section7_glm.jl

### A Pluto.jl notebook ###
# v0.19.46

using Markdown
using InteractiveUtils

# ╔═╡ 88154488-233d-11ed-1072-3dc0dd25d3fd
begin
    using PlutoUI
	using StatsPlots
	using PlutoTeachingTools
	using Plots; default(fontfamily="Computer Modern", framestyle=:box) # LaTex-style
    using Turing
	using DataFrames
    using Random
	using LaTeXStrings
	using Logging; Logging.disable_logging(Logging.Warn);
end;

# ╔═╡ 52431ba0-762b-42d4-ab91-148998c90600
begin
	using CSV
	data_url= "https://exeter-data-analytics.github.io/StatModelling/_data/cuckoo.csv"
	df_cuckoo=CSV.File(download(data_url)) |> DataFrame
end;

# ╔═╡ 89a55d82-d5a9-40cd-9247-9d5355d4fd7c
begin
	using StatsBase
	df_cuckoo[!, :Mass] = zscore(df_cuckoo[!, :Mass])
end;

# ╔═╡ ec93d30f-9d57-4598-8f64-ca706c4f0160
using GLM

# ╔═╡ fbb41799-8ce2-4286-be48-b6977930b6be
using StatsModels

# ╔═╡ 623efa9d-485d-4d9c-8e6d-aedb7dcbc00d
TableOfContents()

# ╔═╡ cc95fefe-9aee-4f16-97bd-9a0b8d09267e
md"[**↩ Home**](https://lf28.github.io/BayesianModelling/) 


[**↪ Next Chapter**](./section8_nonlinear.html)
"

# ╔═╡ 0e52bc27-41e9-4361-a036-77eee3774e46
md"""
# Bayesian generalised linear models

Logistic regression can be viewed as a specific kind of generalised regression when the target ``y_n \in \{1,0\}`` is binary data. There are also other possible data types for ``y_n``, such as count data where the target takes integer values, *i.e.* ``y_n \in \mathbb{N}``, or ``y_n\in \mathbb{R}^+`` when the target can only take positive real values. 



From a data generation perspective, we can simply choose the appropriate likelihood function such that the support matches the target's data type. For example, for real-valued data, a Gaussian is a good choice, while for binary data, a `Bernoulli` is a natural choice. In classific frequentist statistical terms, such a flexible regression model is called **generalised linear model** (GLM).


In this chapter, we are going to see how Bayesian GLMs can be used to do more flexible regression modelling. We will recap the frequentist's models first and then extend them to their Bayesian equivalences.

## A recap on frequentist's GLMs



A **generalised linear model** can be specified as 

```math
p(y_n|\mathbf{x}_n, \boldsymbol{\beta}_1, \beta_0, \theta) = \mathcal{P}(y_n; g^{-1}(\mu_n), \theta)
```
where ``\mu`` as before is a linear function of the input ``\mathbf{x}_n``:
```math
\mu(\mathbf{x}_n) = \beta_0+\boldsymbol{\beta}_1^\top \mathbf{x}_n,
```
and ``\theta`` is some optional parameters associated with the likelihood function ``\mathcal P`` (for example, the observation's noise for the linear regression model).

Based on the above general definition, GLMs have three components:

1. A linear function ``\mu(\mathbf{x}_n)``, that's also why the model is called generalised **linear** models;
2. A suitable likelihood function ``\mathcal{P}``, such as Gaussian, Bernoulli, Poisson, *etc.*
3. A suitable **link function** ``g(\mu)`` that is associated with the requirement of the chosen likelihood function.

**Example (linear regression)** The likelihood function for linear regression is Gaussian: 

```math 
p(y_n|\mathbf{x}_n, \boldsymbol{\beta}) =  \mathcal{N}(y_n; \beta_0+\mathbf{x}_n^\top \boldsymbol{\beta}_1, \sigma^2).
``` The link function here is simply an identity function, *i.e.* ``g^{-1}(x) = x``, since the mean parameter of a Gaussian is unconstrained. An extra parameter is needed for the observation noise ``\sigma^2``. 


