ppl_hbm.jl

### A Pluto.jl notebook ###
# v0.19.6

using Markdown
using InteractiveUtils

# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error).
macro bind(def, element)
    quote
        local iv = try Base.loaded_modules[Base.PkgId(Base.UUID("6e696c72-6542-2067-7265-42206c756150"), "AbstractPlutoDingetjes")].Bonds.initial_value catch; b -> missing; end
        local el = $(esc(element))
        global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : iv(el)
        el
    end
end

# ╔═╡ ab655734-a8b7-47db-9b73-4099c4b11dfc
begin
	using CSV, DataFrames

	#using LinearAlgebra, PDMats
	using Optim, ForwardDiff, Tracker #ReverseDiff#Zygote

	using Turing, DynamicPPL, Distributions
	using MCMCChains
	using FillArrays

	using Plots, StatsPlots, LaTeXStrings#, PairPlots, , ColorSchemes

	using PlutoUI, HypertextLiteral

	# Set a seed for reproduciblity.
	using Random
	Random.seed!(0)
	#using StableRNGs;
	#rng = StableRNG(42);

	Turing.setadbackend(:tracker)

	if false
		# Hide the progress prompt while sampling.
		#Turing.setprogress!(false);
		import Logging
		Logging.disable_logging(Logging.Warn)
	end
end;

# ╔═╡ 1c651c38-d151-11ec-22a6-87180bd6546a
md"""
# Lab 13: Hierarchical Bayesian Modeling with a PPL
#### [Penn State Astroinformatics Summer School 2022](https://sites.psu.edu/astrostatistics/astroinfo-su22-program/)
#### [Eric Ford](https://www.personal.psu.edu/ebf11)
"""

# ╔═╡ c217818e-03c6-4303-b723-3f4e7263dccb
md"""
## Overview

In this lab, we'll analyze the eccentricity distribution of a population of planets using a hierarchical Bayesian Model and the [Turing](https://turing.ml/) probabilistic programming language (PPL).  
You'll be able to generate synthetic data sets from multiple intrinsic distributions, choose the magnitude of measurement uncertainties, and compare the true intrinsic distribution to the inferred distribution.  

The purpose of the lab is to demonstrate how you can implement a hierarchical Bayesian analysis quickly with a probabilistic programming language without having to compute a bunch of conditional probability distributions yourself (either because your model doesn't lend itself to analytic expressions or you just don't want to spend your time doing that).  

Similar to the other lab for hierarchical Bayesian modeling, you'll be able to inspect how the posterior distributions for individual objects shrink in a hierarchical model.  If you have extra time, then you'll also be able to explore what happens when you analyze your population using a model that differs from the true model.  
"""

# ╔═╡ 63a4e3e4-1645-4715-90b1-b708379706ef
md"""
## Set Properties of the Synthetic Data set
First, you'll select a model to use for generating a synthetic population of planets.
Second, you'll set the prior values needed by that model.
Then, you'll set the number of planets to include in our synthetic data set and the magnitude of measurement uncertainties.  
"""

# ╔═╡ 2487706a-f408-4cc4-901a-f2b72fe562d4
md"""
## Generate dataset
The distribution you selected is stored in `dist_gen` and a sampler for that model is stored in `sampling_alg`.
There's some helper code at the bottom of the notebook that collects the inputs above into the variable `param_true`.  The code below conditions the generative model you selected on the true population parameters and draws a realization of the data conditioned on the true population parameters.  That draw provides both the "true" eccentricities and the "observed eccentricities" that you'll analyze later.
"""

# ╔═╡ 90ae54f1-c49d-4d66-82eb-eb783c389e54
md"""
### Plot the data
It's often good to make a quick plot to help understand the characteristics of the data.  

Number of draws:    $(@bind num_draws NumberField(100:100:10000, default=400))
Number of bins for histogram:  $(@bind num_bins NumberField(10:10:200, default=40))
"""

# ╔═╡ def5056e-4817-4602-bd70-97f5798a92f1
md"""
**Question:** For what values of the measurement uncertainty is there a significant difference between the observed distribution and the intrinsic distribution?

**Question:** How would your response above change if you increase the number of planets by a factor of 2?  10?

**Question:** Try changing the parameters of the intrinsic distribution to result in a more/less narrowly peaked intrinsic eccentricity distribution.  How does that affect the number of planets required before the observed and intrinsic distributions are significantly different?
"""

# ╔═╡ 58b4382b-6a5e-43c0-8f93-7f5602efc230
md"## Analyze with a HBM"

# ╔═╡ df58fcff-bc39-44b5-866d-979b29f23c28
md"""
The model you selected for analyze the data with above is stored in the variable `param_mcmc.dist_infer`.  The next cell conditions that model on the data generated previously.
"""

# ╔═╡ a6a18eea-cca6-4cdd-bfab-a88169d0f84d
md"### Run MCMC"

# ╔═╡ 41308bbf-d857-427d-b83b-0b0527b86ce0
md"""Once you've checked the "Ready to run MCMC" box above and clicked submit, it will running a Markov chain.  This can take anywhere from several seconds to minutes or hours, depending on the choice of model, number of observations and length of the Markov chain.  Therefore, it's suggested that you start with a short Markov chain.  For the purposes of this lab, you may be able to work with short Markov chains, even if there's signs that they haven't fully converged."""

