # start 65 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:06

     66.550151 seconds (607.98 M allocations: 67.205 GiB, 6.46% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress:   4%|██                                       |  ETA: 0:00:02

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.252674 seconds (21.97 M allocations: 1.655 GiB, 7.16% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress:   0%|█                                        |  ETA: 0:07:10

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:21
    Progress:  65%|███████████████████████████              |  ETA: 6:04:32

    561.575298 seconds (2.41 G allocations: 465.250 GiB, 16.16% gc time)
    65,0.002332856071180049,0.0023295654694880947,4.885181465629113,0.04113901983135414
    # start 66 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:18

     78.781085 seconds (608.00 M allocations: 67.206 GiB, 8.57% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress:   3%|██                                       |  ETA: 0:00:03

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.486239 seconds (21.97 M allocations: 1.655 GiB, 8.23% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress:   0%|█                                        |  ETA: 0:08:36

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:09

    549.509350 seconds (2.41 G allocations: 465.245 GiB, 16.09% gc time)
    66,0.0030865755991076166,0.0018429652927237986,4.88585357814733,0.040516481400944354

    Progress:  66%|████████████████████████████             |  ETA: 5:54:10

    # start 67 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress:   0%|█                                        |  ETA: 0:01:01

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:04

     64.663610 seconds (607.97 M allocations: 67.204 GiB, 6.45% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress:   3%|██                                       |  ETA: 0:00:03

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.249735 seconds (21.97 M allocations: 1.655 GiB, 7.33% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress:   0%|█                                        |  ETA: 0:07:05

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:04
    Progress:  67%|████████████████████████████             |  ETA: 5:43:40

    544.938501 seconds (2.41 G allocations: 465.247 GiB, 16.28% gc time)
    67,0.002285804367666749,0.0019757954946822576,4.87653635850855,0.040980401433893976
    # start 68 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress:   0%|█                                        |  ETA: 0:00:56

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:16

     76.678890 seconds (608.00 M allocations: 67.206 GiB, 8.55% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.431291 seconds (21.97 M allocations: 1.655 GiB, 8.03% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress:   0%|█                                        |  ETA: 0:07:01

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:05

    545.928270 seconds (2.41 G allocations: 465.245 GiB, 16.11% gc time)
    68,0.002290373506825949,0.003290466687725393,4.88689376786212,0.041680634706813074

    Progress:  68%|████████████████████████████             |  ETA: 5:33:16

    # start 69 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress:   0%|█                                        |  ETA: 0:00:57

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:04

     64.573978 seconds (607.97 M allocations: 67.204 GiB, 6.45% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.158946 seconds (21.97 M allocations: 1.655 GiB, 7.40% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress:   0%|█                                        |  ETA: 0:06:52

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:07

    547.067383 seconds (2.41 G allocations: 465.247 GiB, 16.26% gc time)
    69,0.002038184217191339,0.0028794226315182903,4.887338884975852,0.04107453752306219

    Progress:  69%|█████████████████████████████            |  ETA: 5:22:46

    # start 70 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress:   0%|█                                        |  ETA: 0:00:54

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:15

     75.511091 seconds (607.99 M allocations: 67.206 GiB, 8.74% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress:   4%|██                                       |  ETA: 0:00:03

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.306151 seconds (21.97 M allocations: 1.655 GiB, 8.32% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:04
    Progress:  70%|█████████████████████████████            |  ETA: 5:12:21

    544.337941 seconds (2.41 G allocations: 465.245 GiB, 16.11% gc time)
    70,0.003863746686793748,0.0012073467222771458,4.889963949605781,0.04056852519021992
    # start 71 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:04

     64.057318 seconds (607.97 M allocations: 67.204 GiB, 6.58% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.160264 seconds (21.97 M allocations: 1.655 GiB, 7.47% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:05
    Progress:  71%|██████████████████████████████           |  ETA: 5:01:52

