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genkuroki/DynamicHMCExamples for DynamicHMC.jl v2.0.0.ipynb

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DynamicHMCExamples for DynamicHMC.jl v2.0.0
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DynamicHMCExamples for DynamicHMC.jl v2.0.0.ipynb
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"source": "# DynamicHMCExamples for DynamicHMC.jl v2.0.0\n\n黒木玄\n\n2019-08-26～2019-09-03\n\n2019年9月3日に\n\n```\npkg> add LogDensityProblems\npkg> add DynamicHMC\n```\n\nした状態で\n\nhttps://github.com/tpapp/DynamicHMCExamples.jl/tree/master/src\n\nにあるexamplesを動くようにしたノートブック.\n\nDynamicHMC.jl は v2.0.0 から仕様が大幅に変化するので, 現在公開されているexamplesがそのままでは動かない. それを修正してみたのが, このノートブックです."
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"cell_type": "code",
"source": "]status LogDensityProblems",
"execution_count": 1,
"outputs": [
{
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"text": "\u001b[32m\u001b[1m Status\u001b[22m\u001b[39m `C:\\Users\\genkuroki\\.julia\\environments\\v1.2\\Project.toml`\n \u001b[90m [6e4b80f9]\u001b[39m\u001b[37m BenchmarkTools v0.4.2\u001b[39m\n \u001b[90m [6fdf6af0]\u001b[39m\u001b[37m LogDensityProblems v0.9.1\u001b[39m\n \u001b[90m [d96e819e]\u001b[39m\u001b[37m Parameters v0.11.0\u001b[39m\n \u001b[90m [84d833dd]\u001b[39m\u001b[37m TransformVariables v0.3.5\u001b[39m\n",
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"source": "]status DynamicHMC",
"execution_count": 2,
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"text": "\u001b[32m\u001b[1m Status\u001b[22m\u001b[39m `C:\\Users\\genkuroki\\.julia\\environments\\v1.2\\Project.toml`\n \u001b[90m [bbc10e6e]\u001b[39m\u001b[37m DynamicHMC v2.0.0\u001b[39m\n \u001b[90m [6fdf6af0]\u001b[39m\u001b[37m LogDensityProblems v0.9.1\u001b[39m\n \u001b[90m [429524aa]\u001b[39m\u001b[37m Optim v0.19.2 [`C:\\Users\\genkuroki\\.julia\\dev\\Optim`]\u001b[39m\n \u001b[90m [d96e819e]\u001b[39m\u001b[37m Parameters v0.11.0\u001b[39m\n \u001b[90m [4c63d2b9]\u001b[39m\u001b[37m StatsFuns v0.8.0\u001b[39m\n",
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"source": "<h1>Table of Contents<span class=\"tocSkip\"></span></h1>\n<div class=\"toc\"><ul class=\"toc-item\"><li><span><a href=\"#Bernoulli\" data-toc-modified-id=\"Bernoulli-1\"><span class=\"toc-item-num\">1&nbsp;&nbsp;</span>Bernoulli</a></span></li><li><span><a href=\"#Logistic-regression\" data-toc-modified-id=\"Logistic-regression-2\"><span class=\"toc-item-num\">2&nbsp;&nbsp;</span>Logistic regression</a></span></li><li><span><a href=\"#Linear-regression\" data-toc-modified-id=\"Linear-regression-3\"><span class=\"toc-item-num\">3&nbsp;&nbsp;</span>Linear regression</a></span></li></ul></div>"
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"source": "using Distributions\nusing TransformVariables\nusing LogDensityProblems\nusing DynamicHMC\nusing DynamicHMC.Diagnostics\nusing MCMCDiagnostics\nusing Parameters\nusing Statistics\nusing Random\nusing Plots\ngr(size=(400, 250), titlefontsize=12)\nusing ForwardDiff\nusing StatsFuns\nusing LinearAlgebra",
"execution_count": 3,
"outputs": []
},
{
"metadata": {},
"cell_type": "markdown",
"source": "## Bernoulli\n\nhttps://github.com/tpapp/DynamicHMCExamples.jl/blob/master/src/example_independent_bernoulli.jl"
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"source": "# # Estimate Bernoulli draws probabilility\n\n# We estimate a simple model of ``n`` independent Bernoulli draws, with\n# probability ``α``. First, we load the packages we use.\n\n# using TransformVariables\n# using LogDensityProblems\n# using DynamicHMC\n# using MCMCDiagnostics\n# using Parameters\n# using Statistics\n\n# Then define a structure to hold the data.\n# For this model, the number of draws equal to `1` is a sufficient statistic.\n\n\"\"\"\nToy problem using a Bernoulli distribution.\nWe model `n` independent draws from a ``Bernoulli(α)`` distribution.\n\"\"\"\nstruct BernoulliProblem\n \"Total number of draws in the data.\"\n n::Int\n \"Number of draws `==1` in the data\"\n s::Int\nend\n\n# Then make the type callable with the parameters *as a single argument*. We\n# use decomposition in the arguments, but it could be done inside the function,\n# too.\n\nfunction (problem::BernoulliProblem)((α, )::NamedTuple{(:α, )})\n @unpack n, s = problem # extract the data\n # log likelihood: the constant log(combinations(n, s)) term\n # has been dropped since it is irrelevant to sampling.\n s * log(α) + (n-s) * log(1-α)\nend\n\n# We should test this, also, this would be a good place to benchmark and\n# optimize more complicated problems.\n\np = BernoulliProblem(30, 10)\np((α = 0.5, ))",
"execution_count": 4,
"outputs": [
{
"output_type": "execute_result",
"execution_count": 4,
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"source": "# Recall that we need to\n#\n# 1. transform from ``ℝ`` to the valid parameter domain `(0,1)` for more efficient sampling, and\n#\n# 2. calculate the derivatives for this transformed mapping.\n#\n# The helper packages `TransformVariables` and `LogDensityProblems` take care of\n# this. We use a flat prior (the default, omitted)\n\n\ntrans = as((α = as𝕀,))\nP = TransformedLogDensity(trans, p)\n∇P = ADgradient(:ForwardDiff, P);\n\ndisplay(trans)\ndisplay(P)\ndisplay(∇P)",
"execution_count": 5,
"outputs": [
{
"output_type": "display_data",
"data": {
"text/plain": "TransformVariables.TransformTuple{NamedTuple{(:α,),Tuple{TransformVariables.ScaledShiftedLogistic{Float64}}}}((α = TransformVariables.ScaledShiftedLogistic{Float64}(1.0, 0.0),), 1)"
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"metadata": {}
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"output_type": "display_data",
"data": {
"text/plain": "TransformedLogDensity of dimension 1"
},
"metadata": {}
},
{
"output_type": "display_data",
"data": {
"text/plain": "ForwardDiff AD wrapper for TransformedLogDensity of dimension 1, w/ chunk size 1"
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"metadata": {}
}
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"source": "# Finally, we sample from the posterior. `chain` holds the chain (positions and\n# diagnostic information), while the second returned value is the tuned sampler\n# which would allow continuation of sampling.\n\nL = 1000\n\n#chain, NUTS_tuned = NUTS_init_tune_mcmc(∇P, L)\n@time results = mcmc_with_warmup(Random.GLOBAL_RNG, ∇P, L; reporter = NoProgressReport());\n\ndisplay(results.chain)",
"execution_count": 6,
"outputs": [
{
"output_type": "display_data",
"data": {
"text/plain": "1000-element Array{Array{Float64,1},1}:\n [-1.2619343876166276] \n [-1.121137277131965] \n [-1.0184918317286016] \n [-0.5210269173776014] \n [-0.15134045615471098]\n [-0.15134045615471098]\n [-0.8074622309673839] \n [-0.4907752827826125] \n [-0.7850937898582242] \n [-0.7492371814744815] \n [-0.7698574432684676] \n [-0.2195251634463975] \n [-0.2841568577069278] \n ⋮ \n [-0.8963377811725822] \n [-1.044385179146749] \n [-0.7552283724996616] \n [-1.2453799599897892] \n [-1.2027729435300634] \n [-0.8395568174707463] \n [-0.4101687627491608] \n [-0.13230512550921467]\n [-0.27965954377652136]\n [-0.34008766804867313]\n [-0.9083683674072884] \n [-0.34196802124649434]"
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"text": " 7.601452 seconds (20.24 M allocations: 1.027 GiB, 9.15% gc time)\n",
"name": "stdout"
}
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"trusted": true
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"cell_type": "code",
"source": "# To get the posterior for ``α``, we need to use `get_position` and\n# then transform\n\n#posterior = transform.(Ref(t), get_position.(chain));\nposterior = transform.(trans, results.chain)",
"execution_count": 7,
"outputs": [
{
"output_type": "execute_result",
"execution_count": 7,
"data": {
"text/plain": "1000-element Array{NamedTuple{(:α,),Tuple{Float64}},1}:\n (α = 0.22064107790277224,)\n (α = 0.24580039126086983,)\n (α = 0.2653212775178784,) \n (α = 0.3726121374286622,) \n (α = 0.4622369353562233,) \n (α = 0.4622369353562233,) \n (α = 0.308431542381885,) \n (α = 0.379710947859802,) \n (α = 0.31322309745466015,)\n (α = 0.3209875378758236,) \n (α = 0.31650994495423057,)\n (α = 0.44533805220704364,)\n (α = 0.4294349630127374,) \n ⋮ \n (α = 0.28980366548273423,)\n (α = 0.26030475729653973,)\n (α = 0.3196831334740925,) \n (α = 0.22350091434551958,)\n (α = 0.23098229261173608,)\n (α = 0.30162813156942286,)\n (α = 0.39887165570067545,)\n (α = 0.46697188336413625,)\n (α = 0.4305372454152998,) \n (α = 0.4157881816046156,) \n (α = 0.28733383574674703,)\n (α = 0.41533150052405743,)"
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"metadata": {
"trusted": true
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"cell_type": "code",
"source": "# Extract the parameter.\n\nposterior_α = first.(posterior)",
"execution_count": 8,
"outputs": [
{
"output_type": "execute_result",
"execution_count": 8,
"data": {
"text/plain": "1000-element Array{Float64,1}:\n 0.22064107790277224\n 0.24580039126086983\n 0.2653212775178784 \n 0.3726121374286622 \n 0.4622369353562233 \n 0.4622369353562233 \n 0.308431542381885 \n 0.379710947859802 \n 0.31322309745466015\n 0.3209875378758236 \n 0.31650994495423057\n 0.44533805220704364\n 0.4294349630127374 \n ⋮ \n 0.28980366548273423\n 0.26030475729653973\n 0.3196831334740925 \n 0.22350091434551958\n 0.23098229261173608\n 0.30162813156942286\n 0.39887165570067545\n 0.46697188336413625\n 0.4305372454152998 \n 0.4157881816046156 \n 0.28733383574674703\n 0.41533150052405743"
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]
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "# check the mean\n\nmean(posterior_α)",
"execution_count": 9,
"outputs": [
{
"output_type": "execute_result",
"execution_count": 9,
"data": {
"text/plain": "0.3386421555658641"
},
"metadata": {}
}
]
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"metadata": {
"trusted": true
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"cell_type": "code",
"source": "# check the effective sample size\n\ness_α = effective_sample_size(posterior_α)",
"execution_count": 10,
"outputs": [
{
"output_type": "execute_result",
"execution_count": 10,
"data": {
"text/plain": "347.55933483592884"
},
"metadata": {}
}
]
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "# NUTS-specific statistics\n\n#NUTS_statistics(chain)\nsummarize_tree_statistics(results.tree_statistics)",
"execution_count": 11,
"outputs": [
{
"output_type": "execute_result",
"execution_count": 11,
"data": {
"text/plain": "Hamiltonian Monte Carlo sample of length 1000\n acceptance rate mean: 0.93, 5/25/50/75/95%: 0.71 0.91 0.97 1.0 1.0\n termination: divergence => 0%, max_depth => 0%, turning => 100%\n depth: 0 => 0%, 1 => 63%, 2 => 37%"
},
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}
]
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "P1 = plot(posterior_α)\nP2 = histogram(posterior_α)\nplot(P1, P2, size=(800, 250), title=\"alpha\", legend=false)",
"execution_count": 12,
"outputs": [
{
"output_type": "execute_result",
"execution_count": 12,
"data": {
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"metadata": {}
}
]
},
{
"metadata": {},
"cell_type": "markdown",
"source": "## Logistic regression\n\nhttps://github.com/tpapp/DynamicHMCExamples.jl/blob/master/src/example_logistic_regression.jl"
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "# # Logistic regression\n\n#using TransformVariables, LogDensityProblems, DynamicHMC, MCMCDiagnostics, Parameters,\n# Distributions, Statistics, StatsFuns, ForwardDiff\n\n\"\"\"\nLogistic regression.\nFor each draw, ``logit(Pr(yᵢ == 1)) ∼ Xᵢ β``. Uses a `β ∼ Normal(0, σ)` prior.\n`X` is supposed to include the `1`s for the intercept.\n\"\"\"\nstruct LogisticRegression{Ty, TX, Tσ}\n y::Ty\n X::TX\n σ::Tσ\nend\n\nfunction (problem::LogisticRegression)(θ)\n @unpack y, X, σ = problem\n @unpack β = θ\n loglik = sum(logpdf.(Bernoulli.(logistic.(X*β)), y))\n logpri = sum(logpdf.(Ref(Normal(0, σ)), β))\n loglik + logpri\nend\n\n# Make up parameters, generate data using random draws.\n\nN = 1000\nβ = [1.0, 2.0]\nX = hcat(ones(N), randn(N))\ny = rand.(Bernoulli.(logistic.(X*β)));",
"execution_count": 13,
"outputs": []
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "# Create a problem, apply a transformation, then use automatic differentiation.\n\np = LogisticRegression(y, X, 10.0) # data and (vague) priors",