"""

# ╔═╡ f113ba37-1eeb-4101-8162-5402af63677a
md"
**Example (logistic regression)** Logistic regression assumes a ``\texttt{Bernoulli}`` likelihood: 

```math
p(y_n|\mathbf{x}_n, \boldsymbol{\beta}) = \texttt{Bernoulli}(y_n; \sigma(\beta_0+\mathbf{x}_n^\top \boldsymbol{\beta}_1)).
``` 

Since the bias of a Bernoulli distribution should be between 0 and 1, a suitable (inverse) link function is logistic function: ``g^{-1}(\mu) = \sigma(\mu)``, which is a ``\mathbb{R} \rightarrow [0,1]`` deterministic transformation.
"

# ╔═╡ 73c61d37-9c26-43b7-ae99-9a79c6fe9804
md"""

The following table has summarised some commonly used GLMs.


|Distribution | Support | Typical uses | Inverse Link ``g^{-1}(\cdot)`` |
|:------------|:---------|:--------------|:--------------|
|Normal  | ``\mathbb{R}, (-\infty, +\infty)``| Real valued regression|``\texttt{identity}(x) =x``|
| Bernoulli| ``\{0, 1\}``| binary classification| ``\texttt{logistic}(x)=\frac{1}{1+e^{-x}}``|
| Binomial| ``\{0, 1, \ldots, N\}``| count of # of positives out of N binary choices| ``\texttt{logistic}(x)=\frac{1}{1+e^{-x}}``|
| Categorical| ``\{1, 2,\ldots, K\}``| K class classification| ``\texttt{softmax}_i(\mathbf{x})= \frac{e^{x_i}}{\sum_{k=1}^K e^{x_k}}``|
| Poisson| ``\{0, 1,2,\ldots\}``| count of occurrences in fixed amount of time/space| ``\exp(x) = e^x``|
| Gamma/Exponential |  ``(0, +\infty)``| scale parameters | ``\texttt{negInv}(x)=-(x^{-1})``|

"""

# ╔═╡ 5d01251e-5ea2-4d2c-82c8-d9c576264887
md"""
## Bayesian GLMs 

The frequentist's GLMs have specified a likelihood function for the targets ``\mathbf{y}``:
```math
p(\mathbf{y}|\mathbf{X},  \beta_0, \boldsymbol{\beta}_1).
```
To fully specify a Bayesian model, we only need to add a prior distribution for the unknown regression parameters: ``p(\beta_0, \boldsymbol{\beta})``. Since the regression parameters take unconstrained real values, a sensible prior choice is therefore Gaussian, which was also used in the Bayesian linear and logistic regression models discussed earlier.

A general Bayesian GLM model therefore can be specified as:

!!! infor "Bayesian GLMs"
	```math
	\begin{align}
	\text{Priors: }\;\;\;\;\;\;\beta_0 &\sim \mathcal{N}(m_0^{\beta_0}, v_0^{\beta_0})\\
	\boldsymbol{\beta}_1 &\sim \mathcal{N}(\mathbf{m}_0^{\beta_1}, \mathbf{V}_0^{\beta_1})\\
	\text{Likelihood: } \;\; \text{for } n &= 1,2,\ldots, N:\\
	\mu_n &=\beta_0 + \boldsymbol{\beta}_1^\top \mathbf{x}_n \\
	y_n &\sim \mathcal{P}(y_n; g^{-1}(\mu_n)).
	\end{align}
	```


In the following section, we are going to use Poisson regression as a concrete example to demonstrate how to do GLMs in a Bayesian way in `Turing`.

"""

# ╔═╡ 8b8471cd-6a77-4d8a-bbd9-ee28e0aab1f6
md"""

## Bayesian Poisson Regression

To make the illustration easier, we will consider a popular GLM: Poisson regression. The regression model is a popular choice for counting data which can only take non-negative integer values, such as the daily visitor number of a website.