    545.035974 seconds (2.41 G allocations: 465.247 GiB, 16.36% gc time)
    71,0.0016220356926448843,0.003003076826960202,4.886233647774786,0.04016664881445194
    # start 72 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:15

     75.136250 seconds (607.99 M allocations: 67.206 GiB, 8.74% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress:   4%|██                                       |  ETA: 0:00:03

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.334475 seconds (21.97 M allocations: 1.655 GiB, 8.98% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress:   0%|█                                        |  ETA: 0:07:19

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:03

    543.982253 seconds (2.41 G allocations:

    Progress:  72%|██████████████████████████████           |  ETA: 4:51:26

    465.245 GiB, 16.13% gc time)
    72,0.0015826894385968134,0.0031026634536314495,4.8834436372810845,0.03970300100692144
    # start 73 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress:   0%|█                                        |  ETA: 0:01:01

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:03

     63.473599 seconds (607.97 M allocations: 67.204 GiB, 6.58% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress:   4%|██                                       |  ETA: 0:00:03

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.128953 seconds (21.97 M allocations: 1.655 GiB, 7.40% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:04
    Progress:  73%|██████████████████████████████           |  ETA: 4:40:57

    544.665796 seconds (2.41 G allocations: 465.247 GiB, 16.31% gc time)
    73,0.0011481263823953953,0.004179052406392856,4.88952220100036,0.03962436307923654
    # start 74 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:15

     75.294657 seconds (607.99 M allocations: 67.206 GiB, 8.73% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress:   4%|██                                       |  ETA: 0:00:03

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.258243 seconds (21.97 M allocations: 1.655 GiB, 8.99% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:03
    Progress:  74%|███████████████████████████████          |  ETA: 4:30:32

    543.862378 seconds (2.41 G allocations: 465.245 GiB, 16.14% gc time)
    74,0.0024063751465417398,0.0025364897708553425,4.890243048519236,0.03909187769443901
    # start 75 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress:   0%|█                                        |  ETA: 0:00:55

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:04

     64.034199 seconds (607.97 M allocations: 67.204 GiB, 6.53% gc time)

    Progress:   3%|██                                       |  ETA: 0:00:03

    Start burn-in and thining from 50000 to 2000

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.121143 seconds (21.97 M allocations: 1.655 GiB, 7.18% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress:   0%|█                                        |  ETA: 0:07:56

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:05
    Progress:  75%|███████████████████████████████          |  ETA: 4:20:04

    545.543321 seconds (2.41 G allocations: 465.247 GiB, 16.32% gc time)
    75,0.004856745979368095,0.0005539175267718546,4.897181987954178,0.039186380943478274
    # start 76 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress:   0%|█                                        |  ETA: 0:00:54

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:14

     74.378296 seconds (607.99 M allocations: 67.205 GiB, 8.59% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.321229 seconds (21.97 M allocations: 1.655 GiB, 8.24% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:09
    Progress:  76%|████████████████████████████████         |  ETA: 4:09:41

    549.789043 seconds (2.41 G allocations: 465.246 GiB, 16.08% gc time)
    76,0.002734385768859047,0.002187733050386575,4.897510104155382,0.03866872522252329
    # start 77 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:04

     64.204826 seconds (607.97 M allocations: 67.204 GiB, 6.53% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.200969 seconds (21.97 M allocations: 1.655 GiB, 7.30% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress:   0%|█                                        |  ETA: 0:07:01

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:10
    Progress:  77%|████████████████████████████████         |  ETA: 3:59:15

    550.139580 seconds (2.41 G allocations: 465.248 GiB, 16.18% gc time)
    77,0.003640921485192852,0.0015282597386773776,4.901038300515314,0.03832597409082284
    # start 78 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:13

     73.934913 seconds (607.99 M allocations: 67.205 GiB, 8.72% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress:   4%|██                                       |  ETA: 0:00:03

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.260136 seconds (21.97 M allocations: 1.655 GiB, 8.25% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:01
    Progress:  78%|████████████████████████████████         |  ETA: 3:48:50

