### A Pluto.jl notebook ###
# v1.0.3

#> [frontmatter]
#> title = "Classic Urn Problem"
#> description = "Bayesian approach to the urn problem, with and without replacement."
#> 
#>     [[frontmatter.author]]
#>     name = "Kacper Topolnicki"
#>     url = "https://kacpertopol.github.io/"

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)
    #! format: off
    return 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
    #! format: on
end

# ╔═╡ 8f358f56-eddc-4a85-a8cb-4561e0529d41
using Random, Distributions, Plots, PlutoUI

# ╔═╡ df18f176-86ee-4453-8258-f48dc87ef88e
md"""
# Setup

We have an urn with a given number ``n`` of black and white balls. We are allowed to:
1. draw on ball at random, 
2. note it's color, and then 
3. return the ball (we will consider the case without replacement later) 

## Bayes law

First some mathematics. We would like to work out
```math
P(W = w | O_{m} = d_{m} \,\,\, O_{m-1} = d_{m-1} \,\,\, \ldots \,\,\, O_{1} = d_{1} \,\,\, X)
```
Where 
- ``W = w`` stands for "the number of white balls is ``w``" (the number of black balls is just ``N - w``),
- ``O_{m} = d_{m} `` stands for "at observation m we saw a ball with color ``d_{m}``",
- and ``X`` symbolizes implicit assumptions.

## Updating the posteriour

To simplify things, let's call:
```math
X' \equiv O_{m-1} = d_{m-1} \,\,\, \ldots \,\,\, O_{1} = d_{1} \,\,\, X
```
then
```math
P(W = w | O_{m} = d_{m} \,\,\, X') = P(W = w | X') \frac{P(O_{m} = d_{m} | W = w \,\,\, X')}{P(O_{m} = d_{m} | X')}
```
Where 
- ``W = w`` stands for "the number of white balls is ``w``",
- ``O_{m} = d_{m} `` stands for "at observation m we saw a ball with color ``d_{m}``" (``0`` means black, ``1`` means white),
- and ``X`` symbolizes implicit assumptions.

## Likelihood

The probability is normalized:
```math
\Sigma_{w = 0}^{n} P(W = w | O_{m} = d_{m} \,\,\, X') = \Sigma_{w = 0}^{n} P(W = w | X')\frac{P(O_{m} = d_{m} | W = w \,\,\, X')}{P(O_{m} = d_{m} | X')} = 1
```
and since and the denominator does not depend on ``w``:
```math
P(O_{m} = d_{m} | X') = \Sigma_{w = 0}^{n} P(W = w | X') P(O_{m} = d_{m} | W = w \,\,\, X')
```
we see that the for of the likelihood is key.

The case with replacement is the simplest, here:
```math
P(O_{m} = d_{m} | W = w \,\,\, X') = 
    \begin{cases}
      d_{m} = 0 & 1 - \frac{w}{n}\\
      d_{m} = 1 & \frac{w}{n}
    \end{cases}  
```
"""

# ╔═╡ d4f3e190-1fa4-4079-ae4d-753230b01b88
md"""
# Simulation

Now for some numerical experiments ...
"""

# ╔═╡ fd331135-00ce-4a6e-b298-1221b14e38c0
md"""
## Loading packages

We will use some Julia packages. 
- `Random` is for ... using random numbers, 
- `Distributions` is for the Bernoulli distribution,
- `Plots` is for plotting,
- `PlutoUI` is for the user interface in Pluto.
"""

# ╔═╡ 3d393ba0-7454-40cb-8acb-d7f156ed5dae
md"""
## Known information

We will assume that the only known information is the total number of balls
in the urn. 
"""

# ╔═╡ 52faacb8-a2d0-4749-86e4-8c0d5b1c53d1
n = 50;

# ╔═╡ 095de98f-1d52-4132-8234-16b12de38eea
md"""
## Information to be inferred 

The true numbers of the black and white balls. 
"""