# ╔═╡ 6fae3f66-9e37-4617-aac0-6d3625f338c7
md"### Results for the population parameters"

# ╔═╡ e881c5af-b1da-45e6-8e43-fd81ac8f0daf
md"The sampler returns the results of one (or more) MCMC chains that can be used for inference.  For example, we can get some summary statistics about the model parameters and their convergence diagnostics. "

# ╔═╡ 3fb2aca3-e903-4488-b67f-15ff9b040b67
md"### Results for inferred population"

# ╔═╡ f6618ef6-10e0-4d5f-ad8c-16c86c3854c0
md"Compute inferred distribution $(@bind inferred_ready CheckBox() )"

# ╔═╡ b3ff4257-4758-4fc9-9a74-5150c41866f1
md"""
## Questions to Guide your Exploration
Once you've thought through the results for a first calculation, it's time to do some numerical experiment.  

### Effects of sample size and measurement uncertainty
First, choose the same model to generate a sample and to perform inference.  

**Question:**  Is the inferred distribution a good approximation to the intrinsic distribution?  Is it significantly better than treating the observed distribution as an approximation to the intrinsic distribution?  

**Question:**  How does the above change if you increase the number of planets by a factor of 2?  

**Question:**  Try increasing the measurement uncertainty to 0.2.  Can you still recover the intrinsic distribution accurately by increasing the number of planets?
"""

# ╔═╡ 667a3efc-1746-4997-8b85-04e96a0183cf
md"""
### Posterior for each Planet's Eccentricity
We can also look at the posterior distribution for each planet's eccentricity (blue) and compare it to the measurement for that planet (red) and the true value (green).  Make sure to use the slider to inspect several planets.
"""

# ╔═╡ d322b5de-62dd-4c6e-af05-3452af511c62
md"""
**Question:**  How does the width of the posterior compare to the width of the measurement?  How do you expect that would change if you vary the measurement precision or the number of of planets in your sample?

**Question:**  Do you recognize a trend for location of the mode of the posterior relative to the mode of the measurement distribution?  Why is that?  How would the difference change in you chose an intrinsic distribution that was less sharply peaked?

"""

# ╔═╡ 1b85cf6a-6300-4dd2-a764-670570bde946
md"""
### Effects of model misspecification (optional)
If you still have time, consider choosing one model to generate the data and a different model to perform inference with.  

**Question:** How does the distribution inferred with the wrong model compare to the distribution inferred with the correct model?  

**Question:** Under what circumstances are some quantitative or qualitative properties robustly inferred, even when using the wrong model?  Why factor contribute to that?

"""

# ╔═╡ a480bd7a-a4a9-43c7-81f5-77d15ec16289
md"""
# Setup & Helper Code
"""

# ╔═╡ cf44e114-4ac5-4bf1-9b5a-ef6b4f5e6bc9
md"## Code for Turing Models Used"

# ╔═╡ 8c5b0d18-70bd-4fd4-9f94-99c67b3f8bc0
md"### Beta Distribution"

# ╔═╡ 6cb2fbfb-646e-4e5c-ae5c-c3146f19d76c
begin
	@model function eccentricities_beta(e_obs, σₑ, ::Type{T} = Float64) where {T}
    @assert size(σₑ) == size(e_obs)
	N = size(e_obs,1)
	@assert N >= 1
    # Prior for hyperparameter(s)
	α ~ truncated(Exponential(),lower=0.5)
	β ~ truncated(Exponential(),lower=0.5)
	dist_e = Beta(α,β)
	e_latent = Vector{T}(undef, N)
	# Specify liklihood
	for i ∈ eachindex(e_obs)
		e_latent[i] ~ dist_e  # Latent variables
    	e_obs[i]  ~ truncated(Normal(e_latent[i], σₑ[i]),0,1)  # Observed variables
    end
	return (;e_latent, e_obs)
	end

	# Convenience functions
	eccentricities_beta(df::DataFrame) = eccentricities_beta(df.e_obs, df.σₑ)
	eccentricities_beta(n::Integer, σ = 0.0) = eccentricities_beta( missings(n), fill(σ,n) )