    541.929810 seconds (2.41 G allocations: 465.245 GiB, 16.12% gc time)
    78,0.00208012936828543,0.002536612569382142,4.8973934753506,0.03800659736561544
    # start 79 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:03

     63.963214 seconds (607.97 M allocations: 67.204 GiB, 6.58% gc time)

    Progress:   4%|██                                       |  ETA: 0:00:02

    Start burn-in and thining from 50000 to 2000

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.086847 seconds (21.97 M allocations: 1.655 GiB, 7.22% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress:   0%|█                                        |  ETA: 0:07:10

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:00

    540.854908 seconds (2.41 G allocations: 465.247 GiB, 16.33% gc time)
    79,0.002245894552078598,0.0026024617419438565,4.89677275153632,0.03752755100686577

    Progress:  79%|█████████████████████████████████        |  ETA: 3:38:21

    # start 80 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress:   0%|█                                        |  ETA: 0:00:54

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:14

     74.890887 seconds (607.99 M allocations: 67.205 GiB, 8.71% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress:   4%|██                                       |  ETA: 0:00:03

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.252282 seconds (21.97 M allocations: 1.655 GiB, 8.63% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress:   0%|█                                        |  ETA: 0:07:24

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:13

    553.788398 seconds (2.41 G allocations: 465.246 GiB, 15.98% gc time)

    Progress:  80%|█████████████████████████████████        |  ETA: 3:28:00

    80,0.0021307663591854788,0.003290286355515793,4.903326251075882,0.03763053964101003
    # start 81 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress:   0%|█                                        |  ETA: 0:00:54

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:04

     64.252391 seconds (607.97 M allocations: 67.204 GiB, 6.54% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress:   4%|██                                       |  ETA: 0:00:03

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.169225 seconds (21.97 M allocations: 1.655 GiB, 7.24% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress:   0%|█                                        |  ETA: 0:08:19

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:04
    Progress:  81%|██████████████████████████████████       |  ETA: 3:17:33

    544.684797 seconds (2.41 G allocations: 465.247 GiB, 16.38% gc time)
    81,0.002941286576827066,0.002058881926322989,4.904521834434822,0.037182288124699016
    # start 82 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:15

     75.653679 seconds (607.99 M allocations: 67.206 GiB, 8.84% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.195918 seconds (21.97 M allocations: 1.655 GiB, 9.16% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress:   0%|█                                        |  ETA: 0:07:01

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:05
    Progress:  82%|██████████████████████████████████       |  ETA: 3:07:09

    545.680190 seconds (2.41 G allocations: 465.246 GiB, 16.17% gc time)
    82,0.0027841381417134945,0.001933609797348225,4.902244104003441,0.036796610494805815
    # start 83 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress:   0%|█                                        |  ETA: 0:00:55

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:02

     62.907917 seconds (607.97 M allocations: 67.204 GiB, 6.71% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.161350 seconds (21.97 M allocations: 1.655 GiB, 7.20% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:03

    543.400065 seconds (2.41 G allocations: 465.247 GiB, 16.31% gc time)
    83,0.0008201256004500808,0.004516167923984128,4.907473615092969,0.03672481520682606
    # start 84 th simulation
    Gibbs Sampling

    Progress:  83%|███████████████████████████████████      |  ETA: 2:56:43

    Start 50000 iteration for GS

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:14

     74.394876 seconds (607.99 M allocations: 67.205 GiB, 8.75% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.221615 seconds (21.97 M allocations: 1.655 GiB, 8.15% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress:   0%|█                                        |  ETA: 0:08:01

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:02
    Progress:  84%|███████████████████████████████████      |  ETA: 2:46:18

    542.868043 seconds (2.41 G allocations: 465.246 GiB, 16.18% gc time)
    84,0.003128836966189435,0.001454069799792547,4.903609724032124,0.03649012899938692
    # start 85 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:03