### Poisson model
A Poisson distribution is parameterised with a rate parameter ``\lambda>0``, which should be strictly positive. The distribution, with a probability distribution

```math
\texttt{Poisson}(x=k; \lambda) \triangleq p(x=k|\lambda)=  \frac{\lambda^k}{k!}e^{-\lambda}, \;\;\text{for } k= 0,1,2,\ldots,
```
is suitable to describe the number of independent events occurring within a unit time interval. And the rate ``\lambda`` is the average rate of occurrence of the event. 

Some Poissons with increasing rate parameters are plotted below. It can be observed that when ``\lambda`` is larger, the distribution's probability mass shifts to the right and the variance or spread of the distribution gets wider. And this is because a Poisson distribution's mean and variance are both tied to ``\lambda``, which makes a Poisson's error structure less flexible. Careful model checking, therefore, is usually necessary for Poisson-based models.

"""

# ╔═╡ ba72ad42-87d6-4962-b56e-a8d6ba3d9733
let
	λs = [1, 10, 5, 20]
	plts = []
	for i in 1:length(λs)
 		push!(plts, bar(Poisson(λs[i]), xlim =[0, 40], leg=false, c=i, title=L"\lambda=%$(λs[i])"))
	end
	plot(plts..., layout=(2,2))
end

# ╔═╡ ac69616e-14bf-48fe-adc9-7a810aa7a113
md"""



Since the rate parameter is strictly positive, when used in a GLM, we need to introduce an inverse link function that maps from the real number line to the positive part: ``g^{-1}: \mathbb{R} \rightarrow (0, +\infty)``. A common choice for Poisson regression is an exponential function (alternatively, ``\texttt{softplus}`` transformation can also be used). Therefore, a Poisson regression admits a likelihood function:

```math
p(y_n|\mathbf{x}_n, \boldsymbol{\beta}) = \texttt{Poisson}(y_n;\lambda_n=e^{\beta_0+\mathbf{x}_n^\top \boldsymbol{\beta}_1})
```

for ``n=1,2,\ldots, N`` observations. Note that the transformed rate ``e^{\beta_0+\mathbf{x}_n^\top \boldsymbol{\beta}_1}`` is always positive.

"""

# ╔═╡ ad45d197-0bae-4145-92b9-3b28064a692c
md"""
### An real-world counting data problem



To help illustrate the idea, a real-world `Cuckoo` dataset, described and analysed in the book [Statistical modelling in R](https://exeter-data-analytics.github.io/StatModelling/generalised-linear-models.html), is used as a concrete example. 

$(Resource("https://exeter-data-analytics.github.io/StatModelling/_img/03-cuckoo.png", :align=>""))

The question of interest for the Cuckoo data is:

> How does nestling mass affect begging between two different species?





The begging rate is a count data, which takes non-negative integer values. Bayesian Poisson regression, therefore, is a suitable model. 

We first **download** and **import** the data to the notebook's environment, and then carry out some initial data exploration.
"""

# ╔═╡ b2e26c07-43ad-47c1-a157-2368fbe5c707
first(df_cuckoo, 5)

# ╔═╡ e925ca4b-45c2-48fe-9d94-c34c15583a51
describe(df_cuckoo)

# ╔═╡ 39f3b1c0-e8bd-4d11-99ea-af9d74c7d77e
md"""

The dataset has three variables:
* `Mass`: continuous predictor
* `Species` ``\in\{\texttt{Cuckoo, Warbler}\}``: a binary predictor indicating the species of the bird
* `Beg`: non-negative integer count data.

The data is displayed below in a scatter plot.
"""

# ╔═╡ b2e54b06-91ad-4cb2-b398-fd7926f8c387
@df df_cuckoo scatter(:Mass, :Beg, group =:Species, legend=:outerright, xlabel="Mass", ylabel="Beg")

# ╔═╡ f39d0bab-3ce6-44c4-9fe0-77438884852a
md"""
It seems that the begging count is positively correlated with the mass variable for both species. The correlations, however, show different trends, which indicates there might be some interactions between the two predictors: `Mass` and `Species`. A model that accommodates such an interaction is:

```math
\ln(\mathbb{E}[\texttt{Beg}|\ldots]) = \beta_0 + \beta_1\cdot \texttt{Mass} + \beta_2 \cdot \texttt{Species} + \beta_3 \cdot (\texttt{Mass} \times \texttt{Species})
```