	# Sampler to use for this model
	sampler_beta = NUTS(0.65)
end

# ╔═╡ 0cef66f6-44c4-4f3e-b901-ac8d33bad4a7
md"### Rayleigh Distribution"

# ╔═╡ b8ab2460-7c59-4e87-8580-e7b49f0576aa
begin
	@model function eccentricities_rayleigh(e_obs, σₑ, ::Type{T} = Float64) where {T}
    @assert size(σₑ) == size(e_obs)
	N = size(e_obs,1)
	@assert N >= 1
	# Prior for hyperparameter(s)
	σᵣ ~ Uniform(1e-4, 0.5)
	e_latent = Vector{T}(undef, N)
	# Specify liklihood
	for i ∈ eachindex(e_obs)
		e_latent[i] ~ truncated(Rayleigh(σᵣ), lower=1e-4, upper=1)  # Latent variables
    	e_obs[i]  ~ truncated(Normal(e_latent[i], σₑ[i]),0,1)  # Observed variables
    end
	return (;e_latent, e_obs)
	end

	# Convenience functions
	eccentricities_rayleigh(df::DataFrame) = eccentricities_rayleigh(df.e_obs, df.σₑ)
	eccentricities_rayleigh(n::Integer, σ = 0.0) = eccentricities_rayleigh( missings(n), fill(σ,n) )

	# Sampler to use for this model
	sampler_rayleigh = NUTS(0.65)
end

# ╔═╡ 0b987316-b554-4497-9731-765dfe3a514d
md"## Appearances"

# ╔═╡ d78db46d-1e20-4cd1-be53-81f635561b27
TableOfContents()

# ╔═╡ deb1a219-f33c-469c-98a9-5bab181e4a43
function aside(x)
        @htl("""
                <style>


                @media (min-width: calc(700px + 30px + 300px)) {
                        aside.plutoui-aside-wrapper {
                                position: absolute;
                                right: -11px;
                                width: 0px;
                        }
                        aside.plutoui-aside-wrapper > div {
                                width: 300px;
                        }
                }
                </style>

                <aside class="plutoui-aside-wrapper">
                <div>
                $(x)
                </div>
                </aside>

                """)
end


# ╔═╡ e5fd7d7b-92dc-4232-8295-f81e39385ca3
aside(md"""
!!! tip "Tip:  Choices here affect calculation time below"
	Both the number of planets in our sample (set here) and the length of the Markov chains (set later) that you choose will affect the run-time (roughly proportional to the product for a given mode).  

	It's suggested that you start with a small data set, so you get more prompt feedback (even if the Markov chains still show some signs that they haven't fully converged and you wouldn't use them in a science paper). Near the end of the lab, you may want to try analyzing a larger data set and/or running a longer MCMC calculations (e.g., while you take a coffee break).
""")

# ╔═╡ c8ea0fb7-f8e6-4cdf-b81b-b1eb9d3a5f7d
nbsp = html"&nbsp;"

# ╔═╡ cede1027-5425-448a-9c86-70d27a0942de
br = html"<br />"

# ╔═╡ 57bd6ed3-1382-40e7-8422-8ec9cb50e23f
begin
	max_num_planets = 500
	default_num_planets = 50
	default_σₑ = 0.1
end;

# ╔═╡ c62ef364-ae3a-4b41-b808-567fef6f895e
function pick_param_uniform()
	@bind param_uniform confirm(
	PlutoUI.combine() do Child
md"""
min e: $(Child("min_e",NumberField(0.0:0.001:0.2,default=0.0)))  
max e: $(Child("max_e",NumberField(0.01:0.01:1,default=0.5)))  

Number of Planets: $(Child("num_pl",NumberField(1:max_num_planets,default=default_num_planets)))  
Measurement uncertainty:  $(Child("σₑ",NumberField(0:0.01:1,default=default_σₑ)))  
"""
	end
	)
end

# ╔═╡ 230ad18a-c830-4e86-9edb-83e541994006
function pick_param_beta()
	# Default values from Kipping 2013
	@bind param_beta confirm(
	PlutoUI.combine() do Child
md"""
α: $(Child("α",NumberField(0.5:0.01:10,default=0.867)))
β: $(Child("β",NumberField(0.5:0.01:10,default=3.03)))  

Number of Planets: $(Child("num_pl",NumberField(1:max_num_planets,default=default_num_planets)))  
Measurement uncertainty:  $(Child("σₑ",NumberField(0:0.01:1,default=default_σₑ)))  
"""
	end
	)
end

# ╔═╡ 981a9b4d-d4cd-4497-99c9-f0747e042f0b
function pick_param_rayleigh()
	@bind param_rayleigh confirm(
	PlutoUI.combine() do Child
md"""
σᵣ: $(Child("σᵣ",NumberField(0.005:0.005:1,default=0.01)))  