     63.082251 seconds (607.97 M allocations: 67.204 GiB, 6.56% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.084616 seconds (21.97 M allocations: 1.655 GiB, 7.88% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress: 100%|█████████████████████████████████████████| Time: 0:09:03
    Progress:  85%|███████████████████████████████████      |  ETA: 2:35:52

    543.116117 seconds (2.41 G allocations: 465.247 GiB, 16.32% gc time)
    85,0.0022105164523947528,0.002694124301667834,4.903621853797189,0.036058279982579766
    # start 86 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:13

     73.922279 seconds (607.99 M allocations: 67.205 GiB, 8.72% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.214351 seconds (21.97 M allocations: 1.655 GiB, 8.45% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress: 100%|█████████████████████████████████████████| Time: 0:08:57

    537.564839 seconds (2.41 G allocations: 465.245 GiB, 16.14% gc time)
    86,0.0027553466879853138,0.0026636421060872916,4.9096144926376,0.03613687972989935

    Progress:  86%|████████████████████████████████████     |  ETA: 2:25:28

    # start 87 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress:   0%|█                                        |  ETA: 0:00:57

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:03

     63.051634 seconds (607.97 M allocations: 67.204 GiB, 6.50% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress:   4%|██                                       |  ETA: 0:00:03

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.056493 seconds (21.97 M allocations: 1.655 GiB, 7.25% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress: 100%|█████████████████████████████████████████| Time: 0:08:56

    536.478142 seconds (2.41 G allocations: 465.247 GiB, 16.35% gc time)
    87,0.002298743479879147,0.0031040294216095393,4.915282980095658,0.036166085397235495

    Progress:  87%|████████████████████████████████████     |  ETA: 2:15:01

    # start 88 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:14

     74.275337 seconds (607.99 M allocations: 67.205 GiB, 8.69% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.217216 seconds (21.97 M allocations: 1.655 GiB, 8.39% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error

    Progress:   0%|█                                        |  ETA: 0:07:39

    400000

    Progress: 100%|█████████████████████████████████████████| Time: 0:08:56
    Progress:  88%|█████████████████████████████████████    |  ETA: 2:04:37

    536.386360 seconds (2.41 G allocations: 465.245 GiB, 16.07% gc time)
    88,0.0016028311077355526,0.003508260844507576,4.91750808205188,0.03582191875373712
    # start 89 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress:   0%|█                                        |  ETA: 0:00:55

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:02

     62.497445 seconds (607.97 M allocations: 67.204 GiB, 6.59% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.145716 seconds (21.97 M allocations: 1.655 GiB, 7.70% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error

    Progress:   0%|█                                        |  ETA: 0:08:02

    400000

    Progress: 100%|█████████████████████████████████████████| Time: 0:08:56
    Progress:  89%|█████████████████████████████████████    |  ETA: 1:54:11

    536.899291 seconds (2.41 G allocations: 465.246 GiB, 16.37% gc time)
    89,0.0013637141285463418,0.0033856323134605513,4.915618625422161,0.03546750248965698
    # start 90 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress:   0%|█                                        |  ETA: 0:00:55

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:14

     74.560680 seconds (607.99 M allocations: 67.205 GiB, 8.62% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress:   4%|██                                       |  ETA: 0:00:02

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.207505 seconds (21.97 M allocations: 1.655 GiB, 9.42% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress: 100%|█████████████████████████████████████████| Time: 0:08:57

    537.263192 seconds (2.41 G allocations: 465.245 GiB, 16.09% gc time)
    90,0.000923850267428807,0.0038200706992439068,4.913710873658277,0.03512305431581787

    Progress:  90%|█████████████████████████████████████    |  ETA: 1:43:48

    # start 91 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:02

     62.909995 seconds (607.97 M allocations: 67.204 GiB, 6.63% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.155545 seconds (21.97 M allocations: 1.655 GiB, 7.68% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress: 100%|█████████████████████████████████████████| Time: 0:08:59
    Progress:  91%|██████████████████████████████████████   |  ETA: 1:33:23