"execution_count": 14,
"outputs": [
{
"output_type": "execute_result",
"execution_count": 14,
"data": {
"text/plain": "LogisticRegression{BitArray{1},Array{Float64,2},Float64}(Bool[0, 0, 1, 1, 0, 1, 1, 0, 1, 1 … 1, 1, 0, 1, 1, 0, 0, 1, 1, 1], [1.0 -1.409851853645631; 1.0 -0.31668020646081496; … ; 1.0 -0.250238586579046; 1.0 1.0507380471167176], 10.0)"
},
"metadata": {}
}
]
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "trans = as((β = as(Array, length(β)), )) # identity transformation, just to get the dimension\nP = TransformedLogDensity(trans, p) # transformed\n∇P = ADgradient(:ForwardDiff, P)\n\ndisplay(trans)\ndisplay(P)\ndisplay(∇P)",
"execution_count": 15,
"outputs": [
{
"output_type": "display_data",
"data": {
"text/plain": "TransformVariables.TransformTuple{NamedTuple{(:β,),Tuple{TransformVariables.ArrayTransform{TransformVariables.Identity,1}}}}((β = TransformVariables.ArrayTransform{TransformVariables.Identity,1}(TransformVariables.Identity(), (2,)),), 2)"
},
"metadata": {}
},
{
"output_type": "display_data",
"data": {
"text/plain": "TransformedLogDensity of dimension 2"
},
"metadata": {}
},
{
"output_type": "display_data",
"data": {
"text/plain": "ForwardDiff AD wrapper for TransformedLogDensity of dimension 2, w/ chunk size 2"
},
"metadata": {}
}
]
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "# Sample using NUTS, random starting point.\n\n#chain, NUTS_tuned = NUTS_init_tune_mcmc(∇P, 1000);\n\nL = 10^3\n@time results = mcmc_with_warmup(Random.GLOBAL_RNG, ∇P, L; reporter = NoProgressReport());\ndisplay(results.chain)",
"execution_count": 16,
"outputs": [
{
"output_type": "display_data",
"data": {
"text/plain": "1000-element Array{Array{Float64,1},1}:\n [0.9888810809602057, 1.9648187473705152]\n [0.9865863979912161, 2.1524603523534527]\n [0.997725791405208, 2.0248339485303806] \n [0.9839452912602459, 2.0230147623492964]\n [1.0380973911442024, 1.848938618763244] \n [0.6767613791844072, 1.8244038226366002]\n [0.7427005544302488, 1.6629613668292775]\n [0.9251238834196535, 1.9299437733047207]\n [0.8472427664488742, 1.7963849503330402]\n [0.9020233150652416, 1.9350616499813538]\n [0.9437704199469645, 1.8939439825516062]\n [0.936113915554645, 1.8262646021845097] \n [0.9187904874396151, 1.9833868585880243]\n ⋮ \n [0.7421418938422703, 1.945245151964625] \n [0.790626575004855, 1.952375831902059] \n [0.8902640618451707, 1.7254231492846523]\n [0.7290201965251333, 2.0742243937253058]\n [0.8447826086930605, 1.9610305747326597]\n [0.8731721080219732, 1.9771005984873642]\n [0.9873620125471696, 1.9013418981615862]\n [0.967385090712671, 1.9453663201161355] \n [0.8575402985566797, 1.8810040502055434]\n [0.9257926679646773, 1.92239609407548] \n [0.7841151861691669, 1.735722767592694] \n [0.9438672358262866, 2.084715348760898] "
},
"metadata": {}
},
{
"output_type": "stream",
"text": " 2.746705 seconds (4.44 M allocations: 618.676 MiB, 11.72% gc time)\n",
"name": "stdout"
}
]
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "# Extract the posterior. Here the transformation was not really necessary.\n\nposterior = transform.(trans, results.chain)",
"execution_count": 17,
"outputs": [
{
"output_type": "execute_result",
"execution_count": 17,
"data": {
"text/plain": "1000-element Array{NamedTuple{(:β,),Tuple{Array{Float64,1}}},1}:\n (β = [0.9888810809602057, 1.9648187473705152],)\n (β = [0.9865863979912161, 2.1524603523534527],)\n (β = [0.997725791405208, 2.0248339485303806],) \n (β = [0.9839452912602459, 2.0230147623492964],)\n (β = [1.0380973911442024, 1.848938618763244],) \n (β = [0.6767613791844072, 1.8244038226366002],)\n (β = [0.7427005544302488, 1.6629613668292775],)\n (β = [0.9251238834196535, 1.9299437733047207],)\n (β = [0.8472427664488742, 1.7963849503330402],)\n (β = [0.9020233150652416, 1.9350616499813538],)\n (β = [0.9437704199469645, 1.8939439825516062],)\n (β = [0.936113915554645, 1.8262646021845097],) \n (β = [0.9187904874396151, 1.9833868585880243],)\n ⋮ \n (β = [0.7421418938422703, 1.945245151964625],) \n (β = [0.790626575004855, 1.952375831902059],) \n (β = [0.8902640618451707, 1.7254231492846523],)\n (β = [0.7290201965251333, 2.0742243937253058],)\n (β = [0.8447826086930605, 1.9610305747326597],)\n (β = [0.8731721080219732, 1.9771005984873642],)\n (β = [0.9873620125471696, 1.9013418981615862],)\n (β = [0.967385090712671, 1.9453663201161355],) \n (β = [0.8575402985566797, 1.8810040502055434],)\n (β = [0.9257926679646773, 1.92239609407548],) \n (β = [0.7841151861691669, 1.735722767592694],) \n (β = [0.9438672358262866, 2.084715348760898],) "
},
"metadata": {}
}
]
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "β_posterior = first.(posterior)",
"execution_count": 18,
"outputs": [
{
"output_type": "execute_result",
"execution_count": 18,
"data": {
"text/plain": "1000-element Array{Array{Float64,1},1}:\n [0.9888810809602057, 1.9648187473705152]\n [0.9865863979912161, 2.1524603523534527]\n [0.997725791405208, 2.0248339485303806] \n [0.9839452912602459, 2.0230147623492964]\n [1.0380973911442024, 1.848938618763244] \n [0.6767613791844072, 1.8244038226366002]\n [0.7427005544302488, 1.6629613668292775]\n [0.9251238834196535, 1.9299437733047207]\n [0.8472427664488742, 1.7963849503330402]\n [0.9020233150652416, 1.9350616499813538]\n [0.9437704199469645, 1.8939439825516062]\n [0.936113915554645, 1.8262646021845097] \n [0.9187904874396151, 1.9833868585880243]\n ⋮ \n [0.7421418938422703, 1.945245151964625] \n [0.790626575004855, 1.952375831902059] \n [0.8902640618451707, 1.7254231492846523]\n [0.7290201965251333, 2.0742243937253058]\n [0.8447826086930605, 1.9610305747326597]\n [0.8731721080219732, 1.9771005984873642]\n [0.9873620125471696, 1.9013418981615862]\n [0.967385090712671, 1.9453663201161355] \n [0.8575402985566797, 1.8810040502055434]\n [0.9257926679646773, 1.92239609407548] \n [0.7841151861691669, 1.735722767592694] \n [0.9438672358262866, 2.084715348760898] "
},
"metadata": {}
}
]
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "# Check that we recover the parameters.\n\nmean(β_posterior)",
"execution_count": 19,
"outputs": [
{
"output_type": "execute_result",
"execution_count": 19,
"data": {
"text/plain": "2-element Array{Float64,1}:\n 0.8926900403974772\n 1.876057009842002 "
},
"metadata": {}
}
]
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "# Quantiles\n\nqs = [0.05, 0.25, 0.5, 0.75, 0.95]\nquantile(first.(β_posterior), qs)",
"execution_count": 20,
"outputs": [
{
"output_type": "execute_result",
"execution_count": 20,
"data": {
"text/plain": "5-element Array{Float64,1}:\n 0.7434034018383008\n 0.8298992590839214\n 0.8915406924683416\n 0.9524208250686986\n 1.0385879439099674"
},
"metadata": {}
}
]
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "quantile(last.(β_posterior), qs)",
"execution_count": 21,
"outputs": [
{
"output_type": "execute_result",
"execution_count": 21,
"data": {
"text/plain": "5-element Array{Float64,1}:\n 1.6736359101409377\n 1.7939892325911293\n 1.8753538742089106\n 1.9531333646283606\n 2.089498530870487 "
},
"metadata": {}
}
]
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "# Check that mixing is good.\n\ness = vec(mapslices(effective_sample_size, reduce(hcat, β_posterior); dims = 2))",
"execution_count": 22,
"outputs": [
{
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"execution_count": 22,
"data": {
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},
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"source": "P1 = plot(first.(β_posterior))\nP2 = histogram(first.(β_posterior))\nplot(P1, P2, size=(800, 250), title=\"beta1\", legend=false)",
"execution_count": 23,
"outputs": [
{
"output_type": "execute_result",
"execution_count": 23,
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"source": "P1 = plot(last.(β_posterior))\nP2 = histogram(last.(β_posterior))\nplot(P1, P2, size=(800, 250), title=\"beta2\", legend=false)",
"execution_count": 24,
"outputs": [
{
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},
{
"metadata": {},
"cell_type": "markdown",
"source": "## Linear regression\n\nhttps://github.com/tpapp/DynamicHMCExamples.jl/blob/master/src/example_linear_regression.jl"
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "Random.seed!(4649);",
"execution_count": 25,
"outputs": []
},
{
"metadata": {
"trusted": true
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"cell_type": "code",
"source": "# # Linear regression\n\n# We estimate simple linear regression model with a half-T prior.\n# First, we load the packages we use.\n\n# using TransformVariables, LogDensityProblems, DynamicHMC, MCMCDiagnostics,\n# Parameters, Statistics, Distributions, ForwardDiff\n\n# Then define a structure to hold the data: observables, covariates, and the degrees of freedom for the prior.\n\n\"\"\"\nLinear regression model ``y ∼ Xβ + ϵ``, where ``ϵ ∼ N(0, σ²)`` IID.\nFlat prior for `β`, half-T for `σ`.\n\"\"\"\nstruct LinearRegressionProblem{TY <: AbstractVector, TX <: AbstractMatrix,\n Tν <: Real}\n \"Observations.\"\n y::TY\n \"Covariates\"\n X::TX\n \"Degrees of freedom for prior.\"\n ν::Tν\nend\n\n# Then make the type callable with the parameters *as a single argument*.\n\nfunction (problem::LinearRegressionProblem)(θ)\n @unpack y, X, ν = problem # extract the data\n @unpack β, σ = θ # works on the named tuple too\n loglikelihood(Normal(0, σ), y .- X*β) + logpdf(TDist(ν), σ)\nend\n\n# We should test this, also, this would be a good place to benchmark and\n# optimize more complicated problems.\n\n#N = 100\nN = 5\n#X = hcat(ones(N), randn(N, 2));\nx = rand(Uniform(-2, 2), N)\nX = hcat(ones(N), x, x.^2);\nβ = [1.0, 2.0, -1.0]\n#σ = 0.5\nσ = 1.0\ny = X*β .+ randn(N) .* σ;\np = LinearRegressionProblem(y, X, 1.0);\np((β = β, σ = σ))",
"execution_count": 26,
"outputs": [
{
"output_type": "execute_result",
"execution_count": 26,
"data": {
"text/plain": "-8.66067388411468"
},
"metadata": {}
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{
"metadata": {
"trusted": true
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"cell_type": "code",
"source": "law_true(β, x) = β[1] + β[2]*x + β[3]*x^2",
"execution_count": 27,
"outputs": [
{
"output_type": "execute_result",
"execution_count": 27,
"data": {
"text/plain": "law_true (generic function with 1 method)"
},
"metadata": {}
}
]
},
{
"metadata": {
"trusted": true
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"cell_type": "code",
"source": "# For this problem, we write a function to return the transformation (as it varies with the number of covariates).\n\nproblem_transformation(p::LinearRegressionProblem) =\n as((β = as(Array, size(p.X, 2)), σ = asℝ₊))\n\n# Wrap the problem with a transformation, then use Flux for the gradient.\n\ntrans = problem_transformation(p)\nP = TransformedLogDensity(trans, p)\n∇P = ADgradient(:ForwardDiff, P);\n\ndisplay(trans)\ndisplay(P)\ndisplay(∇P)",
"execution_count": 28,
"outputs": [
{
"output_type": "display_data",
"data": {
"text/plain": "TransformVariables.TransformTuple{NamedTuple{(:β, :σ),Tuple{TransformVariables.ArrayTransform{TransformVariables.Identity,1},TransformVariables.ShiftedExp{true,Float64}}}}((β = TransformVariables.ArrayTransform{TransformVariables.Identity,1}(TransformVariables.Identity(), (3,)), σ = TransformVariables.ShiftedExp{true,Float64}(0.0)), 4)"
},
"metadata": {}
},
{
"output_type": "display_data",
"data": {
"text/plain": "TransformedLogDensity of dimension 4"
},
"metadata": {}
},
{
"output_type": "display_data",
"data": {
"text/plain": "ForwardDiff AD wrapper for TransformedLogDensity of dimension 4, w/ chunk size 4"
},
"metadata": {}
}
]
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "# Finally, we sample from the posterior. `chain` holds the chain (positions and\n# diagnostic information), while the second returned value is the tuned sampler\n# which would allow continuation of sampling.\n\nL = 2^11\n\n#chain, NUTS_tuned = NUTS_init_tune_mcmc(∇P, 1000);\n@time results = mcmc_with_warmup(Random.GLOBAL_RNG, ∇P, L; reporter = NoProgressReport());\n\ndisplay(results.chain)",
"execution_count": 29,
"outputs": [
{
"output_type": "display_data",
"data": {