Number of Planets: $(Child("num_pl",NumberField(1:max_num_planets,default=default_num_planets)))  
Measurement uncertainty:  $(Child("σₑ",NumberField(0:0.01:1,default=default_σₑ)))  
"""
	end
	)
end

# ╔═╡ ef766433-4510-424c-b63a-ab9a81a7461e
begin  # Set choices for distributions
	model_list = [
		eccentricities_rayleigh,
		eccentricities_beta,
			]
	model_name_list = [
		"Rayleigh",
		"Beta",
		]
	model_choices = model_list .=> model_name_list
end;

# ╔═╡ b6ecfca2-05c9-4f6b-b09a-6118557ba616
md"Generative Model: $(@bind dist_gen Select(model_choices))"
#=
@bind param_gen confirm(
	PlutoUI.combine() do Child
md"""
Generative Model: $(Child("dist_gen",Select(model_choices)))  
"""
	end
)
=#

# ╔═╡ 32193e43-cbd6-4414-9044-f8bc58d13d90
if dist_gen == eccentricities_rayleigh
	sampling_alg = sampler_rayleigh
	pick_param_rayleigh()
elseif dist_gen == eccentricities_rayleigh_mixture
	sampling_alg = sampler_rayleigh_mixture
	pick_param_rayleigh_mixture()
elseif dist_gen == eccentricities_beta
	sampling_alg = sampler_beta
	pick_param_beta()
elseif dist_gen == eccentricities_uniform
	sampling_alg = sampler_uniform
	pick_param_uniform()
else
	md"""
	!!! error "Notebook doesn't have code to set generative model parameters."
	"""
end

# ╔═╡ b0c94cb0-f992-4ebf-86f0-66d3ebc4aa9e
@bind param_mcmc confirm(
	PlutoUI.combine() do Child
md"""
Inferential Model:  $(Child("dist_infer",Select(model_choices)))  

Length of Markov chains:  $(Child("num_iterations",NumberField(100:100:5000,default=100)))  

Ready to run MCMC?   $(Child("ready",CheckBox()))  
"""
	end
)
#Try to initialize from MAP? $(Child("init_map",CheckBox(default=false)))  

# ╔═╡ 921a7aa8-f96a-4593-962b-d97be4d7014c
if dist_gen == eccentricities_rayleigh
	num_pl = param_rayleigh.num_pl
	param_true = param_rayleigh
elseif dist_gen == eccentricities_beta
	num_pl = param_beta.num_pl
	param_true = param_beta
else
	num_pl = nothing
	param_true = nothing
	md"""
	!!! error "Notebook doesn't have code to set generative model parameters."
	"""
end

# ╔═╡ 106f5b0a-ab06-450f-b541-56ca64d7347f
begin # Generate synthetic dataset
	# Create model conditioned on _missing_ values for e_obs, but numerical values for σₑ
	model_use = dist_gen(num_pl, param_true.σₑ)    

	# Create model that also conditions on the true population parameter values
	model_true = condition(model_use, param_true )

	# Draw data conditioned on true population parameters
	data = model_true()       

	# Cosmetic details
	data = DataFrame(data, copycols=false)         # Repackage data into DataFrame
	rename!(data,:e_latent=>:e_true)               # Rename column
	data.σₑ .= param_true.σₑ                       # Add measurement uncertainties
	data
end;

# ╔═╡ ffae4c63-78db-4ce0-bf25-ab53845db6e8
begin
	samples_true = mapreduce(i->DataFrame(model_true()),append!, 1:num_draws, init=DataFrame())
end;

# ╔═╡ efcabcc4-40ed-4492-b3e4-e190be248fce
let
	max_e_plt = max(maximum(samples_true.e_obs),maximum(samples_true.e_latent))
	bins_plt = range(0,stop=max_e_plt,length=num_bins)
	plt1 = plot()
	histogram!(plt1,data.e_true, bins=bins_plt, alpha=0.8, label="Actual e's")
	histogram!(plt1,data.e_obs, bins=bins_plt, alpha=0.8, label="Observed e's")
	ylabel!(plt1,"Count")

	plt2 = plot()
	histogram!(plt2,samples_true.e_latent,
		weights=repeat([num_bins/(max_e_plt*length(samples_true.e_latent))],length(samples_true.e_latent)),
		bins=bins_plt, alpha=0.25, label="Intrinsic Distribution")
	histogram!(plt2,samples_true.e_obs,
		weights=repeat([num_bins/(max_e_plt*length(samples_true.e_latent))],length(samples_true.e_latent)),
		bins=bins_plt, alpha=0.25, label="Observed Distribution")
	ylabel!(plt2,"PDF")
	xlabel!(plt2, "Eccentricity")
	l = @layout [a;b]
	plot(plt1,plt2, layout=l)
end