Since `Species` is a binary variable, *i.e.* `Species` ``\in \{0,1\}``, the model for `Species` ``=0`` is


```math
\ln(\mathbb{E}[\texttt{Beg}|\texttt{Species} = 0]) = \beta_0 + \beta_1\cdot \texttt{Mass} ,

```

On the other hand, when `Species` ``=1``,

```math
\ln(\mathbb{E}[\texttt{Beg}|\texttt{Species} = 1]) = (\beta_0 + \beta_2)  + (\beta_1+\beta_3) \cdot \texttt{Mass}.

```


Therefore, ``\beta_3``  can be interpreted as the *add-on* effect of `Mass` on `Beg` when an observation is a ``\texttt{Warbler}``, assuming species ``\texttt{Warbler}`` is encoded as 1 and ``\texttt{Cuckoo}`` is encoded as 0.
"""

# ╔═╡ 0475637e-d092-472d-82f4-fac65f34024e
md"""
### Data preprocessing

"""

# ╔═╡ 3b02812e-bd5f-47d2-89e2-3deb6f243562
md"""

Most GLMs employ non-linear link functions, which can be problematic for a MCMC sampler to mix. It is therefore a common practice to standardise the numerical features such that they have zero means and unit variances.
"""

# ╔═╡ a422af52-c350-45a0-a886-59a8e71b62ba
md"""
### Frequentist inference

It is usually a good idea to fit a frequentist model as a reference. This can be easily accomplished with `GLM.jl`. To accommodate the two different trends of the two species, we have added an interaction term.
"""

# ╔═╡ d4f7d14c-2710-4d6c-869f-3c0ab75d3d51
begin
	glm(@formula(Beg~Mass*Species), df_cuckoo, Poisson())
end

# ╔═╡ b6aa0ffe-c51f-4279-94b2-13c37797acf1
md"""
### Bayesian inference with `Turing`
"""

# ╔═╡ eac559ce-b05f-4b91-84b8-ed1f57a53c66
md"""

Based on the general Bayesian GLM model, we only need to subsitute in the required Poisson likelihood for ``\mathcal{P}`` and its associated (inverse) link function ``g^{-1}``, which is an exponential function. The final model is listed below.

!!! infor "Bayesian Poisson regression"
	```math
	\begin{align}
	\text{Priors: }\;\;\;\;\;\;\beta_0 &\sim \mathcal{N}(m_0^{\beta_0}, v_0^{\beta_0})\\
	\boldsymbol{\beta}_1 &\sim \mathcal{N}(\mathbf{m}_0^{\beta_1}, \mathbf{V}_0^{\beta_1})\\
	\text{Likelihood: } \;\; \text{for } n &= 1,2,\ldots, N:\\
	\mu_n &=\beta_0 + \boldsymbol{\beta}_1^\top \mathbf{x}_n \\
	\lambda_n &= e^{\mu_n}\\
	y_n &\sim \texttt{Poisson}(\lambda_n).
	\end{align}
	```

The model can be easily ported to `Turing`. Here we have assumed 
* the prior mean are all zeros, i.e. ``m_0^{\beta_0}=0``, ``\mathbf{m}_0^{\beta}=\mathbf{0}``
* the prior variances are set to ``1``; 
  * since all features have been standardised, the prior variance should be weak-informative enough (we will carry out predictive checks at the end to make sure the prior parameters are reasonable)
"""

# ╔═╡ 47cafa8d-6eb9-40c8-a9a7-56c9e7909c3d
@model function bayesian_poisson_regression(X, y; v₀=1, V₀ = 1)
	# priors
	β₀ ~ Normal(0, sqrt(v₀))
	nfeatures = size(X)[2]
	β ~ MvNormal(zeros(nfeatures), sqrt(V₀))
	# Likelihood
	μs = β₀ .+ X * β
	# inverse link function for Poisson is exponential
	λs = exp.(μs)
	for i in eachindex(y)
		y[i] ~ Poisson(λs[i])
	end
	return (; μs, λs)
end

# ╔═╡ 8af86af4-5144-4c36-b185-1c91189c9061
md"""

#### Prepare design matrix ``\mathbf{X}``

Instead of preparing the input design matrix ``\mathbf{X}`` manually, we can use `StatsModels.jl` to do the hard work for us. We only need to specify the model by the familiar `@formula` format and the `ModelFrame` object can prepare the required design matrix for us. It will typically do dummy encoding for the categorical variables and encode the interaction terms automatically. 