"text/plain": "2048-element Array{Array{Float64,1},1}:\n [2.155236637715731, 2.276885382687765, -2.405799991161668, -0.026423651643289886] \n [0.8820321795383079, 3.0212328765154846, -0.5618133793781922, -0.004902014225516546]\n [2.4194049021984343, 2.55065541878323, -1.5632131693136735, -0.5622425312719688] \n [2.3968779790619363, 2.547629848709332, -1.8933837910342768, -0.7875575047524692] \n [1.8108060493982405, 2.6194940163512688, -1.5599553724130153, -0.8356061154124103] \n [1.6352654905648003, 2.2287819978326167, -0.8514132082678891, -0.3172430261277607] \n [1.3805405147788343, 2.5364486664376904, -1.0544689245288086, -0.3515475318997308] \n [1.7542226751642724, 2.9090911507435044, -0.834325244521887, -0.9076302213569245] \n [1.7542226751642724, 2.9090911507435044, -0.834325244521887, -0.9076302213569245] \n [1.7932355670131965, 2.9631232626530988, -0.8476248978752932, -0.7381073563444712] \n [1.7366224656644038, 3.002286653916178, -0.8320695984275652, -0.782626556716755] \n [1.8991116615238328, 2.3409011393973977, -1.4646188489578165, -0.6647316333384548] \n [1.890704956627292, 2.2777254740897224, -1.331159664760753, -0.6026764932487678] \n ⋮ \n [4.1508940004638974, 2.510220063518032, -2.98526624859857, 0.10249095370564984] \n [2.3621115888626028, 2.0709737435932993, -1.0705169598440039, 0.09389708792505873] \n [2.0454211822619706, 2.770322209445818, -1.6151342825970258, -0.26509812466907484] \n [2.9405749822922207, 2.302490221137054, -2.261839345947534, -0.5687591844214646] \n [2.8334031624447076, 2.3227990155088745, -2.4716962283272568, -0.4893024117861301] \n [1.5495180747253443, 2.8273734699686046, -0.8487656758888462, -0.5240829839346783] \n [1.61034569463466, 2.530091965694977, -0.9514025654974505, -0.3091507559980067] \n [1.8277110689516278, 2.6682499677386975, -1.0011795282216827, -0.8334723663918548] \n [1.8102999389810182, 2.537206469370479, -1.2127724929430534, -0.8789143973310541] \n [2.0270562727347268, 2.457747868350051, -1.2062162691780038, -0.9533101574473668] \n [2.1663237400199105, 2.4362818776625925, -1.5824159555623636, -0.9984672066602703] \n [2.1663237400199105, 2.4362818776625925, -1.5824159555623636, -0.9984672066602703] "
},
"metadata": {}
},
{
"output_type": "stream",
"text": " 1.751440 seconds (4.72 M allocations: 277.835 MiB, 6.81% gc time)\n",
"name": "stdout"
}
]
},
{
"metadata": {
"trusted": true
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"cell_type": "code",
"source": "# We use the transformation to obtain the posterior from the chain.\n\n#posterior = transform.(Ref(t), get_position.(chain));\nposterior = transform.(trans, results.chain)",
"execution_count": 30,
"outputs": [
{
"output_type": "execute_result",
"execution_count": 30,
"data": {
"text/plain": "2048-element Array{NamedTuple{(:β, :σ),Tuple{Array{Float64,1},Float64}},1}:\n (β = [2.155236637715731, 2.276885382687765, -2.405799991161668], σ = 0.9739223983717544) \n (β = [0.8820321795383079, 3.0212328765154846, -0.5618133793781922], σ = 0.9951099810378956) \n (β = [2.4194049021984343, 2.55065541878323, -1.5632131693136735], σ = 0.5699295448771747) \n (β = [2.3968779790619363, 2.547629848709332, -1.8933837910342768], σ = 0.4549546639106049) \n (β = [1.8108060493982405, 2.6194940163512688, -1.5599553724130153], σ = 0.43361158310415593)\n (β = [1.6352654905648003, 2.2287819978326167, -0.8514132082678891], σ = 0.7281537732290381) \n (β = [1.3805405147788343, 2.5364486664376904, -1.0544689245288086], σ = 0.7035984057982967) \n (β = [1.7542226751642724, 2.9090911507435044, -0.834325244521887], σ = 0.4034792484944173) \n (β = [1.7542226751642724, 2.9090911507435044, -0.834325244521887], σ = 0.4034792484944173) \n (β = [1.7932355670131965, 2.9631232626530988, -0.8476248978752932], σ = 0.47801777722045496)\n (β = [1.7366224656644038, 3.002286653916178, -0.8320695984275652], σ = 0.4572035617594826) \n (β = [1.8991116615238328, 2.3409011393973977, -1.4646188489578165], σ = 0.5144115600972012) \n (β = [1.890704956627292, 2.2777254740897224, -1.331159664760753], σ = 0.547344709440526) \n ⋮ \n (β = [4.1508940004638974, 2.510220063518032, -2.98526624859857], σ = 1.1079272792281842) \n (β = [2.3621115888626028, 2.0709737435932993, -1.0705169598440039], σ = 1.0984466966714403) \n (β = [2.0454211822619706, 2.770322209445818, -1.6151342825970258], σ = 0.7671306718392836) \n (β = [2.9405749822922207, 2.302490221137054, -2.261839345947534], σ = 0.5662275870021227) \n (β = [2.8334031624447076, 2.3227990155088745, -2.4716962283272568], σ = 0.6130539042321534) \n (β = [1.5495180747253443, 2.8273734699686046, -0.8487656758888462], σ = 0.5920980789404265) \n (β = [1.61034569463466, 2.530091965694977, -0.9514025654974505], σ = 0.7340700962141409) \n (β = [1.8277110689516278, 2.6682499677386975, -1.0011795282216827], σ = 0.43453778918916014)\n (β = [1.8102999389810182, 2.537206469370479, -1.2127724929430534], σ = 0.4152334456237136) \n (β = [2.0270562727347268, 2.457747868350051, -1.2062162691780038], σ = 0.3854629662278223) \n (β = [2.1663237400199105, 2.4362818776625925, -1.5824159555623636], σ = 0.368443756707755) \n (β = [2.1663237400199105, 2.4362818776625925, -1.5824159555623636], σ = 0.368443756707755) "
},
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},
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"metadata": {
"scrolled": false,
"trusted": true
},
"cell_type": "code",
"source": "# Extract the parameter posterior means: `β`,\n\nposterior_β = first.(posterior)\n@show mean_posterior_β = mean(posterior_β)\n\n# then `σ`:\n\nposterior_σ = last.(posterior)\n@show mean_posterior_σ = mean(posterior_σ)\n\ndisplay(posterior_β)\ndisplay(posterior_σ)",
"execution_count": 31,
"outputs": [
{
"output_type": "display_data",
"data": {
"text/plain": "2048-element Array{Array{Float64,1},1}:\n [2.155236637715731, 2.276885382687765, -2.405799991161668] \n [0.8820321795383079, 3.0212328765154846, -0.5618133793781922]\n [2.4194049021984343, 2.55065541878323, -1.5632131693136735] \n [2.3968779790619363, 2.547629848709332, -1.8933837910342768] \n [1.8108060493982405, 2.6194940163512688, -1.5599553724130153]\n [1.6352654905648003, 2.2287819978326167, -0.8514132082678891]\n [1.3805405147788343, 2.5364486664376904, -1.0544689245288086]\n [1.7542226751642724, 2.9090911507435044, -0.834325244521887] \n [1.7542226751642724, 2.9090911507435044, -0.834325244521887] \n [1.7932355670131965, 2.9631232626530988, -0.8476248978752932]\n [1.7366224656644038, 3.002286653916178, -0.8320695984275652] \n [1.8991116615238328, 2.3409011393973977, -1.4646188489578165]\n [1.890704956627292, 2.2777254740897224, -1.331159664760753] \n ⋮ \n [4.1508940004638974, 2.510220063518032, -2.98526624859857] \n [2.3621115888626028, 2.0709737435932993, -1.0705169598440039]\n [2.0454211822619706, 2.770322209445818, -1.6151342825970258] \n [2.9405749822922207, 2.302490221137054, -2.261839345947534] \n [2.8334031624447076, 2.3227990155088745, -2.4716962283272568]\n [1.5495180747253443, 2.8273734699686046, -0.8487656758888462]\n [1.61034569463466, 2.530091965694977, -0.9514025654974505] \n [1.8277110689516278, 2.6682499677386975, -1.0011795282216827]\n [1.8102999389810182, 2.537206469370479, -1.2127724929430534] \n [2.0270562727347268, 2.457747868350051, -1.2062162691780038] \n [2.1663237400199105, 2.4362818776625925, -1.5824159555623636]\n [2.1663237400199105, 2.4362818776625925, -1.5824159555623636]"
},
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},
{
"output_type": "display_data",
"data": {
"text/plain": "2048-element Array{Float64,1}:\n 0.9739223983717544 \n 0.9951099810378956 \n 0.5699295448771747 \n 0.4549546639106049 \n 0.43361158310415593\n 0.7281537732290381 \n 0.7035984057982967 \n 0.4034792484944173 \n 0.4034792484944173 \n 0.47801777722045496\n 0.4572035617594826 \n 0.5144115600972012 \n 0.547344709440526 \n ⋮ \n 1.1079272792281842 \n 1.0984466966714403 \n 0.7671306718392836 \n 0.5662275870021227 \n 0.6130539042321534 \n 0.5920980789404265 \n 0.7340700962141409 \n 0.43453778918916014\n 0.4152334456237136 \n 0.3854629662278223 \n 0.368443756707755 \n 0.368443756707755 "
},
"metadata": {}
},
{
"output_type": "stream",
"text": "mean_posterior_β = mean(posterior_β) = [2.013516797685997, 2.654850591438265, -1.4369117371295437]\nmean_posterior_σ = mean(posterior_σ) = 0.9789830905249814\n",
"name": "stdout"
}
]
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "# Effective sample sizes (of untransformed draws)\n\ness = mapslices(effective_sample_size, hcat(results.chain...); dims = 2)",
"execution_count": 32,
"outputs": [
{
"output_type": "execute_result",
"execution_count": 32,
"data": {
"text/plain": "4×1 Array{Float64,2}:\n 791.3528955331897 \n 871.4063925694975 \n 950.9268389580166 \n 273.55168445607535"
},
"metadata": {}
}
]
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "# NUTS-specific statistics\n\n#NUTS_statistics(chain)\nsummarize_tree_statistics(results.tree_statistics)",
"execution_count": 33,
"outputs": [
{
"output_type": "execute_result",
"execution_count": 33,
"data": {
"text/plain": "Hamiltonian Monte Carlo sample of length 2048\n acceptance rate mean: 0.81, 5/25/50/75/95%: 0.12 0.75 0.93 0.98 1.0\n termination: divergence => 0%, max_depth => 0%, turning => 100%\n depth: 0 => 0%, 1 => 2%, 2 => 14%, 3 => 47%, 4 => 34%, 5 => 2%, 6 => 0%"
},
"metadata": {}
}
]
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "scatter(X[:,2], y, label=\"sample\", legend=:bottomright)\nplot!(x -> law_true(β, x), label=\"true law\", ls=:dash)",
"execution_count": 34,
"outputs": [
{
"output_type": "execute_result",
"execution_count": 34,
"data": {
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}
]
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "P1 = plot((x->x[1]).(posterior_β))\nP2 = histogram((x->x[1]).(posterior_β))\nplot(P1, P2, size=(800, 250), title=\"beta1\", legend=false)",
"execution_count": 35,
"outputs": [
{
"output_type": "execute_result",
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549.254,576.468 549.893,540.424 550.532,521.155 551.17,575.572 \n 551.809,575.572 552.448,503.972 553.087,411.269 553.726,394.145 554.365,368.97 555.003,501.989 555.642,603.978 556.281,499.451 556.92,424.363 557.559,453.949 \n 558.197,639.448 558.836,654.986 559.475,616.272 560.114,620.354 560.753,609.885 561.392,438.444 562.03,557.086 562.669,555.338 563.308,523.301 563.947,623.585 \n 564.586,536.289 565.224,507.478 565.863,488.97 566.502,602.848 567.141,615.25 567.78,739.609 568.418,672.143 569.057,503.866 569.696,523.621 570.335,528.851 \n 570.974,542.731 571.613,533.137 572.251,539.853 572.89,569.539 573.529,493.225 574.168,510.29 574.807,548.645 575.445,519.064 576.084,543.455 576.723,608.788 \n 577.362,652.95 578.001,536.366 578.64,521.102 579.278,566.899 579.917,613.017 580.556,590.019 581.195,509.947 581.834,512.566 582.472,506.83 583.111,479.271 \n 583.75,525.255 584.389,567.1 585.028,527.565 585.666,486.475 586.305,490.993 586.944,486.007 587.583,505.797 588.222,519.796 588.861,565.747 589.499,573.838 \n 590.138,575.564 590.777,572.863 591.416,535.631 592.055,533.055 592.693,625.154 593.332,627.131 593.971,501.305 594.61,616.05 595.249,520.991 595.887,530.154 \n 596.526,561.604 597.165,534.472 597.804,543.816 598.443,545.053 599.082,553.871 599.72,549.248 600.359,587.604 600.998,612.593 601.637,605.076 602.276,570.82 \n 602.914,492.886 603.553,523.388 604.192,480.35 604.831,454.156 605.47,424.775 606.109,483.838 606.747,415.56 607.386,410.223 608.025,473.649 608.664,494.442 \n 609.303,523.053 609.941,494.358 610.58,521.349 611.219,516.576 611.858,519.041 612.497,567.868 613.135,591.114 613.774,523.115 614.413,519.656 615.052,543.39 \n 615.691,559.995 616.33,568.208 616.968,528.911 617.607,518.906 618.246,580.097 618.885,647.565 619.524,581.57 620.162,488.146 620.801,491.018 621.44,437.717 \n 622.079,486.925 622.718,463.191 623.357,447.728 623.995,586.422 624.634,618.277 625.273,554.557 625.912,592.583 626.551,578.03 627.189,624.982 627.828,474.786 \n 628.467,498.194 629.106,528.622 629.745,508.182 630.383,506.133 631.022,487.04 631.661,497.603 632.3,486.855 632.939,519.102 633.578,540.877 634.216,520.376 \n 634.855,528.589 635.494,530.332 636.133,509.609 636.772,522.957 637.41,539.323 638.049,539.323 638.688,553.563 639.327,558.978 639.966,493.194 640.604,487.764 \n 641.243,511.766 641.882,529.677 642.521,527.407 643.16,548.899 643.799,556.037 644.437,564.932 645.076,564.932 645.715,529.857 646.354,568.991 646.993,566.732 \n 647.631,514.343 648.27,514.343 648.909,636.882 649.548,494.643 650.187,500.953 650.826,437.393 651.464,496.42 652.103,536.32 652.742,508.741 653.381,572.414 \n 654.02,537.916 654.658,550.693 655.297,517.482 655.936,528.806 656.575,521.14 657.214,604.791 