# ╔═╡ 47bb8337-19fd-4ee1-855a-c9794922896f
black = 10;

# ╔═╡ 733bf7de-7e2d-4615-b7a0-6762cf0a97e8
white = n - black

# ╔═╡ 06964312-31d8-4e53-a2d9-152f4aac357b
md"""
## Simulation

We will simulate random draws from a population of `black` black balls and `white` white balls. The number 0 will represent the draw of a black ball and the number 1 will represent the draw of a white ball. The `data` is ``d_{1} , d_{2} , \ldots , d_{m - 1} d_{m}``:
"""

# ╔═╡ 79595dd5-350a-4f99-812e-8bf8f84f1008
begin
	m = 1000;
	data = convert(Vector{Int64} , rand(Bernoulli(white / n) , m))
end;

# ╔═╡ ff0b189e-f3f8-4da4-9d6b-85d1daa0bb1c
md"""
and we will check the resuls of the inference with `data[:1]` , `data[:2]` , `data[:3]` , ... Note that we will have to change this later for the case without replacement.

## The likelihood:
"""

# ╔═╡ 7bd47297-7127-4baf-8ca6-75ab04b1d3d4
md"""
This is a the most important element. Note that for simplicity we dont separate the data, the last element is treated as the current observation.
"""

# ╔═╡ d5e162d0-33ab-4424-a231-826eea8f506a
function l_with_replacement(w :: Int64 , data :: Vector{Int})
	d_m = data[end] # the last data point is assumed to be the currently drawn ball
	# ((n - w) / n) + (w / n) = 1
	if d_m == 0
		return (n - w) / n
	else
		return w / n
	end
end;

# ╔═╡ 3968a33c-d1f3-4d15-b2a4-bf19a1bc9243
md"""
## Iterations 

Function implementing the iterative update of the probability:
"""

# ╔═╡ 20c2d523-4288-43bd-a3fa-efd8c7338a6c
function iterate(prior :: Vector{Float64} , l :: Function , data :: Vector{Int} ; n = n)
	# P(W = 0 | ...) = 0   P(W = 1 | ...) = 0 ...:
	new = zeros(Float64 , n + 1)
	for w = 0 : n 
		# carefull, in Julia indexing starts from 1:
		new[w + 1] = prior[w + 1] * l(w , data)
	end
	# normalize:
	new / sum(new)
end;

# ╔═╡ 4e2ce0f2-db67-463e-b02e-c80b5b2eacd0
md"""
And the function running `iter` iterations. Note that to make this more efficient we could store all the iterations calculated in the loop.
"""

# ╔═╡ f9828ebe-ac4d-4be3-b944-24b86ada2ec0
function run_iterations(l , iter , prior , data; n = n)
	p = copy(prior)
	for i = 1 : iter
		p = iterate(p , l , data[1:i] ; n = n)
	end
	if iter == 0
		return p , -1
	else
		return p , data[iter]
	end
end;

# ╔═╡ 1c784008-d9af-4bd1-8401-2ad006731988
md"""
## Plots
"""

# ╔═╡ 10fe2007-efe8-4333-a2f7-805d3bd99634
md"""
We will consider three choices for the prior probability:
"""

# ╔═╡ a2d5731b-4594-4963-8305-4a48a1d055dc
begin
	prior1 = ones(Float64 , n + 1);
	prior2 = [1.0 * w for w in 0:n];
	prior3 = [exp(-0.005 * (w - 0.5 * n) * (w - 0.5 * n)) for w in 0:n];
	#
	prior1 = prior1 / sum(prior1);
	prior2 = prior2 / sum(prior2);
	prior3 = prior3 / sum(prior3);
end;

# ╔═╡ 60dced7e-9a2c-4afc-9d9f-b862a812e8f7
@bind prior Slider([prior1 , prior2 , prior3] , default = prior1 , show_value = _ -> "chose prior")

# ╔═╡ ded5000d-158b-48fa-9fae-efd81ff10b09
@bind i Slider(0 : 1 : m , default = 0 , show_value = x -> string(x) * " iterations")