The prepared data are then passed to the `Turing` model to carry out the inference.
"""

# ╔═╡ 03064a4c-6445-446a-a4b5-f50d7d2332d5
begin
	XX_cuckoo = ModelFrame(@formula(Beg~Mass*Species), df_cuckoo) |> modelmatrix
	# remove the first column, the intercept 
	XX_cuckoo = XX_cuckoo[:, 2:end]
end;

# ╔═╡ 44fa5546-4abb-4e0b-bde3-d444aa912377
chain_poisson_cuckoo = let 
	yy = df_cuckoo[!, :Beg]
	Random.seed!(100)
	chain = sample(
		bayesian_poisson_regression(XX_cuckoo, yy; v₀=1, V₀=1),
		NUTS(),
		MCMCThreads(),
		1500,
		3
	)
	chain
end;

# ╔═╡ 7fe35625-4796-4240-bcdd-03aef6ade093
md"""

The inference results look similar to the frequentist method. However, the posterior means are shrinked a little bit towards zero, the prior's mean. The credible intervals also show that all three predictors' parameters are not likely to be zeros, which is in agreement with the frequentist's test result.
"""

# ╔═╡ 1a949d5a-2795-4974-8ae9-83a4e3c7fc90
describe(chain_poisson_cuckoo)

# ╔═╡ fd583678-f289-4075-9dee-8bdd334ce17f
plot(chain_poisson_cuckoo)

# ╔═╡ 6456936d-423e-4c56-931e-15e23fcbd82d
md"""

Different from the frequentist's method, the Bayesian method provides us with uncertain measurements of all parameters within the model. For the Poisson regression model, we are interested in the posterior predictive of the expected begging count: ``\lambda_\ast = e^{\beta_0+x_\ast^\top \beta_1}``. The expected value has its posterior distribution, which can be approximated by the Monte Carlo method:

```math
p(\lambda_\ast | \mathcal{D}) \approx \frac{1}{R} \sum_r e^{\beta_0^{(r)}+x_\ast^\top \beta_1^{(r)}},
```

where ``\{\beta_0^{(r)}, \boldsymbol{\beta}_1^{(r)}\}_{r=1}^R`` are the posterior MCMC samples. The Monte Carlo formula can be interpreted as an *ensemble method*: multiple models, each indexed by one of the ``R`` posterior samples, all contribute to the final prediction.

We can use `Julia`'s `generated_quantities()` method to find the approximation.
"""

# ╔═╡ 6ea6d1a1-fde0-4eb8-aa4c-a99a4f6a22e2
begin
	xs_test = collect(-1.5:0.1:3)
	test_df = DataFrame(Mass = xs_test, Species = trues(length(xs_test)), Beg=zeros(length(xs_test)))
	XX_test = ModelFrame(@formula(Beg~Mass*Species), test_df) |> modelmatrix
	# remove the first column, the intercept 
	XX_test = XX_test[:, 2:end]
	ys_ = Vector{Union{Missing, Int64}}(undef, size(XX_test)[1])
	pred_model_poisson = bayesian_poisson_regression(XX_test, ys_; v₀=1, V₀=1)
	gen_true = generated_quantities(pred_model_poisson, chain_poisson_cuckoo)
	Xs_false =[xs_test falses(length(xs_test)) xs_test .* falses(length(xs_test))]
	gen_false = generated_quantities(bayesian_poisson_regression(Xs_false, ys_; v₀=1, V₀=1), chain_poisson_cuckoo)
	λs_true = map(x -> x.λs, gen_true)
	λs_false = map(x -> x.λs, gen_false)
end;