657.852,621.95 658.491,617.224 659.13,635.468 659.769,560.863 \n 660.408,483.929 661.047,482.925 661.685,502.523 662.324,582.786 662.963,579.506 663.602,564.844 664.241,605.709 664.879,584.818 665.518,628.97 666.157,560.713 \n 666.796,496.604 667.435,525.193 668.074,538.149 668.712,448.647 669.351,412.857 669.99,659.113 670.629,439.03 671.268,403.867 671.906,574.754 672.545,562.352 \n 673.184,544.382 673.823,564.846 674.462,558.345 675.1,626.38 675.739,630.513 676.378,513.405 677.017,588.38 677.656,539.112 678.295,566.861 678.933,558.043 \n 679.572,558.043 680.211,548.325 680.85,550.049 681.489,566.196 682.127,506.925 682.766,583.003 683.405,562.162 684.044,562.162 684.683,521.998 685.322,469.852 \n 685.96,476.011 686.599,422.845 687.238,414.143 687.877,578.534 688.516,642.572 689.154,608.379 689.793,553.122 690.432,491.51 691.071,497.237 691.71,510.361 \n 692.348,528.793 692.987,516.511 693.626,514.977 694.265,525.121 694.904,573.858 695.543,549.558 696.181,581.357 696.82,588.767 697.459,571.978 698.098,541.608 \n 698.737,541.608 699.375,564.201 700.014,579.376 700.653,530.126 701.292,565.72 701.931,549.778 702.569,540.547 703.208,511.03 703.847,525.547 704.486,589.375 \n 705.125,604.287 705.764,483.631 706.402,577.098 707.041,523.828 707.68,556.06 708.319,557.541 708.958,538.135 709.596,539.767 710.235,547.701 710.874,581.961 \n 711.513,537.178 712.152,570.494 712.791,557.586 713.429,523.379 714.068,502.202 714.707,527.478 715.346,498.758 715.985,528.172 716.623,543.864 717.262,564.911 \n 717.901,554.184 718.54,531.153 719.179,523.447 719.817,517.528 720.456,571.575 721.095,542.328 721.734,561.511 722.373,564.267 723.012,550.368 723.65,524.202 \n 724.289,498.611 724.928,613.102 725.567,602.709 726.206,696.124 726.844,696.266 727.483,681.938 728.122,373.777 728.761,445.201 729.4,432.685 730.039,484.748 \n 730.677,626.174 731.316,598.07 731.955,515.147 732.594,536.633 733.233,522.465 733.871,539.373 734.51,535.204 735.149,570.529 735.788,541.589 736.427,525.658 \n 737.065,529.351 737.704,559.967 738.343,560.118 738.982,560.357 739.621,385.245 740.26,258.254 740.898,285.026 741.537,660.034 742.176,556.526 742.815,561.168 \n 743.454,594.738 744.092,610.84 744.731,557.637 745.37,517.223 746.009,587.099 746.648,576.585 747.287,544.044 747.925,526.635 748.564,556.38 749.203,546.676 \n 749.842,596.606 750.481,572.364 751.119,652.959 751.758,554.642 752.397,533.877 753.036,550.46 753.675,548.992 754.313,539.304 754.952,756.041 755.591,625.839 \n 756.23,622.356 756.869,614.275 757.508,736.159 758.146,698.635 758.785,663.796 759.424,673.806 760.063,759.859 760.702,710.906 761.34,663.163 761.979,628.724 \n 762.618,519.046 763.257,573.816 763.896,448.487 764.534,583.719 765.173,552.844 765.812,619.198 766.451,674.193 767.09,633.655 767.729,427.25 768.367,416.624 \n 769.006,809.018 769.645,673.268 770.284,674.345 770.923,561.915 771.561,588.742 772.2,579.293 772.839,737.097 773.478,518.674 774.117,500.775 774.756,604.498 \n 775.394,605.99 776.033,434.351 776.672,569.216 777.311,585.408 777.95,541.182 778.588,600.554 779.227,592.326 779.866,621.519 780.505,602.022 781.144,568.924 \n 781.782,572.79 782.421,552.589 783.06,543.501 783.699,530.298 784.338,513.474 784.977,549.603 785.615,563.647 786.254,477.844 786.893,457.9 787.532,506.439 \n 788.171,539.725 788.809,528.894 789.448,531.873 790.087,531.873 790.726,578.537 791.365,573.695 792.004,568.735 792.642,573.389 793.281,608.106 793.92,506.214 \n 794.559,528.087 795.198,496.573 795.836,530.529 796.475,507.572 797.114,464.63 797.753,363.436 798.392,536.455 799.03,448.309 799.669,601.079 800.308,575.89 \n 800.947,533.175 801.586,568.166 802.225,545.345 802.863,571.106 803.502,551.289 804.141,596.04 804.78,544.205 805.419,570.624 806.057,537.794 806.696,541.909 \n 807.335,587.382 807.974,531.271 808.613,526.475 809.252,516.366 809.89,510.057 810.529,526.059 811.168,599.051 811.807,602.434 812.446,571.847 813.084,581.454 \n 813.723,568.813 814.362,349.462 815.001,419.549 815.64,492.099 816.278,524.734 816.917,523.523 817.556,533.023 818.195,570.436 818.834,567.76 819.473,564.857 \n 820.111,592.322 820.75,544.999 821.389,603.946 822.028,656.354 822.667,584.172 823.305,683.11 823.944,654.013 824.583,572.978 825.222,563.494 825.861,553.477 \n 826.499,515.978 827.138,530.124 827.777,538.043 828.416,550.869 829.055,550.869 829.694,530.094 830.332,554.689 830.971,600.606 831.61,461.913 832.249,470.849 \n 832.888,576.451 833.526,576.401 834.165,600.825 834.804,486.338 835.443,530.635 836.082,554.264 836.721,551.571 837.359,511.244 837.998,527.247 838.637,563.226 \n 839.276,591.622 839.915,591.234 840.553,613.964 841.192,583.718 841.831,624.557 842.47,547.206 843.109,495.731 843.747,483.157 844.386,545.966 845.025,558.356 \n 845.664,593.883 846.303,544.459 846.942,587.017 847.58,612.262 848.219,587.445 848.858,591.716 849.497,566.736 850.136,627.835 850.774,596.142 851.413,494.991 \n 852.052,612.992 852.691,393.991 853.33,418.015 853.969,502.042 854.607,547.434 855.246,630.767 855.885,577.135 856.524,547.899 857.163,521.518 857.801,564.831 \n 858.44,509.444 859.079,515.855 859.718,453.75 860.357,480.69 860.995,529.646 861.634,619.588 862.273,583.381 862.912,559.812 863.551,520.574 864.19,534.34 \n 864.828,566.217 865.467,558.804 866.106,528.064 866.745,515.63 867.384,540.593 868.022,568.086 868.661,536.047 869.3,536.047 869.939,608.012 870.578,652.099 \n 871.217,534.052 871.855,516.504 872.494,539.903 873.133,554.129 873.772,502.397 874.411,545.952 875.049,616.929 875.688,569.169 876.327,598.497 876.966,557.108 \n 877.605,575.293 878.243,566.011 878.882,559.086 879.521,530.749 880.16,562.454 880.799,541 881.438,545.619 882.076,552.461 882.715,585.589 883.354,732.877 \n 883.993,738.485 884.632,691.032 885.27,603.404 885.909,595.849 886.548,563.477 887.187,504.414 887.826,701.324 888.464,737.486 889.103,873.342 889.742,780.284 \n 890.381,698.4 891.02,422.699 891.659,443.781 892.297,482.858 892.936,514.661 893.575,577.326 894.214,508.494 894.853,512.525 895.491,561.777 896.13,512.355 \n 896.769,510.985 897.408,527.143 898.047,550.321 898.686,575.595 899.324,554.867 899.963,542.949 900.602,579.643 901.241,556.705 901.88,549.532 902.518,533.511 \n 903.157,563.811 903.796,546.448 904.435,532.511 905.074,537.281 905.712,529.143 906.351,573.261 906.99,568.916 907.629,579.927 908.268,579.927 908.907,573.654 \n 909.545,566.666 910.184,366.012 910.823,418.547 911.462,464.057 912.101,501.63 912.739,560.33 913.378,558.039 914.017,547.365 914.656,547.365 915.295,547.365 \n 915.934,523.332 916.572,509.473 917.211,526.405 917.85,562.664 918.489,569.255 919.128,569.255 919.766,569.255 920.405,569.255 921.044,575.288 921.683,555.867 \n 922.322,576.089 922.96,624.671 923.599,534.069 924.238,590.612 924.877,575.54 925.516,599.615 926.155,631.298 926.793,495.744 927.432,481.851 928.071,520.659 \n 928.71,523.349 929.349,535.836 929.987,535.472 930.626,535.472 931.265,548.901 931.904,543.426 932.543,543.426 933.182,562.284 933.82,588.226 934.459,608.397 \n 935.098,569.631 935.737,510.356 936.376,632.117 937.014,581.583 937.653,574.247 938.292,556.076 938.931,602.141 939.57,539.366 940.208,563.293 940.847,487.024 \n 941.486,747.968 942.125,443.24 942.764,458.484 943.403,517.817 944.041,505.224 944.68,457.913 945.319,523.654 945.958,512.307 946.597,463.966 947.235,538.552 \n 947.874,539.763 948.513,653.762 949.152,498.624 949.791,589.999 950.429,530.424 951.068,521.29 951.707,606.757 952.346,613.858 952.985,624.331 953.624,568.449 \n 954.262,552.49 954.901,571.984 955.54,554.855 956.179,554.855 956.818,505.793 957.456,539.386 958.095,555.17 958.734,555.17 959.373,548.106 960.012,548.106 \n 960.651,533.582 961.289,529.443 961.928,619.323 962.567,545.213 963.206,657.872 963.845,536.318 964.483,550.111 965.122,547.412 965.761,562.95 966.4,573.545 \n 967.039,546.468 967.677,546.468 968.316,548.464 968.955,561.761 969.594,544.992 970.233,536.751 970.872,546.427 971.51,598.349 972.149,563.493 972.788,535.1 \n 973.427,578.841 974.066,570.385 974.704,539.172 975.343,570.988 975.982,531.871 976.621,512.012 977.26,536.226 977.899,520.043 978.537,555.93 979.176,545.217 \n 979.815,572.116 980.454,566.15 981.093,598.474 981.731,617.064 982.37,693.004 983.009,555.983 983.648,712.303 984.287,608.359 984.925,522.025 985.564,520.628 \n 986.203,489.536 986.842,611.026 987.481,474.888 988.12,480.938 988.758,522.251 989.397,531.944 990.036,522.854 990.675,504.57 991.314,521.661 991.952,536.8 \n 992.591,572.565 993.23,576.327 993.869,576.327 994.508,542.599 995.147,522.641 995.785,500.116 996.424,566.267 997.063,535.442 997.702,557.363 998.341,548.652 \n 998.979,548.652 999.618,548.652 1000.26,548.652 1000.9,544.29 1001.53,523.472 1002.17,569.857 1002.81,561.174 1003.45,618.016 1004.09,581.066 1004.73,527.905 \n 1005.37,562.406 1006.01,537.195 1006.65,545.848 1007.28,480.329 1007.92,632.032 1008.56,606.565 1009.2,454.86 1009.84,576.074 1010.48,546.239 1011.12,570.403 \n 1011.76,555.467 1012.39,535.992 1013.03,587.362 1013.67,531.191 1014.31,547.545 1014.95,566.538 1015.59,563.243 1016.23,542.581 1016.87,533.559 1017.51,595.529 \n 1018.14,595.529 1018.78,579.045 1019.42,579.045 1020.06,583.773 1020.7,583.773 1021.34,582.537 1021.98,555.157 1022.62,534.263 1023.25,537.418 1023.89,503.317 \n 1024.53,574.182 1025.17,556.635 1025.81,537.668 1026.45,524.824 1027.09,549.673 1027.73,549.673 1028.36,554.218 1029,554.959 1029.64,529.393 1030.28,695.183 \n 1030.92,650.755 1031.56,614.67 1032.2,617.605 1032.84,650.699 1033.48,601.134 1034.11,585.087 1034.75,548.952 1035.39,513.403 1036.03,527.132 1036.67,523.622 \n 1037.31,507.947 1037.95,520.729 1038.59,556.819 1039.22,571.776 1039.86,517.029 1040.5,482.77 1041.14,528.038 1041.78,521.499 1042.42,522.82 1043.06,618.62 \n 1043.7,591.887 1044.34,556.731 1044.97,451.465 1045.61,637.95 1046.25,603.615 1046.89,484.685 1047.53,540.59 1048.17,600.38 1048.81,480.698 1049.45,485.406 \n 1050.08,487.34 1050.72,503.429 1051.36,554.957 1052,552.327 1052.64,519.39 1053.28,511.029 1053.92,518.639 1054.56,583.869 1055.2,515.819 1055.83,537.302 \n 1056.47,585.168 1057.11,581.861 1057.75,570.02 1058.39,570.02 1059.03,570.212 1059.67,578.869 1060.31,447.853 1060.94,432.62 1061.58,522.301 1062.22,563.814 \n 1062.86,534.945 1063.5,552.475 1064.14,594.625 1064.78,578.123 1065.42,540.123 1066.05,508.443 1066.69,539.127 1067.33,505.17 1067.97,572.015 1068.61,521.057 \n 1069.25,560.234 1069.89,580.021 1070.53,560.499 1071.17,587.13 1071.8,554.217 1072.44,562.936 1073.08,485.4 1073.72,594.013 1074.36,546.82 1075,531.918 \n 1075.64,520.621 1076.28,566.388 1076.91,587.775 1077.55,548.742 1078.19,564.63 1078.83,584.104 1079.47,550.554 1080.11,598.869 1080.75,552.009 1081.39,523.367 \n 1082.03,568.528 1082.66,558.405 1083.3,613.529 1083.94,548.58 1084.58,553.583 1085.22,580.858 1085.86,546.021 1086.5,529.786 1087.14,574.772 1087.77,577.439 \n 1088.41,459.052 1089.05,564.434 1089.69,480.143 1090.33,475.351 1090.97,496.204 1091.61,500.728 1092.25,614.724 1092.89,502.433 1093.52,574.095 1094.16,576.634 \n 1094.8,587.399 1095.44,518.252 1096.08,502.259 1096.72,521.882 1097.36,537.221 1098,537.221 1098.63,522.992 1099.27,591.292 1099.91,569.052 1100.55,540.111 \n 1101.19,617.421 1101.83,606.916 1102.47,643.751 1103.11,522.65 1103.74,505.606 1104.38,577.915 1105.02,597.756 1105.66,585.335 1106.3,539.33 1106.94,574.548 \n 1107.58,569.579 1108.22,646.517 1108.86,568.703 1109.49,587.082 1110.13,549.775 1110.77,615.887 1111.41,484.252 1112.05,489.218 1112.69,530.491 1113.33,630.807 \n 1113.97,608.399 1114.6,617.57 1115.24,481.759 1115.88,468.927 1116.52,486.164 1117.16,516.061 1117.8,547.327 1118.44,547.327 1119.08,557.02 1119.72,518.356 \n 1120.35,483.904 1120.99,530.761 1121.63,538.829 1122.27,541.737 1122.91,499.283 1123.55,520.499 1124.19,573.143 1124.83,437.681 1125.46,529.721 1126.1,506.503 \n 1126.74,554.36 1127.38,573.635 1128.02,595.363 1128.66,600.099 1129.3,471.017 1129.94,455.985 1130.58,491.261 1131.21,493.193 1131.85,495.59 1132.49,581.801 \n 1133.13,525.862 1133.77,502.161 1134.41,564.852 1135.05,576.129 1135.69,575.066 1136.32,606.923 1136.96,606.694 1137.6,561.108 1138.24,560.161 1138.88,570.399 \n 1139.52,589.015 1140.16,637.893 1140.8,590.476 1141.44,603.365 1142.07,608.069 1142.71,607.65 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]
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "P1 = plot((x->x[2]).(posterior_β))\nP2 = histogram((x->x[2]).(posterior_β))\nplot(P1, P2, size=(800, 250), title=\"beta2\", legend=false)",