# ╔═╡ 0fc7c8ad-80aa-443d-9ac7-5fc7bf2da4a6
begin
	p_fin , ball_fin = run_iterations(l_with_replacement , i , prior , data)
	scatter(
		collect(0:n) , p_fin,
		xlabel = "w", ylabel = "P",
		marker = :circle , markersize = 2, 
		xticks = collect(0:2:n),
		label = false, title = "without replacement, last ball : " * string(ball_fin) , grid = false)
	vline!([white] , label = false)
end

# ╔═╡ 878e7769-febf-4149-90db-cb5e5506bf84
md"""
## Without replacement

First the data, we can only have a limited number of draws:
"""

# ╔═╡ acbe0d92-e3eb-4b82-bb7a-26855ea087f5
data_norep = shuffle(vcat(fill(0 , black) , fill(1 , white)));

# ╔═╡ 3a2111be-886e-4fa9-a691-91f0ec50de23
md"""
Next the prior probabiliy:
"""

# ╔═╡ add9917e-772b-4720-b9fb-bfef185c60ee
begin
	prior_norep1 = ones(Float64 , n + 1);
	prior_norep2 = [1.0 * w for w in 0:n];
	prior_norep3 = [exp(-0.005 * (w - 0.5 * n) * (w - 0.5 * n)) for w in 0:n];
	#
	prior_norep1 = prior_norep1 / sum(prior_norep1);
	prior_norep2 = prior_norep2 / sum(prior_norep2);
	prior_norep3 = prior_norep3 / sum(prior_norep3);
end;

# ╔═╡ d6630bea-b136-46f6-918e-40a541c2b9f6
md"""
Likelihood without replacement:
"""

# ╔═╡ 82befd18-b707-4c0b-9a95-83883983fa88
function l_without_replacement(w :: Int64 , data :: Vector{Int})
	d_previous = data[1:end-1] 			# previous observations
	d_m = data[end] 					# currently drawn ball
	
	white_out = sum(d_previous) 		# number of observed white balls
	black_out = length(d_previous) - white_out

	# Is this correct?
	l = max((w - white_out) / (n - length(d_previous)) , 0)
	# Or this, is this ok?
	#l = (w - white_out) / (n - length(d_previous))
	if d_m == 1
		return l
	else
		return 1.0 - l
	end
end;

# ╔═╡ 751ceed1-0a6d-4108-8f14-689db9ce908b
max(1 , 2)

# ╔═╡ c806c99f-9871-4e77-a7ad-39b1009ef93b
md"""
## Plots
"""

# ╔═╡ b0c3014e-946f-4120-a2e4-1d41b5ef2e9c
@bind prior_norep Slider([prior_norep1 , prior_norep2 , prior_norep3] , default = prior_norep1 , show_value = _ -> "chose prior")

# ╔═╡ 5640ad50-a2dc-4adf-937c-8ce4ce77a027
@bind i_norep Slider(0 : 1 : n , default = 0 , show_value = x -> string(x) * " iterations")

# ╔═╡ c74c75cb-6ab3-4e38-89df-323a9827cf62
begin
	p_fin_norep , ball_fin_norep = run_iterations(l_without_replacement , i_norep , prior_norep , data_norep)
	scatter(
		collect(0:n) , p_fin_norep,
		xlabel = "w", ylabel = "P",
		marker = :circle , markersize = 2, 
		xticks = collect(0:2:n),
		label = false, title = "with replacement, last ball : " * string(ball_fin_norep) , grid = false)
	vline!([white] , label = false)
end

# ╔═╡ fce7e472-473f-4144-a505-b123e1497279
md"""
# Bonus - coin fairness

Inspired by a question on reddit. How many coin tosses does it take to be reasobably certain that the coin is not fair?