# ╔═╡ 2240141a-5c05-4b4c-8c74-0aea660d76fd
(; mean_e_obs = mean(data.e_obs), mean_e_true = mean(data.e_true) )

# ╔═╡ 97c7a26f-635b-416f-a475-fb3cef270b26
if false # Density plots
	plt = plot()
	density!(data.e_true, label="KDE of actual e's")
	density!(data.e_obs, label="KDE of observed e's")
	xlabel!("Eccentricity")
	ylabel!("PDF")
end

# ╔═╡ 614bf8fe-f34e-403e-9399-85045d787ac3
model = param_mcmc.dist_infer(data)

# ╔═╡ 6a23b1e6-fd0f-406d-bc93-c713cecc8fe3
if inferred_ready
	md"Observation number to plot: $(@bind plt_pl_id Slider(1:num_pl))"
end

# ╔═╡ d614f82e-a233-4896-8b9b-26a357f28ff0
if param_mcmc.dist_infer == eccentricities_rayleigh
	param_init = (;σᵣ = 0.3)
elseif param_mcmc.dist_infer == eccentricities_beta
	param_init = (; α = 1, β = 1)
else
	param_init = nothing
	md"""
	!!! error "Notebook doesn't have code to set generative model parameters."
	"""
end

# ╔═╡ 5e002142-7391-4536-aa95-2a422daff27c
begin
	opt_start_param = merge( param_init,
		NamedTuple([Symbol("e_latent[$i]") for i in 1:num_pl]  .=> 	data.e_obs ) )
	param_start_mcmc = opt_start_param
end;

# ╔═╡ 1123ba8b-e638-4a6e-96d3-c51d085b4dc7
param_start_mcmc

# ╔═╡ 741dfd1a-8d46-4458-9f9c-ecd1f62d8da2
if param_mcmc.ready    # Only run MCMC if Ready box is checked above
	# Run MCMC starting from MAP
	if length(param_start_mcmc) > 0

		chain = sample(model, sampling_alg, param_mcmc.num_iterations, init_params=param_start_mcmc)
	else
		chain = sample(model, sampling_alg, param_mcmc.num_iterations)
	end
end;

# ╔═╡ c61b6695-8d7a-4b1c-97b0-c8df28cc6723
if param_mcmc.ready && inferred_ready
	#sample_from_chain = sample(chain,2*num_draws)
	sample_from_chain = sample(chain,min(max(num_draws,100),param_mcmc.num_iterations))
	chain_post_pred = predict(model, sample_from_chain, include_all=true)
end;

# ╔═╡ b6515c1c-8e0c-4084-8be3-8021f5d497bc
if param_mcmc.ready && inferred_ready
	max_e_plt = max(maximum(data.e_obs), maximum(samples_true.e_latent))
	bins_plt = range(0,stop=max_e_plt,length=num_bins)
	samples_post_pred_latent = vec(Array(group(chain_post_pred,"e_latent")))
	plt1 = histogram(samples_true.e_latent,
		weights=repeat([num_bins/(max_e_plt*length(samples_true.e_latent))],length(samples_true.e_latent)),
		bins=bins_plt, alpha=0.5, label="Intrinsic Distribution")
	histogram!(vec(Array(group(chain_post_pred,"e_latent"))),
		weights=repeat([num_bins/(max_e_plt*length(samples_post_pred_latent))],length(samples_post_pred_latent)),
		bins=bins_plt, alpha=0.5, label="Inferred Distribution")
	ylabel!("PDF")
	plt2 = histogram(data.e_obs,
		weights=repeat([num_bins/(max_e_plt*length(data.e_obs))],length(data.e_obs)),
		bins=bins_plt, alpha=0.75, label="Observed e's")
	histogram!(plt2,samples_true.e_obs,
		weights=repeat([num_bins/(max_e_plt*length(samples_true.e_latent))],length(samples_true.e_latent)),
		bins=bins_plt, alpha=0.25, label="Observed e's distribution")
	ylabel!("PDF")
	xlabel!("Eccentricity")
	l = @layout [a;b]
	plot(plt1,plt2,layout=l)
	#histogram!(samples_true.e_obs,nbins=range(0,stop=1,length=num_bins), alpha=0.5, label="Observed Distribution")
end

# ╔═╡ 039fd2fb-ab74-4d11-8144-e4264def99cf
if param_mcmc.ready && inferred_ready
	plot(xlabel=latexstring("e_{" * string(plt_pl_id) * "}"),
		ylabel = "Posterior PDF", xlims=(0,min(1,max(data.e_obs[plt_pl_id]+2.5*data.σₑ[plt_pl_id],maximum(chain_post_pred["e_latent[$plt_pl_id]"])))) )
	density!(chain_post_pred["e_latent[$plt_pl_id]"],
		label="Posterior", nbins=40)
	plot!(Normal(data.e_obs[plt_pl_id], data.σₑ[plt_pl_id]), label="Measurement")
	plot!(fill(data.e_true[plt_pl_id],2),[0,1], label=:none)
end