# ╔═╡ 7a7a5641-2462-46fb-8ab0-5a76788af289
md"""

The Bayesian posterior distributions of ``\lambda`` are shown below. Note that the Bayesion inference not only provides us with a point estimator (the posterior mean) but also uncertainty measurement associated with the estimate.
"""

# ╔═╡ 79a32094-2dd4-484f-b91f-b5f9d99cc4bd
let
	plt =@df df_cuckoo scatter(:Mass, :Beg, group =:Species, legend=:outerright, ylim=[-9,250], xlabel="Mass", ylabel="Beg", legendtitle=L"\texttt{Species}")
	plot!(xs_test, mean(λs_false), lc=1, lw=1.5,label=L"\mathbb{E}[\lambda|\mathcal{D}]"*" Cuckoo")
	plot!(xs_test, mean(λs_true), lc=2, lw=1.5,label=L"\mathbb{E}[\lambda|\mathcal{D}]"*" Warbler")
	for i in 1:80
		plot!(xs_test, λs_false[i, 1], lw=0.2, lc=1, alpha =0.5, label="")
		plot!(xs_test, λs_true[i, 1], lw=0.2, lc=2,alpha=0.5, label="")
	end
	plt
end

# ╔═╡ b4227421-2b35-467f-b11f-f2425b12c9b9
md"""
#### Predictive checks

Lastly, we carry out predictive checks to make sure the Bayesian model's assumptions match the observed data. To simulate data from the posterior predictive distribution, we use the `predict()` method.
"""

# ╔═╡ 4ab38b80-0bae-4ea5-a109-3a88556daac5
pred_ys_poisson = let
	yy_ = Vector{Union{Missing, Float64}}(undef, size(XX_cuckoo)[1])
	gen = predict(
		bayesian_poisson_regression(XX_cuckoo, yy_; v₀=1, V₀=1),
		chain_poisson_cuckoo
	)
end

# ╔═╡ 6ae3eb45-9577-4bfe-8363-0b3c47bd0421
md"""
The mean and (log-transformed) standard deviation of the simulated samples are first calculated; and then plotted to compare against the observed. It can be observed from the plot that the simulated data based on our model assumption matches what we have observed since the observed statistic sits well within the boundary of the simulated statistics.
"""

# ╔═╡ 9d2d1c39-6c64-4deb-a2ef-dbbfbfd88323
let
	pred_y_μ_σ = DataFrame(mean= mean(Array(pred_ys_poisson), dims=2)[:], std=log.(std(Array(pred_ys_poisson), dims=2)[:]));
	@df pred_y_μ_σ scatter(:mean, :std, ms=2, alpha=0.5, label="Predicted", xlabel=L"\mu(\mathbf{y})", ylabel=L"\ln(\sigma(\mathbf{y}))", legend=:topleft)
	scatter!([mean(df_cuckoo.Beg)], [log(std(df_cuckoo.Beg))], ms=8,label="Observed")
end

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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", "Xorg_libX11_jll"]
git-tree-sha1 = "47e45cd78224c53109495b3e324df0c37bb61fbe"
uuid = "ea2f1a96-1ddc-540d-b46f-429655e07cfa"
version = "0.9.11+0"

[[deps.Xorg_libpthread_stubs_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "8fdda4c692503d44d04a0603d9ac0982054635f9"
uuid = "14d82f49-176c-5ed1-bb49-ad3f5cbd8c74"
version = "0.1.1+0"

[[deps.Xorg_libxcb_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "XSLT_jll", "Xorg_libXau_jll", "Xorg_libXdmcp_jll", "Xorg_libpthread_stubs_jll"]
git-tree-sha1 = "bcd466676fef0878338c61e655629fa7bbc69d8e"
uuid = "c7cfdc94-dc32-55de-ac96-5a1b8d977c5b"
version = "1.17.0+0"

[[deps.Xorg_libxkbfile_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libX11_jll"]
git-tree-sha1 = "730eeca102434283c50ccf7d1ecdadf521a765a4"
uuid = "cc61e674-0454-545c-8b26-ed2c68acab7a"
version = "1.1.2+0"

