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-    "markdown": "# Fundamentals of Bayesian Modeling in Julia\n\n![](https://img.shields.io/badge/status-not_started-red)\n\n\n## Very quick intro to Julia and Turing\n\nGoal is to teach just enough so that the reader understands the code.\n\n### Generate Data from Normal Distribution\n\n::: {#6eaf817c .cell execution_count=1}\n``` {.julia .cell-code}\nusing Turing, Distributions, Random\nusing Makie\n\n# μ=100, σ=15\niq = rand(Normal(100, 15), 500)\n```\n\n::: {.cell-output .cell-output-display execution_count=2}\n```\n500-element Vector{Float64}:\n  98.48776743748695\n  86.32452491238901\n  78.83526938610933\n 113.04410457290572\n  97.40745839499718\n  97.97419028900438\n  88.40275967411239\n  92.72164237452724\n 113.32790576148362\n  78.51901725613449\n  89.44848393975448\n 110.37730425103749\n  90.66395248256771\n   ⋮\n 112.27965213239929\n 110.4084690224403\n 117.07901096715132\n 104.56556661142926\n 115.76409270035695\n 102.61126569224227\n  92.7517736224979\n 111.46970868742785\n  93.68344400954143\n 109.85853656837163\n  98.05396623359759\n 121.90982884976155\n```\n:::\n:::\n\n\n::: {#5ebbc89c .cell execution_count=2}\n``` {.julia .cell-code}\nfig = Figure()\nax = Axis(fig[1, 1], title=\"Distribution\")\ndensity!(ax, iq)\nfig\n```\n\n::: {.cell-output .cell-output-stderr}\n```\n┌ Warning: Found `resolution` in the theme when creating a `Scene`. The `resolution` keyword for `Scene`s and `Figure`s has been deprecated. Use `Figure(; size = ...` or `Scene(; size = ...)` instead, which better reflects that this is a unitless size and not a pixel resolution. The key could also come from `set_theme!` calls or related theming functions.\n└ @ Makie C:\\Users\\domma\\.julia\\packages\\Makie\\VRavR\\src\\scenes.jl:220\n```\n:::\n\n::: {.cell-output .cell-output-display execution_count=3}\n![](1_introduction_files/figure-html/cell-3-output-2.svg){}\n:::\n:::\n\n\n### Recover Distribution Parameters with Turing\n\n::: {#c3c318fd .cell execution_count=3}\n``` {.julia .cell-code}\n@model function model_gaussian(x)\n    # Priors\n    μ ~ Uniform(0, 200)\n    σ ~ Uniform(0, 30)\n\n    # Check against data\n    for i in 1:length(x)\n        x[i] ~ Normal(μ, σ)\n    end\nend\n\nmodel = model_gaussian(iq)\nsampling_results = sample(model, NUTS(), 400)\n\n# Summary (95% CI)\nsummarystats(sampling_results)\n```\n\n::: {.cell-output .cell-output-stderr}\n```\n┌ Info: Found initial step size\n└   ϵ = 0.025\n\rSampling:   0%|█                                        |  ETA: 0:00:32\rSampling: 100%|█████████████████████████████████████████| Time: 0:00:01\n```\n:::\n\n::: {.cell-output .cell-output-display execution_count=4}\n\n::: {.ansi-escaped-output}\n```{=html}\n<pre>Summary Statistics\n <span class=\"ansi-bold\"> parameters </span> <span class=\"ansi-bold\">    mean </span> <span class=\"ansi-bold\">     std </span> <span class=\"ansi-bold\">    mcse </span> <span class=\"ansi-bold\"> ess_bulk </span> <span class=\"ansi-bold\"> ess_tail </span> <span class=\"ansi-bold\">    rhat </span> <span class=\"ansi-bold\"> e</span> ⋯\n <span class=\"ansi-bright-black-fg\">     Symbol </span> <span class=\"ansi-bright-black-fg\"> Float64 </span> <span class=\"ansi-bright-black-fg\"> Float64 </span> <span class=\"ansi-bright-black-fg\"> Float64 </span> <span class=\"ansi-bright-black-fg\">  Float64 </span> <span class=\"ansi-bright-black-fg\">  Float64 </span> <span class=\"ansi-bright-black-fg\"> Float64 </span> <span class=\"ansi-bright-black-fg\">  </span> ⋯\n           μ   99.7656    0.6604    0.0334   396.7968   287.6290    1.0019     ⋯\n           σ   15.8141    0.5585    0.0280   390.8097   254.6258    1.0045     ⋯\n<span class=\"ansi-cyan-fg\">                                                                1 column omitted</span>\n</pre>\n```\n:::\n\n:::\n:::\n\n\n## Linear Models\n\nUnderstand what the parameters mean (intercept, slopes, sigma).\n\n3. Boostrapping: Introduce concepts related to pseudo-posterior distribution description\n4. Hierarchical Models: Simpson's paradox, random effects, how to leverage them to model interindividual differences\n5. Bayesian estimation: introduce Bayesian estimation and priors over parameters\n6. Bayesian mixed linear regression: put everything together\n\n",
