Turing example with updating belief after single coinflip

pluto-single-coinflip.jl

### A Pluto.jl notebook ###
# v0.19.3

using Markdown
using InteractiveUtils

# ╔═╡ 193f0cae-53a4-48d0-98ed-6eb94aad5712
begin
using Distributions
using Random
Random.seed!(12)
	using Plots
using StatsPlots
end

# ╔═╡ 7a8c4f4b-6741-4542-9ea3-2178916d9a8d
begin
	using Turing
	using MCMCChains
end

# ╔═╡ 1b8ee8b9-e8d8-4b80-b610-fc9843d05b31
begin 
    using PlutoUI
	EL=PlutoUI.ExperimentalLayout
end

# ╔═╡ c6030b7a-bfd7-4fba-82b2-16ac637d6cb8
md"""
After [https://turing.ml/dev/tutorials/00-introduction/](https://turing.ml/dev/tutorials/00-introduction/)
"""

# ╔═╡ e3edcefe-9d05-46cd-9122-43fc6c32e2ad
md"""
## Basic example
"""

# ╔═╡ 0bb322e1-c730-4e43-8c7f-aedfad29790a
md"""
Assume that we are unsure about the probability of heads in a coin flip. To get an intuitive understanding of what "updating our beliefs" is, we will visualize the probability of heads in a coin flip after each observed evidence.

First, let us load some packages that we need to simulate a coin flip
"""

# ╔═╡ faf423b4-b9b5-4b66-a8e6-48128c4f2d43
md"""
We simulate N coin flips by drawing N random samples from the Bernoulli distribution with success probability p_true.

Configure the data generating model. Let us set the true probability that a coin flip turns up heads.
"""

# ╔═╡ d69cfc8f-e48e-4e3e-9076-07ee07736e4e
p_true = 0.4;

# ╔═╡ 0159ae15-ac81-4adc-8be5-b7ba9a640203
N = 100;

# ╔═╡ 314fec7b-827b-49d5-b9ec-df7ee87f7724
data = rand(Bernoulli(p_true), N)

# ╔═╡ 9589511e-8f23-4611-9fe3-eefbd4d554aa
md"""
Next, we specify a prior belief about the distribution of heads and tails in a coin toss. Here we choose a Beta distribution as prior distribution for the probability of heads. Before any coin flip is observed, we assume a uniform distribution U⁡(0,1)=Beta⁡(1,1) of the probability of heads. I.e., every probability is equally likely initially.
"""

# ╔═╡ c2f70add-e0e6-44a2-ae59-7d8a667be1e2
prior_belief = Beta(1, 1);

# ╔═╡ 23f9c22e-6ea8-4602-9eff-661c0b3163c7
md"""
With our priors set and our data at hand, we can perform Bayesian inference.

This is a fairly simple process. We expose one additional coin flip to our model every iteration, such that the first run only sees the first coin flip, while the last iteration sees all the coin flips. In each iteration we update our belief to an updated version of the original Beta distribution that accounts for the new proportion of heads and tails. The update is particularly simple since our prior distribution is a conjugate prior. Note that a closed-form expression for the posterior (implemented in the updated_belief expression below) is not accessible in general and usually does not exist for more interesting models.
"""

# ╔═╡ 62e88f44-2903-49d1-a6e6-9ec4877c4ecb
function updated_belief(prior_belief::Beta, flip)
    # Count the number of heads and tails.
    head = flip
    tail = 1-flip

    # Update our prior belief in closed form (this is possible because we use a conjugate prior).
    return Beta(prior_belief.α + head, prior_belief.β + tail)
end

# ╔═╡ 6dd570b1-0b3d-476e-a172-8686a29c1f0f
function learn(belief)
gif=@gif for n in 1:N	
    belief=updated_belief(belief, data[n]);
	m=round(mean(belief),digits=5)
	σ=round(std(belief),digits=5)
    plot(
		belief,
        size=(400, 300),
        title="N=$N\nmean=$(m)\nσ=$(σ)",
        xlabel="probability of heads",
        ylabel="",
        legend=nothing,
        xlim=(0, 1),
        fill=0,
        α=0.3,
        w=3,
    )
    p=vline!([p_true])
	end
	belief,gif
end