[[deps.Xorg_xcb_util_cursor_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xcb_util_image_jll", "Xorg_xcb_util_jll", "Xorg_xcb_util_renderutil_jll"]
git-tree-sha1 = "04341cb870f29dcd5e39055f895c39d016e18ccd"
uuid = "e920d4aa-a673-5f3a-b3d7-f755a4d47c43"
version = "0.1.4+0"

[[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", "Xorg_libxkbfile_jll"]
git-tree-sha1 = "330f955bc41bb8f5270a369c473fc4a5a4e4d3cb"
uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4"
version = "1.4.6+0"

[[deps.Xorg_xkeyboard_config_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xkbcomp_jll"]
git-tree-sha1 = "691634e5453ad362044e2ad653e79f3ee3bb98c3"
uuid = "33bec58e-1273-512f-9401-5d533626f822"
version = "2.39.0+0"

[[deps.Xorg_xtrans_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "e92a1a012a10506618f10b7047e478403a046c77"
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
version = "1.5.0+0"

[[deps.Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"
version = "1.2.13+1"

[[deps.Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "555d1076590a6cc2fdee2ef1469451f872d8b41b"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.6+1"

[[deps.ZygoteRules]]
deps = ["ChainRulesCore", "MacroTools"]
git-tree-sha1 = "27798139afc0a2afa7b1824c206d5e87ea587a00"
uuid = "700de1a5-db45-46bc-99cf-38207098b444"
version = "0.2.5"

[[deps.eudev_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "gperf_jll"]
git-tree-sha1 = "431b678a28ebb559d224c0b6b6d01afce87c51ba"
uuid = "35ca27e7-8b34-5b7f-bca9-bdc33f59eb06"
version = "3.2.9+0"

[[deps.fzf_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "936081b536ae4aa65415d869287d43ef3cb576b2"
uuid = "214eeab7-80f7-51ab-84ad-2988db7cef09"
version = "0.53.0+0"

[[deps.gperf_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "3516a5630f741c9eecb3720b1ec9d8edc3ecc033"
uuid = "1a1c6b14-54f6-533d-8383-74cd7377aa70"
version = "3.1.1+0"

[[deps.libaom_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "1827acba325fdcdf1d2647fc8d5301dd9ba43a9d"
uuid = "a4ae2306-e953-59d6-aa16-d00cac43593b"
version = "3.9.0+0"

[[deps.libass_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Zlib_jll"]
git-tree-sha1 = "e17c115d55c5fbb7e52ebedb427a0dca79d4484e"
uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
version = "0.15.2+0"

[[deps.libblastrampoline_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"
version = "5.11.0+0"

[[deps.libdecor_jll]]
deps = ["Artifacts", "Dbus_jll", "JLLWrappers", "Libdl", "Libglvnd_jll", "Pango_jll", "Wayland_jll", "xkbcommon_jll"]
git-tree-sha1 = "9bf7903af251d2050b467f76bdbe57ce541f7f4f"
uuid = "1183f4f0-6f2a-5f1a-908b-139f9cdfea6f"
version = "0.2.2+0"

[[deps.libevdev_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "141fe65dc3efabb0b1d5ba74e91f6ad26f84cc22"
uuid = "2db6ffa8-e38f-5e21-84af-90c45d0032cc"
version = "1.11.0+0"

[[deps.libfdk_aac_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "8a22cf860a7d27e4f3498a0fe0811a7957badb38"
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
version = "2.0.3+0"

[[deps.libinput_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "eudev_jll", "libevdev_jll", "mtdev_jll"]
git-tree-sha1 = "ad50e5b90f222cfe78aa3d5183a20a12de1322ce"
uuid = "36db933b-70db-51c0-b978-0f229ee0e533"
version = "1.18.0+0"

[[deps.libpng_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Zlib_jll"]
git-tree-sha1 = "b70c870239dc3d7bc094eb2d6be9b73d27bef280"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.44+0"