+    "markdown": "# Fundamentals of Bayesian Modeling in Julia\n\n![](https://img.shields.io/badge/status-not_started-red)\n\n\n## Very quick intro to Julia and Turing\n\nGoal is to teach just enough so that the reader understands the code.\n\n### Generate Data from Normal Distribution\n\n::: {#86de7312 .cell execution_count=1}\n``` {.julia .cell-code}\nusing Turing, Distributions, Random\nusing Makie\n\n# Random sample from a Normal(μ=100, σ=15)\niq = rand(Normal(100, 15), 500)\n```\n:::\n\n\n::: {#8df07ece .cell execution_count=2}\n``` {.julia .cell-code}\nfig = Figure()\nax = Axis(fig[1, 1], title=\"Distribution\")\ndensity!(ax, iq)\nfig\n```\n\n::: {.cell-output .cell-output-stderr}\n```\n┌ Warning: Found `resolution` in the theme when creating a `Scene`. The `resolution` keyword for `Scene`s and `Figure`s has been deprecated. Use `Figure(; size = ...` or `Scene(; size = ...)` instead, which better reflects that this is a unitless size and not a pixel resolution. The key could also come from `set_theme!` calls or related theming functions.\n└ @ Makie C:\\Users\\domma\\.julia\\packages\\Makie\\VRavR\\src\\scenes.jl:220\n```\n:::\n\n::: {.cell-output .cell-output-display execution_count=3}\n![](1_introduction_files/figure-html/cell-3-output-2.svg){}\n:::\n:::\n\n\n### Recover Distribution Parameters with Turing\n\n::: {#3276026a .cell execution_count=3}\n``` {.julia .cell-code}\n@model function model_gaussian(x)\n    # Priors\n    μ ~ Uniform(0, 200)\n    σ ~ Uniform(0, 30)\n\n    # Check against each datapoint\n    for i in 1:length(x)\n        x[i] ~ Normal(μ, σ)\n    end\nend\n\nmodel = model_gaussian(iq)\nsampling_results = sample(model, NUTS(), 400)\n\n# Summary (95% CI)\nsummarystats(sampling_results)\n```\n\n::: {.cell-output .cell-output-stderr}\n```\n┌ Info: Found initial step size\n└   ϵ = 0.05\n\rSampling:   0%|█                                        |  ETA: 0:00:29\rSampling: 100%|█████████████████████████████████████████| Time: 0:00:00\n```\n:::\n\n::: {.cell-output .cell-output-display execution_count=4}\n\n::: {.ansi-escaped-output}\n```{=html}\n<pre>Summary Statistics\n <span class=\"ansi-bold\"> parameters </span> <span class=\"ansi-bold\">    mean </span> <span class=\"ansi-bold\">     std </span> <span class=\"ansi-bold\">    mcse </span> <span class=\"ansi-bold\"> ess_bulk </span> <span class=\"ansi-bold\"> ess_tail </span> <span class=\"ansi-bold\">    rhat </span> <span class=\"ansi-bold\"> e</span> ⋯\n <span class=\"ansi-bright-black-fg\">     Symbol </span> <span class=\"ansi-bright-black-fg\"> Float64 </span> <span class=\"ansi-bright-black-fg\"> Float64 </span> <span class=\"ansi-bright-black-fg\"> Float64 </span> <span class=\"ansi-bright-black-fg\">  Float64 </span> <span class=\"ansi-bright-black-fg\">  Float64 </span> <span class=\"ansi-bright-black-fg\"> Float64 </span> <span class=\"ansi-bright-black-fg\">  </span> ⋯\n           μ   99.7082    0.6935    0.0367   349.6040   232.8456    1.0045     ⋯\n           σ   15.2626    0.4713    0.0255   337.7052   265.8666    1.0002     ⋯\n<span class=\"ansi-cyan-fg\">                                                                1 column omitted</span>\n</pre>\n```\n:::\n\n:::\n:::\n\n\n## Linear Models\n\nUnderstand what the parameters mean (intercept, slopes, sigma).\n\n## Boostrapping\n\nIntroduce concepts related to pseudo-posterior distribution description\n\n## Hierarchical Models\n\nSimpson's paradox, random effects, how to leverage them to model interindividual differences\n\n## Bayesian estimation\n\nintroduce Bayesian estimation and priors over parameters\n\n## Bayesian mixed linear regression\n\nput everything together\n\n",
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