We will measure the fairess of the coin using ``q`` - the probability that the a coin toss returns heads. Here we are dealing with discrete probability so we will simplify and approximate:
```math
q = \frac{0}{n_{q}}, \frac{1}{n_{q}} , \frac{2}{n_{q}} , \ldots , \frac{n_{q}}{n_{q}}
```
This approximation deserves carefull consideration. Is it fine to use here?
"""

# ╔═╡ 8ca81647-b9c6-4c17-8900-1ad64641c0cd
nq = 100;

# ╔═╡ 39292a20-3f29-4a87-a26d-66d495564da3
md"""
The likelihood:
"""

# ╔═╡ 1ea2a9e4-e223-4826-a2a7-7699cbdaf23d
function l_coin_toss(w :: Int64 , data :: Vector{Int})
	q = w / nq
	# in the data 1 - heads, 0 - tails
	d_previous = data[1:end-1] 			# previous observations
	d_m = data[end] 					# currently drawn ball
	if d_m == 1
		return q
	else
		return 1 - q
	end
end;

# ╔═╡ 2922bb8d-f727-47de-8f66-0d49023772ac
md"""
The results of coin tosses:
"""

# ╔═╡ 016d7082-78a0-4d64-b49e-8a181d44725e
#coinTosses = [1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 0 , 0 , 1 , 0 , 1 , 1 , 0 , 1 , 0 , 1 , 1 , 0 , 0 ,0 , 1 , 0 , 1 , 1 , 0 , 0];
coinTosses = convert(Vector{Int64} , rand(Bernoulli(0.7) , 2000));
#coinTosses = vcat(convert(Vector{Int64} , rand(Bernoulli(0.5) , 200)) , [1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1] , convert(Vector{Int64} , rand(Bernoulli(0.5) , 200)));

# ╔═╡ 3868f3e5-fc37-4561-88e3-04350f5c5345
md"""
The priors:
"""

# ╔═╡ ff67a7e7-c1f3-4792-ab74-875be62e4ed8
begin
	prior_coin1 = ones(Float64 , nq + 1);
	prior_coin2 = [1.0 * w for w in 0:nq];
	prior_coin3 = [exp(-0.005 * (w - 0.5 * nq) * (w - 0.5 * nq)) for w in 0:nq];
	#
	prior_coin1 = prior_coin1 / sum(prior_coin1);
	prior_coin2 = prior_coin2 / sum(prior_coin2);
	prior_coin3 = prior_coin3 / sum(prior_coin3);
end;

# ╔═╡ 5373ac64-ef2e-4233-bb7a-2c96a9b72a0b
md"""
The probabilities:
"""

# ╔═╡ d2f490a7-0b8f-49e6-a8b1-c99eb9a4538f
@bind prior_coin Slider([prior_coin1 , prior_coin2 , prior_coin3] , default = prior_coin1 , show_value = _ -> "chose prior")

# ╔═╡ 2f6c1196-7d4e-4e10-a505-37efb33f1b32
@bind i_coin Slider(0 : 1 : length(coinTosses) , default = 0 , show_value = x -> string(x) * " iterations")

# ╔═╡ ef1e5e12-5c9f-4e44-a2e3-99686b412fa3
begin
	p_fin_coin , ball_fin_coin = run_iterations(l_coin_toss , i_coin , prior_coin , coinTosses ; n = nq)
	scatter(
		collect(0:nq) ./ nq , p_fin_coin,
		xlabel = "w", ylabel = "P",
		marker = :circle , markersize = 2, 
		label = false, title = "coin toss, last side : " * string(ball_fin_coin) , grid = false)
end

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uuid = "0c0b7dd1-d40b-584c-a123-a41640f87eec"
version = "1.0.13+0"

[[deps.Xorg_libXcursor_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libXfixes_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "6c74ca84bbabc18c4547014765d194ff0b4dc9da"
uuid = "935fb764-8cf2-53bf-bb30-45bb1f8bf724"
version = "1.2.4+0"

[[deps.Xorg_libXdmcp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "52858d64353db33a56e13c341d7bf44cd0d7b309"
uuid = "a3789734-cfe1-5b06-b2d0-1dd0d9d62d05"
version = "1.1.6+0"