    539.846331 seconds (2.41 G allocations: 465.247 GiB, 16.32% gc time)
    91,0.0016568961263016507,0.003553097769243307,4.916966731041648,0.03488720148877536
    # start 92 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress:   0%|█                                        |  ETA: 0:00:56

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:13

     73.934068 seconds (607.99 M allocations: 67.205 GiB, 8.64% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.006332 seconds (21.97 M allocations: 1.655 GiB, 8.53% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress: 100%|█████████████████████████████████████████| Time: 0:08:18
    Progress:  92%|██████████████████████████████████████   |  ETA: 1:22:57

    498.946897 seconds (2.41 G allocations: 465.241 GiB, 16.29% gc time)
    92,0.0032373109742565866,0.0012805605206567773,4.9126287393446,0.034777520587441484
    # start 93 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress:   0%|█                                        |  ETA: 0:00:51

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:57

     57.601781 seconds (607.96 M allocations: 67.204 GiB, 6.47% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress:   4%|██                                       |  ETA: 0:00:02

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:02

      2.910604 seconds (21.97 M allocations: 1.655 GiB, 7.60% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress:   0%|█                                        |  ETA: 0:06:48

    Progress: 100%|█████████████████████████████████████████| Time: 0:08:33

    513.251700 seconds (2.41 G allocations: 465.243 GiB, 16.42% gc time)
    93,0.00282886573375201,0.0022205982806971407,4.914100086388736,0.03443298658414458

    Progress:  93%|███████████████████████████████████████  |  ETA: 1:12:31

    # start 94 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:11

     71.327851 seconds (607.99 M allocations: 67.205 GiB, 8.79% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.147933 seconds (21.97 M allocations: 1.655 GiB, 8.62% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress: 100%|█████████████████████████████████████████| Time: 0:08:38
    Progress:  94%|███████████████████████████████████████  |  ETA: 1:02:08

    518.572049 seconds (2.41 G allocations: 465.244 GiB, 16.23% gc time)
    94,0.003230614504990668,0.0019207047243509695,4.916623694292491,0.03415805878148695
    # start 95 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress:   0%|█                                        |  ETA: 0:00:54

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:59

     59.641425 seconds (607.96 M allocations: 67.204 GiB, 6.40% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress:   4%|██                                       |  ETA: 0:00:02

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:02

      2.918453 seconds (21.97 M allocations: 1.655 GiB, 7.73% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress:   0%|█                                        |  ETA: 0:07:10

    Progress: 100%|█████████████████████████████████████████| Time: 0:08:29
    Progress:  95%|███████████████████████████████████████  |  ETA: 0:51:44

    509.976726 seconds (2.41 G allocations: 465.242 GiB, 16.55% gc time)
    95,0.0029832490358978297,0.0019506071596321773,4.916805089042359,0.033797074460617034
    # start 96 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress:   0%|█                                        |  ETA: 0:00:55

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:09

     69.631003 seconds (607.98 M allocations: 67.205 GiB, 8.89% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress:   4%|██                                       |  ETA: 0:00:02

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:03

      3.043443 seconds (21.97 M allocations: 1.655 GiB, 8.81% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress:   0%|█                                        |  ETA: 0:08:13

    Progress: 100%|█████████████████████████████████████████| Time: 0:08:26
    Progress:  96%|████████████████████████████████████████ |  ETA: 0:41:21

    506.094442 seconds (2.41 G allocations: 465.243 GiB, 16.20% gc time)
    96,0.002424661085657162,0.002571657081576243,4.9176333502735154,0.03345342352266087
    # start 97 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:57

     57.051072 seconds (607.96 M allocations: 67.203 GiB, 6.50% gc time)

    Progress:   4%|██                                       |  ETA: 0:00:02

    Start burn-in and thining from 50000 to 2000

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:02

      2.908651 seconds (21.97 M allocations: 1.655 GiB, 7.53% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress:   0%|█                                        |  ETA: 0:06:52