[[deps.libvorbis_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"]
git-tree-sha1 = "490376214c4721cdaca654041f635213c6165cb3"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.7+2"

[[deps.mtdev_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "814e154bdb7be91d78b6802843f76b6ece642f11"
uuid = "009596ad-96f7-51b1-9f1b-5ce2d5e8a71e"
version = "1.1.6+0"

[[deps.nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"
version = "1.59.0+0"

[[deps.oneTBB_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "7d0ea0f4895ef2f5cb83645fa689e52cb55cf493"
uuid = "1317d2d5-d96f-522e-a858-c73665f53c3e"
version = "2021.12.0+0"

[[deps.p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"
version = "17.4.0+2"

[[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 = "9c304562909ab2bab0262639bd4f444d7bc2be37"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "1.4.1+1"
"""

# ╔═╡ Cell order:
# ╟─88154488-233d-11ed-1072-3dc0dd25d3fd
# ╟─623efa9d-485d-4d9c-8e6d-aedb7dcbc00d
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# ╟─0e52bc27-41e9-4361-a036-77eee3774e46
# ╟─f113ba37-1eeb-4101-8162-5402af63677a
# ╟─73c61d37-9c26-43b7-ae99-9a79c6fe9804
# ╟─5d01251e-5ea2-4d2c-82c8-d9c576264887
# ╟─8b8471cd-6a77-4d8a-bbd9-ee28e0aab1f6
# ╟─ba72ad42-87d6-4962-b56e-a8d6ba3d9733
# ╟─ac69616e-14bf-48fe-adc9-7a810aa7a113
# ╟─ad45d197-0bae-4145-92b9-3b28064a692c
# ╠═52431ba0-762b-42d4-ab91-148998c90600
# ╟─b2e26c07-43ad-47c1-a157-2368fbe5c707
# ╟─e925ca4b-45c2-48fe-9d94-c34c15583a51
# ╟─39f3b1c0-e8bd-4d11-99ea-af9d74c7d77e
# ╠═b2e54b06-91ad-4cb2-b398-fd7926f8c387
# ╟─f39d0bab-3ce6-44c4-9fe0-77438884852a
# ╟─0475637e-d092-472d-82f4-fac65f34024e
# ╟─3b02812e-bd5f-47d2-89e2-3deb6f243562
# ╠═89a55d82-d5a9-40cd-9247-9d5355d4fd7c
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# ╠═ec93d30f-9d57-4598-8f64-ca706c4f0160
# ╠═d4f7d14c-2710-4d6c-869f-3c0ab75d3d51
# ╟─b6aa0ffe-c51f-4279-94b2-13c37797acf1
# ╟─eac559ce-b05f-4b91-84b8-ed1f57a53c66
# ╠═47cafa8d-6eb9-40c8-a9a7-56c9e7909c3d
# ╟─8af86af4-5144-4c36-b185-1c91189c9061
# ╠═fbb41799-8ce2-4286-be48-b6977930b6be
# ╠═03064a4c-6445-446a-a4b5-f50d7d2332d5
# ╠═44fa5546-4abb-4e0b-bde3-d444aa912377
# ╟─7fe35625-4796-4240-bcdd-03aef6ade093
# ╠═1a949d5a-2795-4974-8ae9-83a4e3c7fc90
# ╠═fd583678-f289-4075-9dee-8bdd334ce17f
# ╟─6456936d-423e-4c56-931e-15e23fcbd82d
# ╠═6ea6d1a1-fde0-4eb8-aa4c-a99a4f6a22e2
# ╟─7a7a5641-2462-46fb-8ab0-5a76788af289
# ╟─79a32094-2dd4-484f-b91f-b5f9d99cc4bd
# ╟─b4227421-2b35-467f-b11f-f2425b12c9b9
# ╠═4ab38b80-0bae-4ea5-a109-3a88556daac5
# ╟─6ae3eb45-9577-4bfe-8363-0b3c47bd0421
# ╠═9d2d1c39-6c64-4deb-a2ef-dbbfbfd88323
# ╟─00000000-0000-0000-0000-000000000001
# ╟─00000000-0000-0000-0000-000000000002