[[deps.Xorg_libXext_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libX11_jll"]
git-tree-sha1 = "1a4a26870bf1e5d26cd585e38038d399d7e65706"
uuid = "1082639a-0dae-5f34-9b06-72781eeb8cb3"
version = "1.3.8+0"

[[deps.Xorg_libXfixes_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libX11_jll"]
git-tree-sha1 = "75e00946e43621e09d431d9b95818ee751e6b2ef"
uuid = "d091e8ba-531a-589c-9de9-94069b037ed8"
version = "6.0.2+0"

[[deps.Xorg_libXi_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libXext_jll", "Xorg_libXfixes_jll"]
git-tree-sha1 = "dcb316b3ce0941f195537dda56bea4517fcd3ff5"
uuid = "a51aa0fd-4e3c-5386-b890-e753decda492"
version = "1.8.4+0"

[[deps.Xorg_libXinerama_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libXext_jll"]
git-tree-sha1 = "0ba01bc7396896a4ace8aab67db31403c71628f4"
uuid = "d1454406-59df-5ea1-beac-c340f2130bc3"
version = "1.1.7+0"

[[deps.Xorg_libXrandr_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libXext_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "6c174ef70c96c76f4c3f4d3cfbe09d018bcd1b53"
uuid = "ec84b674-ba8e-5d96-8ba1-2a689ba10484"
version = "1.5.6+0"

[[deps.Xorg_libXrender_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libX11_jll"]
git-tree-sha1 = "7ed9347888fac59a618302ee38216dd0379c480d"
uuid = "ea2f1a96-1ddc-540d-b46f-429655e07cfa"
version = "0.9.12+0"

[[deps.Xorg_libpciaccess_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Zlib_jll"]
git-tree-sha1 = "58972370b81423fc546c56a60ed1a009450177c3"
uuid = "a65dc6b1-eb27-53a1-bb3e-dea574b5389e"
version = "0.19.0+0"

[[deps.Xorg_libxcb_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libXau_jll", "Xorg_libXdmcp_jll"]
git-tree-sha1 = "bfcaf7ec088eaba362093393fe11aa141fa15422"
uuid = "c7cfdc94-dc32-55de-ac96-5a1b8d977c5b"
version = "1.17.1+0"

[[deps.Xorg_libxkbfile_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libX11_jll"]
git-tree-sha1 = "ed756a03e95fff88d8f738ebc2849431bdd4fd1a"
uuid = "cc61e674-0454-545c-8b26-ed2c68acab7a"
version = "1.2.0+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 = "9750dc53819eba4e9a20be42349a6d3b86c7cdf8"
uuid = "e920d4aa-a673-5f3a-b3d7-f755a4d47c43"
version = "0.1.6+0"

[[deps.Xorg_xcb_util_image_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xcb_util_jll"]
git-tree-sha1 = "f4fc02e384b74418679983a97385644b67e1263b"
uuid = "12413925-8142-5f55-bb0e-6d7ca50bb09b"
version = "0.4.1+0"

[[deps.Xorg_xcb_util_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libxcb_jll"]
git-tree-sha1 = "68da27247e7d8d8dafd1fcf0c3654ad6506f5f97"
uuid = "2def613f-5ad1-5310-b15b-b15d46f528f5"
version = "0.4.1+0"

[[deps.Xorg_xcb_util_keysyms_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xcb_util_jll"]
git-tree-sha1 = "44ec54b0e2acd408b0fb361e1e9244c60c9c3dd4"
uuid = "975044d2-76e6-5fbe-bf08-97ce7c6574c7"
version = "0.4.1+0"

[[deps.Xorg_xcb_util_renderutil_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xcb_util_jll"]
git-tree-sha1 = "5b0263b6d080716a02544c55fdff2c8d7f9a16a0"
uuid = "0d47668e-0667-5a69-a72c-f761630bfb7e"
version = "0.3.10+0"

[[deps.Xorg_xcb_util_wm_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xcb_util_jll"]
git-tree-sha1 = "f233c83cad1fa0e70b7771e0e21b061a116f2763"
uuid = "c22f9ab0-d5fe-5066-847c-f4bb1cd4e361"
version = "0.4.2+0"