# ╔═╡ 6d4a9e19-9511-45ff-b74a-a37f0caaa16a
begin
	pop_params = param_true[setdiff(keys(param_true), [:num_pl, :σₑ])]
	pop_param_names = collect(keys(pop_params))
	infer_pop_param_names = collect(keys(param_init))
end;

# ╔═╡ 9e24813e-96a1-4fe5-81e7-def708d150c9
if param_mcmc.ready
	chain_pop_param = chain[infer_pop_param_names]
	summarize(chain_pop_param)
end

# ╔═╡ d452c7f4-2f04-4972-ada9-cd4e71cd787f
if param_mcmc.ready
	traceplot(chain_pop_param)
end

# ╔═╡ 8de22fbb-5a4d-40dd-a6e3-9d6789d5d923
if param_mcmc.ready
	histogram(chain_pop_param, nbins=num_bins)
end

# ╔═╡ 00000000-0000-0000-0000-000000000001
PLUTO_PROJECT_TOML_CONTENTS = """
[deps]
CSV = "336ed68f-0bac-5ca0-87d4-7b16caf5d00b"
DataFrames = "a93c6f00-e57d-5684-b7b6-d8193f3e46c0"
Distributions = "31c24e10-a181-5473-b8eb-7969acd0382f"
DynamicPPL = "366bfd00-2699-11ea-058f-f148b4cae6d8"
FillArrays = "1a297f60-69ca-5386-bcde-b61e274b549b"
ForwardDiff = "f6369f11-7733-5829-9624-2563aa707210"
HypertextLiteral = "ac1192a8-f4b3-4bfe-ba22-af5b92cd3ab2"
LaTeXStrings = "b964fa9f-0449-5b57-a5c2-d3ea65f4040f"
Logging = "56ddb016-857b-54e1-b83d-db4d58db5568"
MCMCChains = "c7f686f2-ff18-58e9-bc7b-31028e88f75d"
Optim = "429524aa-4258-5aef-a3af-852621145aeb"
Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8"
Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"
StatsPlots = "f3b207a7-027a-5e70-b257-86293d7955fd"
Tracker = "9f7883ad-71c0-57eb-9f7f-b5c9e6d3789c"
Turing = "fce5fe82-541a-59a6-adf8-730c64b5f9a0"

[compat]
CSV = "~0.10.4"
DataFrames = "~1.3.4"
Distributions = "~0.25.58"
DynamicPPL = "~0.19.1"
FillArrays = "~0.13.2"
ForwardDiff = "~0.10.30"
HypertextLiteral = "~0.9.4"
LaTeXStrings = "~1.3.0"
MCMCChains = "~5.3.0"
Optim = "~1.7.0"
Plots = "~1.29.0"
PlutoUI = "~0.7.38"
StatsPlots = "~0.14.34"
Tracker = "~0.2.20"
Turing = "~0.21.2"
"""

# ╔═╡ 00000000-0000-0000-0000-000000000002
PLUTO_MANIFEST_TOML_CONTENTS = """
# This file is machine-generated - editing it directly is not advised

julia_version = "1.7.0"
manifest_format = "2.0"

[[deps.AbstractFFTs]]
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version = "1.1.0"

[[deps.AbstractMCMC]]
deps = ["BangBang", "ConsoleProgressMonitor", "Distributed", "Logging", "LoggingExtras", "ProgressLogging", "Random", "StatsBase", "TerminalLoggers", "Transducers"]
git-tree-sha1 = "47aca4cf0dc430f20f68f6992dc4af0e4dc8ebee"
uuid = "80f14c24-f653-4e6a-9b94-39d6b0f70001"
version = "4.0.0"

[[deps.AbstractPPL]]
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version = "0.5.2"

[[deps.AbstractPlutoDingetjes]]
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[[deps.AbstractTrees]]
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[[deps.Adapt]]
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[[deps.AdvancedHMC]]
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git-tree-sha1 = "345effa84030f273ee86fcdd706d8484ce9a1a3c"
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version = "0.3.5"