    Progress: 100%|█████████████████████████████████████████| Time: 0:08:28

    508.874380 seconds (2.41 G allocations: 465.242 GiB, 16.40% gc time)
    97,0.0009254042134889023,0.00367321348821641,4.914344529154256,0.03326970111853343

    Progress:  97%|████████████████████████████████████████ |  ETA: 0:30:59

    # start 98 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress:   0%|█                                        |  ETA: 0:00:55

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:12

     72.818996 seconds (607.98 M allocations: 67.205 GiB, 8.34% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress:   4%|██                                       |  ETA: 0:00:02

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:02

      2.968161 seconds (21.97 M allocations: 1.655 GiB, 8.57% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress:   0%|█                                        |  ETA: 0:06:52

    Progress: 100%|█████████████████████████████████████████| Time: 0:08:21
    Progress:  98%|█████████████████████████████████████████|  ETA: 0:20:39

    501.397779 seconds (2.41 G allocations: 465.241 GiB, 16.28% gc time)
    98,0.0037537982815946747,0.0012535272025918365,4.915293314409687,0.03294213048398767
    # start 99 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:57

     57.545704 seconds (607.96 M allocations: 67.204 GiB, 6.43% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress:   4%|██                                       |  ETA: 0:00:03

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:02

      2.908259 seconds (21.97 M allocations: 1.655 GiB, 7.42% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress:   0%|█                                        |  ETA: 0:07:05

    Progress: 100%|█████████████████████████████████████████| Time: 0:08:20

    500.105736 seconds (2.41 G allocations: 465.241 GiB, 16.43% gc time)

    Progress:  99%|█████████████████████████████████████████|  ETA: 0:10:19

    99,0.0036838889565491764,0.0011444475977243525,4.9144149632972,0.03261951184799367
    # start 100 th simulation
    Gibbs Sampling
    Start 50000 iteration for GS

    Progress: 100%|█████████████████████████████████████████| Time: 0:01:07

     67.804963 seconds (607.98 M allocations: 67.205 GiB, 8.77% gc time)
    Start burn-in and thining from 50000 to 2000

    Progress:   4%|██                                       |  ETA: 0:00:02

    Progress: 100%|█████████████████████████████████████████| Time: 0:00:02

      2.987960 seconds (21.97 M allocations: 1.655 GiB, 8.57% gc time)
    Calculation Normalized WAIC
    Calculaton Generalization Error
    400000

    Progress:   0%|█                                        |  ETA: 0:07:39

    Progress: 100%|█████████████████████████████████████████| Time: 0:08:21
    Progress: 100%|█████████████████████████████████████████| Time: 17:10:30

    501.144084 seconds (2.41 G allocations: 465.241 GiB, 16.18% gc time)
    100,0.0038134907433271348,0.0006645132255128205,4.910050853352629,0.03258523220334156
    61830.639760 seconds (303.70 G allocations: 52.215 TiB, 15.13% gc time)
    The numerical RLCT is 4.910050853352629 ± 0.03258523220334156

```julia
println("$M, $N, $H, $r")
λ = rlct_LDA(M,N,H,r)
println("The exact RLCT = $λ")
```

    4, 4, 4, 2
    The exact RLCT = 5.5

```julia
using DelimitedFiles
each_lams_savepath = split(LOG_FILE_PATH, ".")[1]*"_each_lams.csv"
writedlm(each_lams_savepath,  each_lams, ',')
```

```julia
each_lams_mean = mean(each_lams)
each_lams_std = std(each_lams)
upperline = each_lams_mean + each_lams_std
middleline = each_lams_mean
lowerline = each_lams_mean - each_lams_std
trueline = λ
plot(y=each_lams, yintercept = [upperline, middleline, lowerline, trueline],
    Geom.violin, Geom.hline(color=["orange","red","orange","blue"], style=[:dot,:dash,:dot,:solid]))
```

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