[[deps.Xorg_xkbcomp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libxkbfile_jll"]
git-tree-sha1 = "801a858fc9fb90c11ffddee1801bb06a738bda9b"
uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4"
version = "1.4.7+0"

[[deps.Xorg_xkeyboard_config_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xkbcomp_jll"]
git-tree-sha1 = "2e59214e017a55cb87474a00fa76035c82ac0e17"
uuid = "33bec58e-1273-512f-9401-5d533626f822"
version = "2.47.0+2"

[[deps.Xorg_xtrans_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "a63799ff68005991f9d9491b6e95bd3478d783cb"
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
version = "1.6.0+0"

[[deps.Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"
version = "1.3.1+2"

[[deps.Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "446b23e73536f84e8037f5dce465e92275f6a308"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.7+1"

[[deps.eudev_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "c3b0e6196d50eab0c5ed34021aaa0bb463489510"
uuid = "35ca27e7-8b34-5b7f-bca9-bdc33f59eb06"
version = "3.2.14+0"

[[deps.fzf_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "b6a34e0e0960190ac2a4363a1bd003504772d631"
uuid = "214eeab7-80f7-51ab-84ad-2988db7cef09"
version = "0.61.1+0"

[[deps.libaom_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "1210ba774d3427387d307bf1f416d699b7c39417"
uuid = "a4ae2306-e953-59d6-aa16-d00cac43593b"
version = "3.15.1+0"

[[deps.libass_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Zlib_jll"]
git-tree-sha1 = "cb007192783c56d8249db4cf0e3495001edfe414"
uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
version = "0.17.5+0"

[[deps.libblastrampoline_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"
version = "5.15.0+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.libdrm_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libpciaccess_jll"]
git-tree-sha1 = "28e57478e8a160d346a19c28b3fffb9273bcc9c2"
uuid = "8e53e030-5e6c-5a89-a30b-be5b7263a166"
version = "2.4.134+0"

[[deps.libevdev_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "56d643b57b188d30cccc25e331d416d3d358e557"
uuid = "2db6ffa8-e38f-5e21-84af-90c45d0032cc"
version = "1.13.4+0"

[[deps.libfdk_aac_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "646634dd19587a56ee2f1199563ec056c5f228df"
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
version = "2.0.4+0"

[[deps.libinput_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "eudev_jll", "libevdev_jll", "mtdev_jll"]
git-tree-sha1 = "91d05d7f4a9f67205bd6cf395e488009fe85b499"
uuid = "36db933b-70db-51c0-b978-0f229ee0e533"
version = "1.28.1+0"

[[deps.libpng_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Zlib_jll"]
git-tree-sha1 = "e51150d5ab85cee6fc36726850f0e627ad2e4aba"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.58+0"

[[deps.libva_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libX11_jll", "Xorg_libXext_jll", "Xorg_libXfixes_jll", "libdrm_jll"]
git-tree-sha1 = "7dbf96baae3310fe2fa0df0ccbb3c6288d5816c9"
uuid = "9a156e7d-b971-5f62-b2c9-67348b8fb97c"
version = "2.23.0+0"

[[deps.libvorbis_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll"]
git-tree-sha1 = "11e1772e7f3cc987e9d3de991dd4f6b2602663a5"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.8+0"

[[deps.mtdev_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "b4d631fd51f2e9cdd93724ae25b2efc198b059b1"
uuid = "009596ad-96f7-51b1-9f1b-5ce2d5e8a71e"
version = "1.1.7+0"

[[deps.nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"
version = "1.64.0+1"

[[deps.p7zip_jll]]
deps = ["Artifacts", "CompilerSupportLibraries_jll", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"
version = "17.7.0+0"

[[deps.x264_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "14cc7083fc6dff3cc44f2bc435ee96d06ed79aa7"
uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
version = "10164.0.1+0"

[[deps.x265_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "e7b67590c14d487e734dcb925924c5dc43ec85f3"
uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
version = "4.1.0+0"