"execution_count": 36,
"outputs": [
{
"output_type": "execute_result",
"execution_count": 36,
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495.436,527.592 496.083,527.592 496.73,527.592 497.378,600.708 \n 498.025,582.272 498.672,497.033 499.319,563.025 499.966,533.689 500.613,541.996 501.261,546.087 501.908,546.456 502.555,553.967 503.202,533.581 503.849,533.236 \n 504.496,533.236 505.144,543.429 505.791,572.705 506.438,576.491 507.085,521.69 507.732,522.872 508.379,560.693 509.027,577.932 509.674,519.722 510.321,497.667 \n 510.968,515.167 511.615,511.231 512.262,523.41 512.91,360.441 513.557,537.058 514.204,580.373 514.851,279.555 515.498,448.886 516.145,443.152 516.793,508.599 \n 517.44,499.231 518.087,498.247 518.734,502.15 519.381,531.691 520.028,488.861 520.676,395.74 521.323,448.391 521.97,463.109 522.617,522.707 523.264,571.849 \n 523.911,576.613 524.559,483.65 525.206,486.64 525.853,545.506 526.5,574.591 527.147,512.519 527.794,598.141 528.442,465.713 529.089,512.774 529.736,407.974 \n 530.383,472.455 531.03,512.189 531.677,528.57 532.325,531.741 532.972,458.443 533.619,491.496 534.266,585.545 534.913,559.476 535.56,568.772 536.208,569.018 \n 536.855,557.253 537.502,520.23 538.149,475.474 538.796,521.124 539.443,565.109 540.091,545.274 540.738,643.219 541.385,581.109 542.032,465.762 542.679,561.079 \n 543.326,570.128 543.974,520.268 544.621,535.396 545.268,507.685 545.915,578.464 546.562,529.064 547.209,524.443 547.857,513.487 548.504,583.757 549.151,579.67 \n 549.798,505.264 550.445,485.092 551.093,567.133 551.74,477.574 552.387,485.456 553.034,487.222 553.681,508.501 554.328,548.122 554.976,556.1 555.623,491.014 \n 556.27,560.931 556.917,522.652 557.564,485.801 558.211,492.435 558.859,603.855 559.506,638.286 560.153,704.366 560.8,741.474 561.447,720.912 562.094,596.139 \n 562.742,530.797 563.389,544.265 564.036,504.705 564.683,450.23 565.33,492.785 565.977,611.249 566.625,589.517 567.272,502.01 567.919,497.171 568.566,595.325 \n 569.213,578.878 569.86,581.639 570.508,478.117 571.155,430.251 571.802,667.971 572.449,322.05 573.096,660.453 573.743,672.868 574.391,465.312 575.038,471.685 \n 575.685,501.59 576.332,546.214 576.979,572.335 577.626,568.51 578.274,543.426 578.921,610.364 579.568,574.805 580.215,557.683 580.862,482.86 581.509,560.928 \n 582.157,535.907 582.804,603.952 583.451,539.122 584.098,589.039 584.745,658.891 585.392,629.507 586.04,650.542 586.687,510.938 587.334,598.772 587.981,607.296 \n 588.628,575.289 589.275,549.061 589.923,538.469 590.57,522.021 591.217,552.533 591.864,552.533 592.511,513.826 593.158,572.036 593.806,510.167 594.453,530.248 \n 595.1,558.783 595.747,627.973 596.394,657.487 597.041,545.501 597.689,581.843 598.336,602.689 598.983,602.689 599.63,549.16 600.277,518.804 600.924,521.514 \n 601.572,583.104 602.219,583.104 602.866,533.74 603.513,625.384 604.16,625.553 604.807,456.065 605.455,574.822 606.102,554.604 606.749,541.572 607.396,499.994 \n 608.043,536.459 608.69,580.387 609.338,530.185 609.985,535.829 610.632,521.879 611.279,518.657 611.926,463.469 612.574,520.422 613.221,605.739 613.868,457.763 \n 614.515,547.102 615.162,575.169 615.809,549.188 616.457,526.941 617.104,488.667 617.751,456.781 618.398,466.522 619.045,479.468 619.692,520.306 620.34,439.319 \n 620.987,525.866 621.634,579.083 622.281,581.587 622.928,711.36 623.575,713.149 624.223,109.911 624.87,484.3 625.517,444.297 626.164,507.117 626.811,544.595 \n 627.458,552.197 628.106,559.135 628.753,574.164 629.4,522.729 630.047,531.265 630.694,508.497 631.341,561.8 631.989,491.854 632.636,526.54 633.283,532.906 \n 633.93,532.906 634.577,544.104 635.224,543.706 635.872,553.3 636.519,528.757 637.166,549.445 637.813,572.253 638.46,572.253 639.107,538.815 639.755,526.074 \n 640.402,468.183 641.049,662.376 641.696,646.245 642.343,268.581 642.99,745.931 643.638,750.392 644.285,785.777 644.932,499.421 645.579,538.478 646.226,508.495 \n 646.873,611.409 647.521,573.924 648.168,497.697 648.815,493.174 649.462,567.446 650.109,549.79 650.756,517.506 651.404,529.928 652.051,579.691 652.698,592.697 \n 653.345,592.697 653.992,554.196 654.639,545.617 655.287,502.316 655.934,550.019 656.581,510.094 657.228,481.384 657.875,541.495 658.522,534.991 659.17,542.058 \n 659.817,523.251 660.464,600.136 661.111,517.065 661.758,460.594 662.405,602.706 663.053,568.041 663.7,520.142 664.347,572.574 664.994,574.508 665.641,567.173 \n 666.288,567.084 666.936,535.431 667.583,509.17 668.23,513.791 668.877,518.561 669.524,543.558 670.172,560.236 670.819,521.028 671.466,481.833 672.113,568.282 \n 672.76,577.054 673.407,571.792 674.055,489.46 674.702,603.936 675.349,524.153 675.996,420.716 676.643,454.545 677.29,445.317 677.938,431.895 678.585,612.573 \n 679.232,611.238 679.879,523.698 680.526,561.301 681.173,476.137 681.821,474.76 682.468,476.712 683.115,638.445 683.762,623.214 684.409,478.239 685.056,767.029 \n 685.704,675.543 686.351,665.409 686.998,709.009 687.645,646.72 688.292,631.604 688.939,505.117 689.587,524.507 690.234,546.606 690.881,505.147 691.528,527.226 \n 692.175,563.604 692.822,539.673 693.47,528.984 694.117,520.479 694.764,691.929 695.411,599.415 696.058,606.924 696.705,685.609 697.353,401.159 698,365.39 \n 698.647,341.582 699.294,369.804 699.941,462.036 700.588,494.527 701.236,512.966 701.883,540.121 702.53,563.589 703.177,573.141 703.824,554.967 704.471,548.213 \n 705.119,591.42 705.766,544.308 706.413,538.048 707.06,529.039 707.707,573.314 708.354,561.073 709.002,512.711 709.649,569.426 710.296,293.19 710.943,285.993 \n 711.59,345.437 712.237,351.571 712.885,467.26 713.532,474.349 714.179,499.113 714.826,487.083 715.473,584.518 716.12,610.268 716.768,633.333 717.415,605.294 \n 718.062,588.646 718.709,610.753 719.356,663.668 720.003,555.411 720.651,490.552 721.298,543.721 721.945,379.804 722.592,352.916 723.239,675.43 723.886,370.898 \n 724.534,564.538 725.181,509.716 725.828,410.498 726.475,382.012 727.122,425.216 727.769,408.802 728.417,549.525 729.064,553.432 729.711,513.384 730.358,619.316 \n 731.005,622.383 731.653,360.375 732.3,372.712 732.947,379.701 733.594,411.818 734.241,435.995 734.888,489.871 735.536,499.714 736.183,528.18 736.83,551.542 \n 737.477,518.175 738.124,551.666 738.771,530.863 739.419,615.672 740.066,490.281 740.713,519.028 741.36,550.873 742.007,572.864 742.654,574.62 743.302,596.685 \n 743.949,523.828 744.596,537.784 745.243,556.431 745.89,556.431 746.537,531.353 747.185,582.589 747.832,572.646 748.479,539.777 749.126,532.479 749.773,536.399 \n 750.42,527.791 751.068,613.465 751.715,603.518 752.362,626.393 753.009,754.785 753.656,726.365 754.303,586.853 754.951,557.192 755.598,525.23 756.245,499.951 \n 756.892,553.827 757.539,572.382 758.186,553.226 758.834,450.188 759.481,435.731 760.128,468.703 760.775,558.46 761.422,548.095 762.069,513.154 762.717,503.778 \n 763.364,521.233 764.011,521.911 764.658,525.082 765.305,474.699 765.952,493.764 766.6,520.836 767.247,533.464 767.894,550.762 768.541,545.545 769.188,569.51 \n 769.835,608.035 770.483,589.862 771.13,615.592 771.777,591.398 772.424,589.285 773.071,550.919 773.718,578.859 774.366,518.683 775.013,525.834 775.66,535.82 \n 776.307,555.017 776.954,414.669 777.601,434.054 778.249,546.746 778.896,591.586 779.543,470.976 780.19,460.95 780.837,532.947 781.484,533.412 782.132,544.168 \n 782.779,576.925 783.426,542.258 784.073,561.488 784.72,527.986 785.367,527.986 786.015,470.552 786.662,477.382 787.309,570.143 787.956,489.213 788.603,472.237 \n 789.251,418.206 789.898,398.363 790.545,585.545 791.192,510.349 791.839,503.808 792.486,536.511 793.134,542.951 793.781,553.781 794.428,547.284 795.075,548.671 \n 795.722,543.023 796.369,511.977 797.017,530.416 797.664,532.819 798.311,525.17 798.958,593.26 799.605,536.172 800.252,531.4 800.9,597.393 801.547,568.242 \n 802.194,527.878 802.841,546.577 803.488,543.509 804.135,510.408 804.783,559.679 805.43,486.505 806.077,487.381 806.724,440.76 807.371,458.804 808.018,648.731 \n 808.666,313.268 809.313,586.36 809.96,667.56 810.607,586.453 811.254,527.972 811.901,476.837 812.549,385.653 813.196,438.645 813.843,439.236 814.49,464.002 \n 815.137,576.116 815.784,487.218 816.432,597.66 817.079,552.887 817.726,489.871 818.373,567.277 819.02,575.983 819.667,565.719 820.315,529.226 820.962,562.419 \n 821.609,493.371 822.256,525.61 822.903,526.134 823.55,515.935 824.198,539.44 824.845,529.221 825.492,567.837 826.139,567.837 826.786,512.372 827.433,522.274 \n 828.081,445.813 828.728,425.353 829.375,447.688 830.022,486.705 830.669,493.33 831.316,501.189 831.964,507.645 832.611,505.552 833.258,530.389 833.905,497.807 \n 834.552,570.496 835.199,639.397 835.847,546.536 836.494,559.316 837.141,496.264 837.788,588.211 838.435,630.167 839.082,593.074 839.73,674.367 840.377,500.191 \n 841.024,526.849 841.671,559.798 842.318,381.958 842.965,502.622 843.613,512.987 844.26,586.153 844.907,610.41 845.554,568.492 846.201,475.435 846.848,436.408 \n 847.496,209.626 848.143,564.939 848.79,675.593 849.437,679.639 850.084,552.026 850.732,545.386 851.379,482.144 852.026,494.204 852.673,460.801 853.32,427.533 \n 853.967,550.483 854.615,485.066 855.262,614.065 855.909,485.214 856.556,534.442 857.203,541.429 857.85,603.481 858.498,554.474 859.145,532.346 859.792,553.273 \n 860.439,544.306 861.086,513.236 861.733,525.441 862.381,522.861 863.028,544.129 863.675,518.539 864.322,524.49 864.969,535.12 865.616,535.12 866.264,537.734 \n 866.911,499.397 867.558,887.948 868.205,344.639 868.852,419.621 869.499,409.701 870.147,445.557 870.794,526.03 871.441,548.532 872.088,548.532 872.735,548.532 \n 873.382,574.162 874.03,576.17 874.677,505.965 875.324,548.493 875.971,544.198 876.618,544.198 877.265,544.198 877.913,544.198 878.56,535.064 879.207,506.53 \n 879.854,520.062 880.501,482.433 881.148,475.222 881.796,517.049 882.443,504.494 883.09,445.929 883.737,583.198 884.384,511.946 885.031,513.531 885.679,529.997 \n 886.326,561.763 886.973,542.511 887.62,550.64 888.267,550.64 888.914,547.256 889.562,545.993 890.209,545.993 890.856,523.568 891.503,543.204 892.15,507.155 \n 892.797,479.269 893.445,489.166 894.092,427.26 894.739,557.705 895.386,449.597 896.033,455.411 896.68,497.251 897.328,569.531 897.975,324.777 898.622,312.374 \n 899.269,629.795 899.916,446.079 900.563,436.515 901.211,434.931 901.858,434.284 902.505,517.31 903.152,558.04 903.799,548.701 904.446,639.264 905.094,514.679 \n 905.741,516.418 906.388,521.66 907.035,534.474 907.682,483.591 908.33,469.614 908.977,449.564 909.624,548.649 910.271,557.343 910.918,571.152 911.565,512.685 \n 912.213,505.899 912.86,538.945 913.507,548.027 914.154,548.027 914.801,567.858 915.448,521.896 916.096,545.342 916.743,545.342 917.39,568.691 918.037,568.691 \n 918.684,543.139 919.331,500.868 919.979,580.145 920.626,340.846 921.273,437.435 921.92,488.177 922.567,566.88 923.214,512.651 923.862,592.97 924.509,520.26 \n 925.156,558.722 925.803,558.722 926.45,523.567 927.097,536.574 927.745,528.798 928.392,552.248 929.039,602.38 929.686,561.379 930.333,541.334 930.98,580.035 \n 931.628,569.796 932.275,556.225 932.922,502.911 933.569,536.209 934.216,585.238 934.863,566.659 935.511,553.868 936.158,542.294 936.805,533.133 937.452,532.709 \n 938.099,521.528 938.746,520.368 939.394,537.654 940.041,553.794 940.688,477.499 941.335,456.41 941.982,459.46 942.629,517.554 943.277,557.516 943.924,556.147 \n 944.571,520.144 945.218,483.591 945.865,601.632 946.512,599.156 947.16,573.607 947.807,489.96 948.454,480.232 949.101,526.625 949.748,536.294 950.395,562.475 \n 951.043,569.051 951.69,502.431 952.337,502.431 952.984,579.783 953.631,594.51 954.278,564.91 954.926,539.186 955.573,559.832 956.22,517.111 956.867,522.6 \n 957.514,522.6 958.161,522.6 958.809,522.6 959.456,545.472 960.103,527.125 960.75,524.794 961.397,557.058 962.044,539.423 962.692,516.767 963.339,599.835 \n 963.986,579.469 964.633,569.911 965.28,571.271 965.927,547.177 966.575,537.793 967.222,543.886 967.869,568.889 968.516,539.611 969.163,576.491 969.811,521.869 \n 970.458,457.584 971.105,480.083 971.752,522.439 972.399,524.65 973.046,565.284 973.694,547.474 974.341,564.242 974.988,511.165 975.635,524.601 976.282,536.238 \n 976.929,536.238 977.577,533.6 978.224,533.6 978.871,529.517 979.518,529.517 980.165,527.093 980.812,506.45 981.46,519.774 982.107,538.979 982.754,484.816 \n 983.401,448.876 984.048,473.996 984.695,456.538 985.343,515.149 985.99,557.323 986.637,557.323 