[[deps.AdvancedMH]]
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[[deps.AdvancedPS]]
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git-tree-sha1 = "9ff1247be1e2aa2e740e84e8c18652bd9d55df22"
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[[deps.AdvancedVI]]
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git-tree-sha1 = "e743af305716a527cdb3a67b31a33a7c3832c41f"
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[[deps.ArgCheck]]
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[[deps.ArgTools]]
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[[deps.Arpack_jll]]
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[[deps.Artifacts]]
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[[deps.AxisAlgorithms]]
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[[deps.AxisArrays]]
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[[deps.BangBang]]
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[[deps.Base64]]
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[[deps.Baselet]]
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[[deps.Bijectors]]
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[[deps.Bzip2_jll]]
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[[deps.CEnum]]
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[[deps.CSV]]
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[[deps.Cairo_jll]]
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[[deps.Calculus]]
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[[deps.ChainRules]]
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[[deps.ChainRulesCore]]
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[[deps.ColorVectorSpace]]
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[[deps.StatisticalTraits]]
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[[deps.Statistics]]
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[[deps.StatsAPI]]
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[[deps.StatsBase]]
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[[deps.StatsFuns]]
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[[deps.TOML]]
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[[deps.TableOperations]]
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[[deps.TableTraits]]
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[[deps.Tar]]
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[[deps.TensorCore]]
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[[deps.TerminalLoggers]]
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[[deps.Test]]
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[[deps.Tracker]]
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[[deps.TranscodingStreams]]
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[[deps.Transducers]]
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[[deps.TreeViews]]
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[[deps.Tricks]]
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[[deps.Turing]]
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[[deps.URIs]]
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[[deps.UUIDs]]
deps = ["Random", "SHA"]
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[[deps.UnPack]]
git-tree-sha1 = "387c1f73762231e86e0c9c5443ce3b4a0a9a0c2b"
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[[deps.Unicode]]
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[[deps.UnicodeFun]]
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[[deps.Unzip]]
git-tree-sha1 = "34db80951901073501137bdbc3d5a8e7bbd06670"
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[[deps.Wayland_jll]]
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git-tree-sha1 = "3e61f0b86f90dacb0bc0e73a0c5a83f6a8636e23"
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[[deps.Wayland_protocols_jll]]
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git-tree-sha1 = "4528479aa01ee1b3b4cd0e6faef0e04cf16466da"
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[[deps.WeakRefStrings]]
deps = ["DataAPI", "InlineStrings", "Parsers"]
git-tree-sha1 = "b1be2855ed9ed8eac54e5caff2afcdb442d52c23"
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[[deps.Widgets]]
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[[deps.WoodburyMatrices]]
deps = ["LinearAlgebra", "SparseArrays"]
git-tree-sha1 = "de67fa59e33ad156a590055375a30b23c40299d3"
uuid = "efce3f68-66dc-5838-9240-27a6d6f5f9b6"
version = "0.5.5"

[[deps.XML2_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libiconv_jll", "Pkg", "Zlib_jll"]
git-tree-sha1 = "1acf5bdf07aa0907e0a37d3718bb88d4b687b74a"
uuid = "02c8fc9c-b97f-50b9-bbe4-9be30ff0a78a"
version = "2.9.12+0"

[[deps.XSLT_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libgcrypt_jll", "Libgpg_error_jll", "Libiconv_jll", "Pkg", "XML2_jll", "Zlib_jll"]
git-tree-sha1 = "91844873c4085240b95e795f692c4cec4d805f8a"
uuid = "aed1982a-8fda-507f-9586-7b0439959a61"
version = "1.1.34+0"

[[deps.Xorg_libX11_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll", "Xorg_xtrans_jll"]
git-tree-sha1 = "5be649d550f3f4b95308bf0183b82e2582876527"
uuid = "4f6342f7-b3d2-589e-9d20-edeb45f2b2bc"
version = "1.6.9+4"

[[deps.Xorg_libXau_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4e490d5c960c314f33885790ed410ff3a94ce67e"
uuid = "0c0b7dd1-d40b-584c-a123-a41640f87eec"
version = "1.0.9+4"

[[deps.Xorg_libXcursor_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXfixes_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "12e0eb3bc634fa2080c1c37fccf56f7c22989afd"
uuid = "935fb764-8cf2-53bf-bb30-45bb1f8bf724"
version = "1.2.0+4"

[[deps.Xorg_libXdmcp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fe47bd2247248125c428978740e18a681372dd4"
uuid = "a3789734-cfe1-5b06-b2d0-1dd0d9d62d05"
version = "1.1.3+4"

[[deps.Xorg_libXext_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "b7c0aa8c376b31e4852b360222848637f481f8c3"
uuid = "1082639a-0dae-5f34-9b06-72781eeb8cb3"
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[[deps.Xorg_libXfixes_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "0e0dc7431e7a0587559f9294aeec269471c991a4"
uuid = "d091e8ba-531a-589c-9de9-94069b037ed8"
version = "5.0.3+4"

[[deps.Xorg_libXi_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXfixes_jll"]
git-tree-sha1 = "89b52bc2160aadc84d707093930ef0bffa641246"
uuid = "a51aa0fd-4e3c-5386-b890-e753decda492"
version = "1.7.10+4"