[[deps.xkbcommon_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
git-tree-sha1 = "a1fc6507a40bf504527d0d4067d718f8e179b2b8"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "1.13.0+0"
"""

# ╔═╡ Cell order:
# ╟─df18f176-86ee-4453-8258-f48dc87ef88e
# ╟─d4f3e190-1fa4-4079-ae4d-753230b01b88
# ╟─fd331135-00ce-4a6e-b298-1221b14e38c0
# ╠═8f358f56-eddc-4a85-a8cb-4561e0529d41
# ╟─3d393ba0-7454-40cb-8acb-d7f156ed5dae
# ╠═52faacb8-a2d0-4749-86e4-8c0d5b1c53d1
# ╟─095de98f-1d52-4132-8234-16b12de38eea
# ╠═47bb8337-19fd-4ee1-855a-c9794922896f
# ╠═733bf7de-7e2d-4615-b7a0-6762cf0a97e8
# ╟─06964312-31d8-4e53-a2d9-152f4aac357b
# ╠═79595dd5-350a-4f99-812e-8bf8f84f1008
# ╟─ff0b189e-f3f8-4da4-9d6b-85d1daa0bb1c
# ╟─7bd47297-7127-4baf-8ca6-75ab04b1d3d4
# ╠═d5e162d0-33ab-4424-a231-826eea8f506a
# ╟─3968a33c-d1f3-4d15-b2a4-bf19a1bc9243
# ╠═20c2d523-4288-43bd-a3fa-efd8c7338a6c
# ╟─4e2ce0f2-db67-463e-b02e-c80b5b2eacd0
# ╠═f9828ebe-ac4d-4be3-b944-24b86ada2ec0
# ╟─1c784008-d9af-4bd1-8401-2ad006731988
# ╟─10fe2007-efe8-4333-a2f7-805d3bd99634
# ╠═a2d5731b-4594-4963-8305-4a48a1d055dc
# ╟─60dced7e-9a2c-4afc-9d9f-b862a812e8f7
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# ╟─878e7769-febf-4149-90db-cb5e5506bf84
# ╠═acbe0d92-e3eb-4b82-bb7a-26855ea087f5
# ╟─3a2111be-886e-4fa9-a691-91f0ec50de23
# ╠═add9917e-772b-4720-b9fb-bfef185c60ee
# ╟─d6630bea-b136-46f6-918e-40a541c2b9f6
# ╠═82befd18-b707-4c0b-9a95-83883983fa88
# ╠═751ceed1-0a6d-4108-8f14-689db9ce908b
# ╟─c806c99f-9871-4e77-a7ad-39b1009ef93b
# ╟─b0c3014e-946f-4120-a2e4-1d41b5ef2e9c
# ╟─5640ad50-a2dc-4adf-937c-8ce4ce77a027
# ╟─c74c75cb-6ab3-4e38-89df-323a9827cf62
# ╟─fce7e472-473f-4144-a505-b123e1497279
# ╠═8ca81647-b9c6-4c17-8900-1ad64641c0cd
# ╟─39292a20-3f29-4a87-a26d-66d495564da3
# ╠═1ea2a9e4-e223-4826-a2a7-7699cbdaf23d
# ╟─2922bb8d-f727-47de-8f66-0d49023772ac
# ╠═016d7082-78a0-4d64-b49e-8a181d44725e
# ╟─3868f3e5-fc37-4561-88e3-04350f5c5345
# ╠═ff67a7e7-c1f3-4792-ab74-875be62e4ed8
# ╟─5373ac64-ef2e-4233-bb7a-2c96a9b72a0b
# ╟─d2f490a7-0b8f-49e6-a8b1-c99eb9a4538f
# ╟─2f6c1196-7d4e-4e10-a505-37efb33f1b32
# ╠═ef1e5e12-5c9f-4e44-a2e3-99686b412fa3
# ╟─00000000-0000-0000-0000-000000000001
# ╟─00000000-0000-0000-0000-000000000002