987.284,583.932 987.931,564.945 988.578,524.621 989.226,578.613 \n 989.873,565.88 990.52,613.412 991.167,645.26 991.814,706.422 992.461,638.325 993.109,686.672 993.756,518.401 994.403,510.953 995.05,535.579 995.697,576.003 \n 996.344,598.543 996.992,414.394 997.639,462.979 998.286,594.641 998.933,522.722 999.58,534.807 1000.23,548.791 1000.87,529.801 1001.52,550.636 1002.17,584.391 \n 1002.82,572.802 1003.46,633.399 1004.11,519.606 1004.76,557.321 1005.4,539.639 1006.05,558.152 1006.7,530.162 1007.35,497.074 1007.99,589.313 1008.64,553.589 \n 1009.29,541.425 1009.93,549.064 1010.58,522.231 1011.23,529.667 1011.88,508.994 1012.52,534.444 1013.17,550.135 1013.82,564.026 1014.47,492.736 1015.11,511.133 \n 1015.76,532.045 1016.41,522.516 1017.05,488.087 1017.7,488.087 1018.35,497.459 1019,490.756 1019.64,609.92 1020.29,605.61 1020.94,585.963 1021.58,568.778 \n 1022.23,546.24 1022.88,544.443 1023.53,561.462 1024.17,630.286 1024.82,511.952 1025.47,481.216 1026.11,475.74 1026.76,463.005 1027.41,465.136 1028.06,489.732 \n 1028.7,579.673 1029.35,538.336 1030,546.573 1030.64,510.955 1031.29,500.65 1031.94,564.102 1032.59,617.575 1033.23,474.751 1033.88,582.099 1034.53,562.632 \n 1035.17,570.319 1035.82,528.258 1036.47,508.456 1037.12,552.171 1037.76,546.376 1038.41,561.446 1039.06,581.509 1039.7,592.008 1040.35,590.331 1041,565.601 \n 1041.65,491.115 1042.29,466.141 1042.94,576.858 1043.59,508.935 1044.23,558.368 1044.88,577.816 1045.53,500.848 1046.18,494.828 1046.82,593.853 1047.47,528.09 \n 1048.12,556.505 1048.77,634.265 1049.41,584.48 1050.06,590.846 1050.71,577.446 1051.35,532.453 1052,531.373 1052.65,539.88 1053.3,494.104 1053.94,541.177 \n 1054.59,520.804 1055.24,499.855 1055.88,591.179 1056.53,548.212 1057.18,588.195 1057.83,588.195 1058.47,562.953 1059.12,502.572 1059.77,554.003 1060.41,548.997 \n 1061.06,487.371 1061.71,460.179 1062.36,440.314 1063,405.913 1063.65,508.218 1064.3,408.473 1064.94,580.619 1065.59,557.386 1066.24,493.335 1066.89,514.801 \n 1067.53,572.631 1068.18,558.285 1068.83,574.472 1069.47,472.748 1070.12,446.798 1070.77,413.694 1071.42,416.538 1072.06,380.162 1072.71,469.47 1073.36,530.353 \n 1074,533.1 1074.65,535.844 1075.3,551.905 1075.95,547.823 1076.59,633.474 1077.24,629.46 1077.89,478.825 1078.53,478.825 1079.18,624.472 1079.83,614.071 \n 1080.48,617.245 1081.12,528.57 1081.77,551.456 1082.42,582.262 1083.06,533.082 1083.71,495.809 1084.36,762.212 1085.01,635.064 1085.65,505.778 1086.3,440.206 \n 1086.95,412.511 1087.6,409.87 1088.24,480.153 1088.89,501.872 1089.54,642.617 1090.18,668.458 1090.83,673.852 1091.48,660.134 1092.13,597.667 1092.77,443.684 \n 1093.42,547.38 1094.07,517.196 1094.71,506.577 1095.36,492.143 1096.01,498.866 1096.66,609.519 1097.3,607.534 1097.95,519.721 1098.6,519.463 1099.24,515.16 \n 1099.89,589.922 1100.54,615.376 1101.19,703.535 1101.83,749.468 1102.48,784.945 1103.13,706.754 1103.77,721.885 1104.42,670.049 1105.07,427.258 1105.72,418.346 \n 1106.36,423.388 1107.01,601.897 1107.66,516.78 1108.3,540.54 1108.95,539.378 1109.6,585.112 1110.25,544.562 1110.89,526.314 1111.54,536.094 1112.19,552.531 \n 1112.83,499.065 1113.48,423.613 1114.13,281.087 1114.78,324.831 1115.42,302.638 1116.07,332.138 1116.72,296.471 1117.36,602.006 1118.01,682.399 1118.66,638.101 \n 1119.31,662.481 1119.95,446.794 1120.6,434.42 1121.25,574.827 1121.9,502.56 1122.54,615.556 1123.19,594.791 1123.84,536.674 1124.48,523.882 1125.13,584.966 \n 1125.78,473.934 1126.43,552.293 1127.07,547.982 1127.72,520.882 1128.37,535.447 1129.01,551.132 1129.66,543.539 1130.31,557.715 1130.96,529.595 1131.6,537.922 \n 1132.25,495.555 1132.9,538.361 1133.54,541.218 1134.19,548.559 1134.84,525.906 1135.49,500.281 1136.13,487.185 1136.78,514.8 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]
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "P1 = plot((x->x[3]).(posterior_β))\nP2 = histogram((x->x[3]).(posterior_β))\nplot(P1, P2, size=(800, 250), title=\"beta3\", legend=false)",
"execution_count": 37,
"outputs": [
{
"output_type": "execute_result",
"execution_count": 37,
"data": {
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600.703,379.682 \n 601.335,424.439 601.966,417.713 602.598,418.483 603.229,420.048 603.861,337.354 604.492,347.3 605.123,496.394 605.755,378.407 606.386,469.269 607.018,500.559 \n 607.649,348.31 608.281,372.033 608.912,409.641 609.543,458.887 610.175,454.297 610.806,449.516 611.438,374.635 612.069,364.029 612.701,358.801 613.332,393.873 \n 613.964,476.248 614.595,417.025 615.226,513.405 615.858,481.777 616.489,535.564 617.121,542.136 617.752,523.415 618.384,485.893 619.015,525.155 619.647,466.243 \n 620.278,463.137 620.909,438.146 621.541,489.946 622.172,481.464 622.804,446.033 623.435,411.007 624.067,410.033 624.698,467.607 625.329,462.634 625.961,429.201 \n 626.592,431.508 627.224,443.281 627.855,451.337 628.487,429.277 629.118,394.086 629.75,273.341 630.381,405.234 631.012,518.067 631.644,490.447 632.275,504.595 \n 632.907,595.969 633.538,547.897 634.17,462.512 634.801,347.692 635.433,381.972 636.064,393.239 636.695,390.575 637.327,370.834 637.958,360.868 638.59,490.383 \n 639.221,476.833 639.853,465.012 640.484,467.977 641.115,519.188 641.747,469.659 642.378,505.55 643.01,515.075 643.641,473.79 644.273,452.69 644.904,435.499 \n 645.536,452.956 646.167,456.804 646.798,450.929 647.43,423.248 648.061,438.009 648.693,438.009 649.324,387.724 649.956,443.83 650.587,502.141 651.219,460.635 \n 651.85,508.964 652.481,466.16 653.113,464.863 653.744,428.41 654.376,422.869 655.007,437.434 655.639,437.434 656.27,443.113 656.902,386.21 657.533,396.699 \n 658.164,472.22 658.796,472.22 659.427,326.415 660.059,495.68 660.69,500.755 661.322,442.337 661.953,474.247 662.584,452.236 663.216,445.537 663.847,415.943 \n 664.479,422.953 665.11,430.769 665.742,469.876 666.373,459.254 667.005,433.045 667.636,370.151 668.267,340.984 668.899,348.868 669.53,313.474 670.162,367.871 \n 670.793,463.073 671.425,502.042 672.056,470.491 672.688,369.529 673.319,403.044 673.95,365.792 674.582,373.962 675.213,377.827 675.845,339.293 676.476,396.307 \n 677.108,420.748 677.739,540.802 678.37,534.733 679.002,632.142 679.633,547.596 680.265,282.539 680.896,525.578 681.528,544.693 682.159,382.239 682.791,449.223 \n 683.422,430.699 684.053,388.505 684.685,369.574 685.316,377.911 685.948,378.545 686.579,412.228 687.211,407.301 687.842,415.036 688.474,392.475 689.105,425.854 \n 689.736,425.854 690.368,407.047 690.999,405.721 691.631,403.621 692.262,460.25 692.894,393.621 693.525,418.956 694.156,418.956 694.788,436.741 695.419,491.839 \n 696.051,525.675 696.682,523.117 697.314,521.277 697.945,436.136 698.577,298.249 699.208,400.067 699.839,481.184 700.471,482.092 701.102,473.269 701.734,468.894 \n 702.365,446.689 702.997,458.53 703.628,462.022 704.26,429.423 704.891,405.675 705.522,420.133 706.154,395.002 706.785,371.569 707.417,398.209 708.048,428.945 \n 708.68,428.945 709.311,408.923 709.943,403.715 710.574,453.871 711.205,406.467 711.837,410.486 712.468,427.649 713.1,471.099 713.731,479.497 714.363,356.117 \n 714.994,371.373 715.625,497.785 716.257,365.216 716.888,412.917 717.52,404.252 718.151,449.525 718.783,443.37 719.414,429.621 720.046,453.318 720.677,376.652 \n 721.308,472.566 721.94,404.325 722.571,414.093 723.203,454.63 723.834,463.829 724.466,454.093 725.097,470.614 725.729,450.668 726.36,424.606 726.991,416.791 \n 727.623,424.087 728.254,450.513 728.886,437.967 729.517,483.606 730.149,356.731 730.78,426.051 731.411,391.063 732.043,393.794 732.674,399.143 733.306,460.285 \n 733.937,492.581 734.569,387.001 735.2,378.256 735.832,273.012 736.463,267.128 737.094,302.457 737.726,617.147 738.357,559.707 738.989,629.205 739.62,456.611 \n 740.252,416.004 740.883,413.458 741.515,476.472 742.146,491.016 742.777,453.492 743.409,406.191 744.04,424.891 744.672,400.476 745.303,416.843 745.935,431.448 \n 746.566,428.884 747.198,398.23 747.829,414.699 748.46,388.86 749.092,672.206 749.723,713.111 750.355,728.64 750.986,358.961 751.618,279.72 752.249,275.836 \n 752.88,246.421 753.512,327.006 754.143,401.83 754.775,448.753 755.406,385.448 756.038,404.101 756.669,444.398 757.301,425.079 757.932,439.257 758.563,428.67 \n 759.195,398.145 759.826,421.831 760.458,403.128 761.089,444.543 761.721,452.868 762.352,462.182 762.984,402.887 763.615,459.38 764.246,227.035 764.878,230.186 \n 765.509,274.523 766.141,299.537 766.772,310.177 767.404,291.441 768.035,343.345 768.666,310.223 769.298,394.529 769.929,421.923 770.561,263.532 771.192,403.733 \n 771.824,420.203 772.455,457.156 773.087,647.948 773.718,416.713 774.349,395.255 774.981,359.016 775.612,274.206 776.244,311.422 776.875,512.652 777.507,507.866 \n 778.138,305.436 778.77,325.447 779.401,391.481 780.032,336.89 780.664,441.615 781.295,327.962 781.927,259.699 782.558,467.218 783.19,469.223 783.821,377.643 \n 784.452,341.471 785.084,490.217 785.715,404.767 786.347,381.242 786.978,382.323 787.61,353.829 788.241,366.467 788.873,354.017 789.504,357.643 790.135,417.106 \n 790.767,413.385 791.398,437.799 792.03,423.659 792.661,451.039 793.293,460.556 793.924,449.146 794.556,387.219 795.187,532.52 795.818,517.91 796.45,489.529 \n 797.081,411.559 797.713,418.288 798.344,432.123 798.976,432.123 799.607,397.804 800.239,423.653 800.87,420.45 801.501,425.855 802.133,395.358 802.764,471.53 \n 803.396,432.361 804.027,469.671 804.659,477.566 805.29,421.374 805.921,626.389 806.553,594.932 807.184,447.805 807.816,462.469 808.447,343.489 809.079,397.688 \n 809.71,444.269 810.342,436.214 810.973,444.78 811.604,373.423 812.236,367.292 812.867,394.191 813.499,437.478 814.13,428.891 814.762,431.812 815.393,437.104 \n 816.025,374.524 816.656,436.036 817.287,444.337 817.919,451.584 818.55,448.864 819.182,453.404 819.813,370.69 820.445,393.643 821.076,406.839 821.707,384.104 \n 822.339,415.354 822.97,588.327 823.602,571.007 824.233,495.248 824.865,442.13 825.496,460.734 826.128,443.568 826.759,395.549 827.39,390.904 828.022,419.562 \n 828.653,383.116 829.285,401.957 829.916,320.958 830.548,325.476 831.179,385.882 831.811,323.837 832.442,329.92 833.073,431.136 833.705,418.487 834.336,428.802 \n 834.968,475.898 835.599,435.548 836.231,450.514 836.862,429.927 837.493,429.927 838.125,422.127 838.756,432.607 839.388,420.586 840.019,470.434 840.651,518.959 \n 841.282,341.715 841.914,347.2 842.545,368.698 843.176,490.364 843.808,435.559 844.439,421.605 845.071,419.889 845.702,466.017 846.334,436.954 846.965,436.431 \n 847.597,405.339 848.228,398.982 848.859,374.409 849.491,357.517 850.122,371.945 850.754,501.664 851.385,498.382 852.017,500.346 852.648,426.287 853.28,403.016 \n 853.911,410.395 854.542,395.443 855.174,382.818 855.805,354.703 856.437,380.151 857.068,344.785 857.7,414.641 858.331,346.916 858.962,348.127 859.594,575.824 \n 860.225,385.108 860.857,499.239 861.488,513.984 862.12,431.187 862.751,391.801 863.383,367.483 864.014,369.879 864.645,365.886 865.277,329.63 865.908,328.845 \n 866.54,519.51 867.171,496.269 867.803,551.275 868.434,501.672 869.066,465.032 869.697,380.554 870.328,396.109 870.96,414.987 871.591,448.528 872.223,436.316 \n 872.854,414.318 873.486,397.457 874.117,457.887 874.748,425.667 875.38,441.588 876.011,401.511 876.643,450.144 877.274,450.144 877.906,346.144 878.537,348.722 \n 879.169,426.07 879.8,435.378 880.431,448.121 881.063,447.042 881.694,385.521 882.326,399.955 882.957,365.485 883.589,413.081 884.22,365.672 884.852,408.405 \n 885.483,401.382 886.114,441.853 886.746,434.309 887.377,426.072 888.009,374.752 888.64,439.442 889.272,434.155 889.903,393.828 890.535,424.108 891.166,211.972 \n 891.797,241.158 892.429,302.453 893.06,377.559 893.692,355.5 894.323,328.825 894.955,478.087 895.586,320.913 896.217,109.911 896.849,136.048 897.48,142.064 \n 898.112,246.807 898.743,582.728 899.375,513.826 900.006,536.229 900.638,477.061 901.269,400.421 901.9,455.076 902.532,457.503 903.163,394.24 903.795,407.974 \n 904.426,449.252 905.058,410.181 905.689,473.157 906.321,406.412 906.952,395.275 907.583,372.561 908.215,396.666 908.846,393.342 909.478,414.285 910.109,434.097 \n 910.741,403.541 911.372,424.071 912.003,449.393 912.635,453.542 913.266,440.728 913.898,381.07 914.529,397.029 915.161,385.656 915.792,385.656 916.424,394.546 \n 917.055,375.944 917.686,784.718 918.318,453.363 918.949,517.417 919.581,401.9 920.212,434.513 920.844,427.918 921.475,431.98 922.107,431.98 922.738,431.98 \n 923.369,460.409 924.001,461.194 924.632,449.601 