[[deps.Xorg_libXinerama_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll"]
git-tree-sha1 = "26be8b1c342929259317d8b9f7b53bf2bb73b123"
uuid = "d1454406-59df-5ea1-beac-c340f2130bc3"
version = "1.1.4+4"

[[deps.Xorg_libXrandr_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "34cea83cb726fb58f325887bf0612c6b3fb17631"
uuid = "ec84b674-ba8e-5d96-8ba1-2a689ba10484"
version = "1.5.2+4"

[[deps.Xorg_libXrender_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "19560f30fd49f4d4efbe7002a1037f8c43d43b96"
uuid = "ea2f1a96-1ddc-540d-b46f-429655e07cfa"
version = "0.9.10+4"

[[deps.Xorg_libpthread_stubs_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "6783737e45d3c59a4a4c4091f5f88cdcf0908cbb"
uuid = "14d82f49-176c-5ed1-bb49-ad3f5cbd8c74"
version = "0.1.0+3"

[[deps.Xorg_libxcb_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "XSLT_jll", "Xorg_libXau_jll", "Xorg_libXdmcp_jll", "Xorg_libpthread_stubs_jll"]
git-tree-sha1 = "daf17f441228e7a3833846cd048892861cff16d6"
uuid = "c7cfdc94-dc32-55de-ac96-5a1b8d977c5b"
version = "1.13.0+3"

[[deps.Xorg_libxkbfile_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "926af861744212db0eb001d9e40b5d16292080b2"
uuid = "cc61e674-0454-545c-8b26-ed2c68acab7a"
version = "1.1.0+4"

[[deps.Xorg_xcb_util_image_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "0fab0a40349ba1cba2c1da699243396ff8e94b97"
uuid = "12413925-8142-5f55-bb0e-6d7ca50bb09b"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll"]
git-tree-sha1 = "e7fd7b2881fa2eaa72717420894d3938177862d1"
uuid = "2def613f-5ad1-5310-b15b-b15d46f528f5"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_keysyms_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "d1151e2c45a544f32441a567d1690e701ec89b00"
uuid = "975044d2-76e6-5fbe-bf08-97ce7c6574c7"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_renderutil_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "dfd7a8f38d4613b6a575253b3174dd991ca6183e"
uuid = "0d47668e-0667-5a69-a72c-f761630bfb7e"
version = "0.3.9+1"

[[deps.Xorg_xcb_util_wm_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "e78d10aab01a4a154142c5006ed44fd9e8e31b67"
uuid = "c22f9ab0-d5fe-5066-847c-f4bb1cd4e361"
version = "0.4.1+1"

[[deps.Xorg_xkbcomp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxkbfile_jll"]
git-tree-sha1 = "4bcbf660f6c2e714f87e960a171b119d06ee163b"
uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4"
version = "1.4.2+4"

[[deps.Xorg_xkeyboard_config_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xkbcomp_jll"]
git-tree-sha1 = "5c8424f8a67c3f2209646d4425f3d415fee5931d"
uuid = "33bec58e-1273-512f-9401-5d533626f822"
version = "2.27.0+4"

[[deps.Xorg_xtrans_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "79c31e7844f6ecf779705fbc12146eb190b7d845"
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
version = "1.4.0+3"

[[deps.Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"

[[deps.Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "e45044cd873ded54b6a5bac0eb5c971392cf1927"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.2+0"

[[deps.ZygoteRules]]
deps = ["MacroTools"]
git-tree-sha1 = "8c1a8e4dfacb1fd631745552c8db35d0deb09ea0"
uuid = "700de1a5-db45-46bc-99cf-38207098b444"
version = "0.2.2"

[[deps.libass_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "5982a94fcba20f02f42ace44b9894ee2b140fe47"
uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
version = "0.15.1+0"

[[deps.libblastrampoline_jll]]
deps = ["Artifacts", "Libdl", "OpenBLAS_jll"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"

[[deps.libfdk_aac_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "daacc84a041563f965be61859a36e17c4e4fcd55"
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
version = "2.0.2+0"

[[deps.libpng_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "94d180a6d2b5e55e447e2d27a29ed04fe79eb30c"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.38+0"

[[deps.libvorbis_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"]
git-tree-sha1 = "b910cb81ef3fe6e78bf6acee440bda86fd6ae00c"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.7+1"

[[deps.nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"

[[deps.p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"

[[deps.x264_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2"
uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
version = "2021.5.5+0"

[[deps.x265_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9"
uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
version = "3.5.0+0"

[[deps.xkbcommon_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
git-tree-sha1 = "ece2350174195bb31de1a63bea3a41ae1aa593b6"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "0.9.1+5"
"""

# ╔═╡ Cell order:
# ╟─1c651c38-d151-11ec-22a6-87180bd6546a
# ╟─c217818e-03c6-4303-b723-3f4e7263dccb
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