925.264,416.104 925.895,408.164 926.527,408.164 927.158,408.164 927.789,408.164 928.421,387.053 929.052,420.517 \n 929.684,377.122 930.315,344.658 930.947,420.936 931.578,367.987 932.21,386.652 932.841,359.913 933.472,349.199 934.104,519.137 934.735,483.828 935.367,471.933 \n 935.998,447.974 936.63,439.677 937.261,441.174 937.893,441.174 938.524,434.719 939.155,439.038 939.787,439.038 940.418,435.599 941.05,422.768 941.681,360.185 \n 942.313,425.583 942.944,361.803 943.576,344.716 944.207,360.417 944.838,387.497 945.47,407.427 946.101,379.707 946.733,483.703 947.364,418.358 947.996,377.507 \n 948.627,338.967 949.258,480.868 949.89,502.746 950.521,434.88 951.153,451.502 951.784,516.957 952.416,507.908 953.047,503.479 953.679,530.576 954.31,433.915 \n 954.941,418.041 955.573,283.825 956.204,554.841 956.836,384.34 957.467,425.288 958.099,413.704 958.73,345.774 959.362,342.589 959.993,363.383 960.624,349.807 \n 961.256,445.396 961.887,406.407 962.519,428.324 963.15,428.324 963.782,481.161 964.413,425.313 965.044,412.052 965.676,412.052 966.307,401.688 966.939,401.688 \n 967.57,425.064 968.202,408.278 968.833,332.487 969.465,457.266 970.096,313.182 970.727,413.972 971.359,420.988 971.99,415.414 972.622,416.69 973.253,415.647 \n 973.885,437.908 974.516,437.908 975.148,400.604 975.779,420.742 976.41,418.469 977.042,450.92 977.673,445.416 978.305,417.251 978.936,405.949 979.568,441.535 \n 980.199,377.781 980.831,392.378 981.462,414.245 982.093,395.181 982.725,454.129 983.356,446.265 983.988,458.377 984.619,441.701 985.251,429.182 985.882,407.144 \n 986.513,407.245 987.145,405.781 987.776,351.726 988.408,352.647 989.039,265.096 989.671,427.999 990.302,281.399 990.934,301.057 991.565,469.968 992.196,457.7 \n 992.828,492.43 993.459,354.207 994.091,492.284 994.722,495.62 995.354,480.939 995.985,430.188 996.617,415.6 997.248,450.394 997.879,442.671 998.511,451.152 \n 999.142,402.087 999.774,384.023 1000.41,384.023 1001.04,424.586 1001.67,475.664 1002.3,453.783 1002.93,438.346 1003.56,427.797 1004.19,422.661 1004.83,424.575 \n 1005.46,424.575 1006.09,424.575 1006.72,424.575 1007.35,425.692 1007.98,426.874 1008.61,419.498 1009.25,456.031 1009.88,362.148 1010.51,352.627 1011.14,432.173 \n 1011.77,444.818 1012.4,436.026 1013.03,432.531 1013.67,478.371 1014.3,357.441 1014.93,323.639 1015.56,567.623 1016.19,364.903 1016.82,480.474 1017.45,350.03 \n 1018.09,432.883 1018.72,433.62 1019.35,400.697 1019.98,488.245 1020.61,414.826 1021.24,423.445 1021.87,398.213 1022.51,432.671 1023.14,446.331 1023.77,397.284 \n 1024.4,397.284 1025.03,396.322 1025.66,396.322 1026.29,391.151 1026.93,391.151 1027.56,388.623 1028.19,400.571 1028.82,398.161 1029.45,469.706 1030.08,427.714 \n 1030.71,411.849 1031.35,375.019 1031.98,411.476 1032.61,455.931 1033.24,432.461 1033.87,432.461 1034.5,438.113 1035.13,426.457 1035.77,452.431 1036.4,336.821 \n 1037.03,336.333 1037.66,359.444 1038.29,334.069 1038.92,331.697 1039.55,338.357 1040.19,424.907 1040.82,424.106 1041.45,408.31 1042.08,424.732 1042.71,468.265 \n 1043.34,468.744 1043.97,411.371 1044.61,426.053 1045.24,381.536 1045.87,494.065 1046.5,498.638 1047.13,504.569 1047.76,453.124 1048.39,439.502 1049.03,400.82 \n 1049.66,417.352 1050.29,419.569 1050.92,500.368 1051.55,348.001 1052.18,350.701 1052.81,458.11 1053.45,465.863 1054.08,339.168 1054.71,499.663 1055.34,486.571 \n 1055.97,490.511 1056.6,500.393 1057.23,403.375 1057.87,444.313 1058.5,441.555 1059.13,469.053 1059.76,442.179 1060.39,368.475 1061.02,461.135 1061.65,428.362 \n 1062.29,382.217 1062.92,393.511 1063.55,389.708 1064.18,389.708 1064.81,378.711 1065.44,367.223 1066.07,526.735 1066.71,563.066 1067.34,455.667 1067.97,419.805 \n 1068.6,406.265 1069.23,428.452 1069.86,455.998 1070.5,344.529 1071.13,454.395 1071.76,438.515 1072.39,464.695 1073.02,427.585 1073.65,408.691 1074.28,453.757 \n 1074.92,406.375 1075.55,401.04 1076.18,403.752 1076.81,397.163 1077.44,439.774 1078.07,396.18 1078.7,480.71 1079.34,375.359 1079.97,467.175 1080.6,443.16 \n 1081.23,458.835 1081.86,412.085 1082.49,362.309 1083.12,474.852 1083.76,378.095 1084.39,410.268 1085.02,400.268 1085.65,404.182 1086.28,427.738 1086.91,428.794 \n 1087.54,411.1 1088.18,398.284 1088.81,359.142 1089.44,415.161 1090.07,405.398 1090.7,430.899 1091.33,430.32 1091.96,414.729 1092.6,414.094 1093.23,400.167 \n 1093.86,466.695 1094.49,443.691 1095.12,469.605 1095.75,459.315 1096.38,497 1097.02,470.115 1097.65,347.689 1098.28,440.717 1098.91,404.794 1099.54,396.112 \n 1100.17,392.886 1100.8,462.605 1101.44,468.79 1102.07,463.472 1102.7,446.092 1103.33,446.092 1103.96,444.092 1104.59,377.658 1105.22,428.794 1105.86,406.914 \n 1106.49,343.953 1107.12,340.548 1107.75,327.686 1108.38,414.022 1109.01,469.641 1109.64,364.342 1110.28,391.773 1110.91,375.223 1111.54,430.538 1112.17,394.825 \n 1112.8,408.843 1113.43,383.922 1114.06,334.985 1114.7,383.944 1115.33,345.551 1115.96,373.589 1116.59,441.194 1117.22,449.313 1117.85,438.097 1118.48,343.306 \n 1119.12,368.434 1119.75,367.454 1120.38,512.243 1121.01,513.164 1121.64,530.258 1122.27,446.781 1122.9,407.389 1123.54,407.389 1124.17,482.349 1124.8,539.768 \n 1125.43,504.318 1126.06,459.147 1126.69,440.186 1127.32,439.691 1127.96,477.426 1128.59,458.273 1129.22,491.341 1129.85,548.124 1130.48,391.326 1131.11,505.409 \n 1131.74,316.694 1132.38,347.275 1133.01,432.976 1133.64,310.467 1134.27,583.641 1134.9,559.867 1135.53,511.724 1136.16,474.273 1136.8,497.202 1137.43,355.907 \n 1138.06,475.474 1138.69,501.787 1139.32,434.339 1139.95,419.018 1140.58,367.981 1141.22,377.078 1141.85,381.256 1142.48,385.932 1143.11,386.46 1143.74,370.197 \n 1144.37,421.887 1145.01,312.624 1145.64,393.841 1146.27,385.103 1146.9,378.442 1147.53,443.803 1148.16,416.713 1148.79,412.271 1149.43,354.468 1150.06,308.364 \n 1150.69,510.767 1151.32,463.4 1151.95,420.631 1152.58,425.239 1153.21,441.443 1153.85,424.766 1154.48,437.343 1155.11,434.755 1155.74,381.625 1156.37,486.368 \n 1157,434.704 1157.63,375.943 1158.27,385.098 1158.9,320.743 1159.53,312.497 1160.16,149.991 1160.79,257.972 1161.42,887.948 1162.05,363.679 1162.69,430.704 \n 1163.32,405.147 1163.95,472.04 1164.58,543.069 1165.21,253.21 1165.84,268.314 1166.47,310.713 1167.11,335.645 1167.74,488.117 1168.37,423.755 1169,433.832 \n 1169.63,375.053 1170.26,445.061 1170.89,460.855 1171.53,399.394 1172.16,421.248 1172.79,428.669 1173.42,429.537 1174.05,462.838 1174.68,396.417 1175.31,400.95 \n 1175.95,425.251 1176.58,370.32 1177.21,428.736 1177.84,391.765 1178.47,453.671 1179.1,451.404 1179.73,442.062 1180.37,360.58 1181,454.031 1181.63,453.53 \n 1182.26,417.875 1182.89,423.756 1183.52,479.06 1184.15,472.983 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},
{
"metadata": {
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},
"cell_type": "code",
"source": "P1 = plot(posterior_σ)\nP2 = histogram(posterior_σ)\nplot(P1, P2, size=(800, 250), title=\"sigma\", legend=false)",
"execution_count": 38,
"outputs": [
{
"output_type": "execute_result",
"execution_count": 38,
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622.928,554.661 623.575,667.848 624.223,369.814 624.87,519.362 625.517,602.901 626.164,817.24 626.811,808.332 \n 627.458,801.225 628.106,692.008 628.753,756.047 629.4,779.8 630.047,776.123 630.694,801.731 631.341,850.691 631.989,854.199 632.636,819.397 633.283,852.827 \n 633.93,852.827 634.577,864.139 635.224,862.583 635.872,860.235 636.519,854.404 637.166,807.425 637.813,859.974 638.46,859.974 639.107,824.617 639.755,714.881 \n 640.402,793.179 641.049,548.664 641.696,569.739 642.343,690.155 642.99,279.931 643.638,722.142 644.285,548.033 644.932,747.776 645.579,750.584 646.226,796.955 \n 646.873,828.211 647.521,805.659 648.168,827.807 648.815,847.811 649.462,774.715 650.109,793.468 650.756,806.339 651.404,810.562 652.051,778.259 652.698,790.126 \n 653.345,790.126 653.992,694.996 654.639,801.514 655.287,800.234 655.934,856.562 656.581,840.004 657.228,793.125 657.875,836.125 658.522,844.514 659.17,732.135 \n 659.817,727.186 660.464,835.665 661.111,786.621 661.758,827.058 662.405,805.374 663.053,810.03 663.7,857.352 664.347,860.682 664.994,820.65 665.641,797.092 \n 666.288,794.764 666.936,826.247 667.583,819.487 668.23,815.468 668.877,809.697 669.524,798.79 670.172,809.285 670.819,807.129 671.466,853.722 672.113,778.249 \n 672.76,814.347 673.407,801.988 674.055,804.963 674.702,838.461 675.349,627.892 675.996,783.97 676.643,751.715 677.29,800.63 677.938,816.776 678.585,775.123 \n 679.232,797.454 679.879,763.967 680.526,753.76 681.173,761.44 681.821,762.628 682.468,766.079 683.115,703.171 683.762,504.399 684.409,553.122 685.056,604.803 \n 685.704,685.333 686.351,754.983 686.998,615.073 687.645,186.359 688.292,443.045 688.939,820.234 689.587,820.977 690.234,858.913 690.881,836.188 691.528,855.977 \n 692.175,831.283 692.822,826.319 693.47,835.66 694.117,786.753 694.764,407.106 695.411,506.601 696.058,323.991 696.705,362.091 697.353,618.175 698,577.301 \n 698.647,470.578 699.294,518.938 699.941,797.297 700.588,825.847 701.236,841.215 701.883,842.365 702.53,764.392 703.177,786.028 703.824,803.811 704.471,781.409 \n 705.119,800.94 705.766,731.575 706.413,693.332 707.06,840.739 707.707,828.641 708.354,823.597 709.002,810.323 709.649,844.132 710.296,455.653 710.943,624.982 \n 711.59,623.193 712.237,460.815 712.885,629.041 713.532,663.616 714.179,483.342 714.826,408.115 715.473,459.296 716.12,476.849 716.768,641.921 717.415,681.575 \n 718.062,803.1 718.709,734.594 719.356,578.262 720.003,811.542 720.651,821.355 721.298,793.089 721.945,652.789 722.592,666.157 723.239,705.529 723.886,509.583 \n 724.534,561.972 725.181,368.753 725.828,373.922 726.475,787.941 727.122,478.995 727.769,599.195 728.417,730.259 729.064,758.524 729.711,777.745 730.358,789.376 \n 731.005,819.373 731.653,633.26 732.3,647.922 732.947,689.66 733.594,737.929 734.241,784.944 734.888,819.37 735.536,825.247 736.183,815.36 736.83,835.285 \n 737.477,840.392 738.124,848.529 738.771,840.311 739.419,843.757 740.066,833.905 740.713,755.155 741.36,809.184 742.007,817.049 742.654,826.462 743.302,800.837 \n 743.949,858.178 744.596,858.701 745.243,860.933 745.89,860.933 746.537,811.529 747.185,797.277 747.832,792.549 748.479,839.27 749.126,842.958 749.773,813.305 \n 750.42,822.747 751.068,804.08 751.715,788.792 752.362,627.884 753.009,681.873 753.656,523.809 754.303,715.284 754.951,700.79 755.598,791.081 756.245,845.369 \n 756.892,825.13 757.539,859.008 758.186,852.435 758.834,791.653 759.481,820.264 760.128,827.628 760.775,806.142 761.422,845.13 762.069,834.025 762.717,825.332 \n 763.364,826.936 764.011,848.592 764.658,846.29 765.305,819.91 765.952,841.459 766.6,845.869 767.247,827.232 767.894,840.394 768.541,808.399 769.188,793.126 \n 769.835,794.719 770.483,553.596 771.13,764.599 771.777,774.54 772.424,845.689 773.071,839.703 773.718,835.809 774.366,853.782 775.013,840.031 775.66,853.907 \n 776.307,831.313 776.954,682.268 777.601,704.719 778.249,765.11 778.896,794.054 779.543,729.215 780.19,626.935 780.837,846.606 781.484,847.064 782.132,824.899 \n 782.779,811.467 783.426,834.946 784.073,830.543 784.72,845.438 785.367,845.438 786.015,829.527 786.662,830.575 787.309,716.669 787.956,705.37 788.603,763.942 \n 789.251,702.644 789.898,718.408 790.545,847.757 791.192,652.236 791.839,754.518 792.486,846.889 793.134,834.47 793.781,797.047 794.428,801.012 795.075,821.343 \n 795.722,830.949 796.369,805.736 797.017,826.805 797.664,813.353 798.311,812.836 798.958,710.44 799.605,773.545 800.252,776.181 800.9,758.311 801.547,793.384 \n 802.194,807.147 802.841,785.929 803.488,761.23 804.135,843.785 804.783,760.749 805.43,712.984 806.077,742.975 806.724,760.087 807.371,780.715 808.018,616.334 \n 808.666,684.984 809.313,641.303 809.96,696.082 810.607,754.244 811.254,831.435 811.901,723.491 812.549,711.539 813.196,716.404 813.843,692.145 814.49,660.426 \n 815.137,806.631 815.784,688.165 816.432,749.913 817.079,777.91 817.726,840.196 818.373,672.9 819.02,690.59 819.667,823.469 820.315,842.643 820.962,845.681 \n 821.609,791.69 822.256,849.671 822.903,833.887 823.55,805.003 824.198,827.594 824.845,862.776 825.492,873.834 826.139,873.834 826.786,792.855 827.433,827.967 \n 828.081,638.76 828.728,779.765 829.375,447.086 830.022,525.475 830.669,782.222 831.316,768.822 831.964,830.615 832.611,783.309 833.258,794.652 833.905,855.86 \n 834.552,795.031 835.199,832.953 835.847,843.874 836.494,835.438 837.141,767.617 837.788,844.864 838.435,600.974 839.082,657.702 839.73,680.566 840.377,668.909 \n 841.024,692.693 841.671,588.661 842.318,732.295 842.965,675.567 843.613,801.425 844.26,705.258 844.907,758.088 845.554,348.871 846.201,717.623 846.848,661.322 \n 847.496,450.216 848.143,589.184 848.79,770.195 849.437,749.97 850.084,793.405 850.732,747.247 851.379,792.421 852.026,824.809 852.673,804.659 853.32,791.004 \n 853.967,751.152 854.615,830.764 855.262,784.617 855.909,832.826 856.556,788.458 857.203,793.336 857.85,765.095 858.498,799.742 859.145,849.478 859.792,856.204 \n 860.439,847.402 861.086,854.102 861.733,857.419 862.381,855.185 863.028,846.307 863.675,856.696 864.322,819.026 864.969,835.103 865.616,835.103 866.264,819.599 \n 866.911,785.96 867.558,464.532 868.205,614.345 868.852,542.981 869.499,761.124 870.147,650.469 870.794,768.402 871.441,871.885 872.088,871.885 872.735,871.885 \n 873.382,799.499 874.03,788.888 874.677,749.282 875.324,864.823 875.971,861.183 876.618,861.183 877.265,861.183 877.913,861.183 878.56,825.998 879.207,818.859 \n 879.854,796.305 880.501,803.562 881.148,803.416 881.796,819.108 882.443,822.727 883.09,815.264 883.737,733.831 884.384,823.182 885.031,830.192 885.679,844.188 \n 886.326,825.756 886.973,863.248 887.62,867.767 888.267,867.767 888.914,858.019 889.562,862.432 890.209,862.432 890.856,751.66 891.503,741.925 892.15,763.209 \n 892.797,827.45 893.445,664.846 894.092,765.53 894.739,702.749 895.386,802.247 896.033,756.882 896.68,790.281 897.328,821.834 897.975,631.962 898.622,594.842 \n 899.269,595.465 899.916,743.323 900.563,750.319 901.211,730.098 901.858,742.95 902.505,799.572 903.152,746.843 903.799,786.84 904.446,716.564 905.094,842.208 \n 905.741,830.952 906.388,691.489 907.035,746.405 907.682,726.04 908.33,785.534 908.977,759.774 909.624,827.373 910.271,827.368 910.918,832.875 911.565,806.651 \n 912.213,842.576 912.86,868.561 913.507,850.812 914.154,850.812 914.801,838.283 915.448,825.822 916.096,857.303 916.743,857.303 917.39,867.547 918.037,867.547 \n 918.684,844.057 919.331,808.006 919.979,520.928 920.626,723.911 921.273,615.786 921.92,800.56 922.567,869.2 923.214,807.644 923.862,816.022 924.509,805.657 \n 925.156,855.578 925.803,855.578 926.45,870.652 927.097,870.429 927.745,862.718 928.392,842.49 929.039,808.416 929.686,611.397 930.333,730.225 930.98,850.819 \n 931.628,791.656 932.275,802.147 932.922,859.878 933.569,837.413 934.216,825.915 934.863,810.783 935.511,819.592 936.158,837.41 936.805,834.058 937.452,858.011 \n 938.099,835.786 938.746,795.801 939.394,784.399 940.041,790.757 940.688,646.52 941.335,798.1 941.982,674.882 942.629,707.751 943.277,813.893 943.924,821.805 \n 944.571,827.459 945.218,764.781 945.865,818.883 946.512,816.913 947.16,744.348 947.807,814.471 948.454,823.561 949.101,836.863 949.748,815.34 950.395,856.952 \n 951.043,775.772 951.69,845.734 952.337,845.734 952.984,821.417 953.631,816.55 954.278,836.835 954.926,825.049 955.573,822.378 956.22,860.839 956.867,863.173 \n 957.514,863.173 958.161,863.173 958.809,863.173 959.456,804.688 960.103,807.641 960.75,791.26 961.397,786.158 962.044,783.758 962.692,829.683 963.339,791.254 \n 963.986,830.714 964.633,769.661 965.28,773.459 965.927,833.224 966.575,806.02 967.222,830.527 967.869,706.506 968.516,845.736 969.163,754.121 969.811,812.996 \n 970.458,741.843 971.105,793.007 971.752,754.94 972.399,813.757 973.046,848.498 973.694,830.844 974.341,845.581 974.988,852.479 975.635,844.819 976.282,867.169 \n 976.929,867.169 977.577,873.18 978.224,873.18 978.871,866.98 979.518,866.98 980.165,835.828 980.812,795.689 981.46,783.923 982.107,775.614 982.754,798.347 \n 983.401,777.847 984.048,770.563 984.695,781.223 985.343,780.959 985.99,864.721 986.637,864.721 987.284,864.466 987.931,848.759 988.578,845.06 989.226,326.296 \n 989.873,554.823 990.52,790.864 991.167,731.669 991.814,676.072 992.461,650.769 993.109,732.666 993.756,813.726 994.403,822.312 995.05,854.769 995.697,822.119 \n 996.344,788.85 996.992,765.481 997.639,731.669 998.286,762.689 998.933,721.126 999.58,720.084 1000.23,706.674 1000.87,712.069 1001.52,700.767 1002.17,766.915 \n 1002.82,747.131 1003.46,733.545 1004.11,807.439 1004.76,815.598 1005.4,805.047 1006.05,796.271 1006.7,755.363 1007.35,822.426 1007.99,750.209 1008.64,837.296 \n 1009.29,825.807 1009.93,799.234 1010.58,761.11 1011.23,813.559 1011.88,836.912 1012.52,823.364 1013.17,818.548 1013.82,818.625 1014.47,825.678 1015.11,838.695 \n 1015.76,849.999 1016.41,847.085 1017.05,868.592 1017.7,868.592 1018.35,856.269 1019,846.475 1019.64,744.885 1020.29,684.315 1020.94,786.957 1021.58,753.915 \n 1022.23,788.081 1022.88,783.301 1023.53,785.539 1024.17,725.65 1024.82,793.579 1025.47,821.423 1026.11,815.707 1026.76,834.508 1027.41,505.306 1028.06,736.224 \n 1028.7,858.734 1029.35,842.775 1030,837.153 1030.64,814.393 1031.29,797.472 1031.94,810.408 1032.59,794.173 1033.23,833.079 1033.88,805.85 1034.53,843.457 \n 1035.17,810.537 1035.82,835.212 1036.47,802.941 1037.12,769.412 1037.76,817.585 1038.41,833.22 1039.06,824.79 1039.7,775.065 1040.35,641.027 1041,753.616 \n 1041.65,769.835 1042.29,796.846 1042.94,738.21 1043.59,864.407 1044.23,812.096 1044.88,822.786 1045.53,840.791 1046.18,833.361 1046.82,833.869 1047.47,846.451 \n 1048.12,736.669 1048.77,793.051 1049.41,808.682 1050.06,811.047 1050.71,812.794 1051.35,833.683 1052,828.933 1052.65,821.953 1053.3,772.61 1053.94,823.432 \n 1054.59,834.473 1055.24,791.094 1055.88,779.587 1056.53,829.699 1057.18,865.023 1057.83,865.023 1058.47,849.017 1059.12,852.221 1059.77,811.83 1060.41,819.534 \n 1061.06,819.972 1061.71,734.433 1062.36,736.92 1063,722.853 1063.65,597.289 1064.3,754.142 1064.94,808.099 1065.59,832.271 1066.24,835.613 1066.89,789.396 \n 1067.53,773.075 1068.18,713.985 1068.83,706.829 1069.47,659.937 1070.12,680.732 1070.77,568.772 1071.42,774.788 1072.06,729.852 1072.71,776.141 1073.36,824.347 \n 1074,823.464 1074.65,828.546 1075.3,837.436 1075.95,753.538 1076.59,803.907 1077.24,777.6 1077.89,859.764 1078.53,859.764 1079.18,737.596 1079.83,737.149 \n 1080.48,744.633 1081.12,819.663 1081.77,805.833 1082.42,843.354 1083.06,831.122 1083.71,810.899 1084.36,642.981 1085.01,736.354 1085.65,788.329 1086.3,731.762 \n 1086.95,676.358 1087.6,688.769 1088.24,689.253 1088.89,752.033 1089.54,716.759 1090.18,727.869 1090.83,735.746 1091.48,718.731 1092.13,748.216 1092.77,810.222 \n 1093.42,743.671 1094.07,748.827 1094.71,678.611 1095.36,630.734 1096.01,738.52 1096.66,814.458 1097.3,827.943 1097.95,805.951 1098.6,813.661 1099.24,829.086 \n 1099.89,565.129 1100.54,733.63 1101.19,542.113 1101.83,549.331 1102.48,606.623 1103.13,685.982 1103.77,694.918 1104.42,700.186 1105.07,806.081 1105.72,790.704 \n 1106.36,591.841 1107.01,804.085 1107.66,821.585 1108.3,817.841 1108.95,833.714 1109.6,865.703 1110.25,843.949 1110.89,789.935 1111.54,837.61 1112.19,822.396 \n 1112.83,770.607 1113.48,597.93 1114.13,703.391 1114.78,470.138 1115.42,681.107 1116.07,616.157 1116.72,507.226 1117.36,523.561 1118.01,803.567 1118.66,775.602 \n 1119.31,768.715 1119.95,572.576 1120.6,763.347 1121.25,741.312 1121.9,545.293 1122.54,758.687 1123.19,727.539 1123.84,841.453 1124.48,718.299 1125.13,778.379 \n 1125.78,841.791 1126.43,823.133 1127.07,824.265 1127.72,850.245 1128.37,829.578 1129.01,856.083 1129.66,812.246 1130.31,842.485 1130.96,850.573 1131.6,819.182 \n 1132.25,862.171 1132.9,793.973 1133.54,781.488 1134.19,732.477 1134.84,817.94 1135.49,809.406 1136.13,799.645 1136.78,753.641 1137.43,800.465 1138.07,763.034 \n 1138.72,711.682 1139.37,791.96 1140.02,822.286 1140.66,809.147 1141.31,799.143 1141.96,778.29 1142.6,797.844 1143.25,830.141 1143.9,793.709 1144.55,809.951 \n 1145.19,861.255 1145.84,854.502 1146.49,794.131 1147.13,800.821 1147.78,760.305 1148.43,797.81 1149.08,828.179 1149.72,841.705 1150.37,814.504 1151.02,795.298 \n 1151.66,620.306 1152.31,672.064 1152.96,767.187 1153.61,770.366 1154.25,714.629 1154.9,783.104 1155.55,802.154 1156.2,790.45 1156.84,760.881 1157.49,763.884 \n 1158.14,568.61 1158.78,673.698 1159.43,650.194 1160.08,583.71 1160.73,624.952 1161.37,799.139 1162.02,692.639 1162.67,756.849 1163.31,715.633 1163.96,822.145 \n 1164.61,823.7 1165.26,822.226 1165.9,779.057 1166.55,798.801 1167.2,822.491 1167.84,821.816 1168.49,813.749 1169.14,809.848 1169.79,798.927 1170.43,834.273 \n 1171.08,818.903 1171.73,813.84 1172.37,818.392 1173.02,806.135 1173.67,776.096 1174.32,810.137 1174.96,852.661 1175.61,852.661 1176.26,791.117 1176.9,762.234 \n 1177.55,671.956 1178.2,639.506 1178.85,755.23 1179.49,811.593 1180.14,824.981 1180.79,808.956 1181.43,809.436 1182.08,873.223 1182.73,875.656 1183.38,875.656 \n 1184.02,875.656 1184.67,813.696 1185.32,856.2 1185.96,855.232 1186.61,855.232 1187.26,855.232 1187.91,840.501 1188.55,789.875 1189.2,798.8 1189.85,788.248 \n 1190.49,787.468 1191.14,769.836 1191.79,711.749 1192.44,810.133 1193.08,781.129 1193.73,809.298 1194.38,856.371 1195.03,858 1195.67,851.827 1196.32,859.401 \n 1196.97,835.479 1197.61,822.147 1198.26,821.589 1198.91,849.464 1199.56,812.298 1200.2,830.318 1200.85,857.483 1201.5,784.76 1202.14,822.902 1202.79,835.573 \n 1203.44,741.943 1204.09,791.95 1204.73,785.497 1205.38,635.573 1206.03,446.553 1206.67,513.007 1207.32,654.716 1207.97,744.91 1208.62,815.024 1209.26,783.76 \n 1209.91,828.441 1210.56,775.475 1211.2,781.653 1211.85,704.91 1212.5,672.146 1213.15,698.646 1213.79,716.86 1214.44,594.262 1215.09,598.574 1215.73,659.973 \n 1216.38,781.927 1217.03,783.437 1217.68,765.662 1218.32,840.186 1218.97,832.262 1219.62,848.214 1220.26,778.672 1220.91,769.973 1221.56,776.087 1222.21,773.385 \n 1222.85,782.418 1223.5,787.676 1224.15,822.149 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"source": "function pred_linreg(posterior, x, y)\n L = length(posterior)\n β = first.(posterior)\n σ = last.(posterior)\n mean(pdf(Normal(0.0, σ[i]), y - (β[i][1] + β[i][2]*x + β[i][3]*x^2)) for i in 1:L)\nend",
"execution_count": 39,
"outputs": [
{
"output_type": "execute_result",
"execution_count": 39,
"data": {
"text/plain": "pred_linreg (generic function with 1 method)"
},
"metadata": {}
}
]
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{
"metadata": {
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"source": "@show a = (X'X)\\X'*y\n@show s = fit_mle(Normal, y - X*a).σ\npred_linreg_mle(a, s, x, y) = pdf(Normal(0.0, s), y - (a[1] + a[2]*x + a[3]*x^2))",
"execution_count": 40,
"outputs": [
{
"output_type": "stream",
"text": "a = ((X' * X) \\ X') * y = [1.9898026341763784, 2.630342130669235, -1.4074843029322044]\ns = (fit_mle(Normal, y - X * a)).σ = 0.42454239399788624\n",
"name": "stdout"
},
{
"output_type": "execute_result",
"execution_count": 40,
"data": {
"text/plain": "pred_linreg_mle (generic function with 1 method)"
},
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]
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"metadata": {
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"source": "xs = range(-4, 4, length=100)\nys = range(-23, 7, length=100)\n\n@time zs = [pred_linreg(posterior, x, y) for y in ys, x in xs]\nP1 = plot(; size=(400, 300))\nheatmap!(xs, ys, zs, colorbar=false)\nscatter!(X[:,2], y, label=\"sample\", legend=false, markersize=4, color=:cyan, alpha=0.7)\nplot!(xs, [law_true(β, x) for x in xs], ls=:dash, lw=2)\ntitle!(\"Bayes\")\n\n@time zs = [pred_linreg_mle(a, s, x, y) for y in ys, x in xs]\nP2 = plot(; size=(400, 300))\nheatmap!(xs, ys, zs, colorbar=false)\nscatter!(X[:,2], y, label=\"sample\", legend=false, markersize=4, color=:cyan, alpha=0.7)\nplot!(xs, [law_true(β, x) for x in xs], ls=:dash, lw=2)\ntitle!(\"MLE\")\n\nplot(P1, P2, size=(800, 300))",
"execution_count": 41,
"outputs": [
{
"output_type": "stream",
"text": " 1.494529 seconds (528.88 k allocations: 339.035 MiB, 6.88% gc time)\n 0.072744 seconds (183.94 k allocations: 7.868 MiB)\n",
"name": "stdout"
},
{
"output_type": "execute_result",
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"source": "2N*mean(-log(pred_linreg(posterior, X[i,2], y[i])) for i in 1:N) ",
"execution_count": 42,
"outputs": [
{
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"cell_type": "code",
"source": "2N*mean(-log(pred_linreg_mle(a, s, X[i,2], y[i])) for i in 1:N) ",
"execution_count": 43,
"outputs": [
{
"output_type": "execute_result",
"execution_count": 43,
"data": {
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"metadata": {}
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]
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