# ╔═╡ 59ae1cb9-0dfc-4ae4-b8a0-c2428a5921cc
final_belief,gif=learn(prior_belief);

# ╔═╡ a7f72a2a-b3ab-4bbf-8e4d-34f6ba7f796a
EL.hbox([
	md"""
The animation shows that with increasing evidence our belief about the probability of heads in a coin flip slowly adjusts towards the true value. The orange line in the animation represents the true probability of seeing heads on a single coin flip, while the mode of the distribution shows what the model believes the probability of a heads is given the evidence it has seen.
""",
	gif])

# ╔═╡ c43efead-6daf-45c3-9c97-a4bfec45dd6d
md"""
## Turing.jl
"""

# ╔═╡ 69fb63fd-9f74-4329-9663-e990f764bf42
md"""
In the Turing model the prior distribution of the variable p, the probability of heads in a coin toss, and the distribution of the observations y[n] are specified on the right-hand side of the ~ expressions. The @model macro modifies the body of the Julia function coinflip(y) and, e.g., replaces the ~ statements with internal function calls that are used for sampling.
"""

# ╔═╡ aca0d9e6-32be-4cd9-8252-624ec94b1879
@model function coinflip(y)
    # Our prior belief about the probability of heads in a coin toss.
    p ~ Beta(1, 1)

    # The number of observations.
    N = length(y)
    for n in 1:N
        # Heads or tails of a coin are drawn from a Bernoulli distribution.
        y[n] ~ Bernoulli(p)
    end
end;

# ╔═╡ 9d965e5c-b294-4777-9042-51b5db7ed7b0
md"""
We combine the model with the observations:
"""

# ╔═╡ 3afa6935-af80-41a8-b4c6-33441185cea2
model = coinflip(data);

# ╔═╡ 64210e98-d289-4a4a-ae7f-a5748358d5bf
md"""
After defining the model, we can approximate the posterior distribution by drawing samples from the distribution. In this example, we use a Hamiltonian Monte Carlo sampler to draw these samples. 
"""

# ╔═╡ 3f4591c7-f7ec-4a65-bfc7-06415ffe4a01
sampler = HMC(0.05, 10);

# ╔═╡ 5014a882-a8ca-471c-8e5f-807e126c159a
chain = sample(model, sampler, N)

# ╔═╡ 97bc750b-ba7a-40d2-a6a4-b2ad6e2adf63
md"""
The sample function and common keyword arguments are explained more extensively in the documentation of AbstractMCMC.jl.

After finishing the sampling process, we can visually compare the closed-form posterior distribution with the approximation obtained with Turing.
"""

# ╔═╡ 2f0e9798-055b-478b-b65c-9004954802f9
histogram(chain,bins=100)

# ╔═╡ 71fa56fb-a685-48fb-8789-57335ea51cf5
plot(chain,size=(600,200))

# ╔═╡ 16611fc5-c577-4fbf-961b-f3407dce63a5
final_belief

# ╔═╡ 1999803a-45bf-4bef-8a6a-5941b3316aff
beta_fit=fit(Beta,Array(chain))

# ╔═╡ be9f297f-c24e-4ece-b1e0-93c42262d50b
mean(chain)

# ╔═╡ d077b77d-d6ae-4ad1-a8a5-c235241fe190
function chainplot()
	
pl=density(chain; xlim=(0, 1), legend=:best, w=2, c=:blue,label="Chain density",size=(400,300))
plot!(pl,
    beta_fit,
	color=:red,
	linewidth=2,
	label="Chain fit"
)

plot!(pl,
    final_belief,
    xlabel="probability of heads",
    ylabel="",
    title="",
    xlim=(0, 1),
    label="Closed-form",
    fill=0,
    α=0.1,
    linewidth=3,
	color=:green,
    fillcolor=:green,
)

vline!([p_true]; label="True probability", c=:orange)
end

# ╔═╡ 521f0bf0-60c6-4291-867b-123e44d11d4f
EL.hbox([
md"""
- Blue: density plot of the approximate posterior distribution using HMC
- Green:  density plot of the posterior distribution in closed-form obtained from
   the first example
- Red: densitiy plot of fit of chain samples to beta distribution
- Orange: true probability
""",
chainplot()])

# ╔═╡ 211033f2-587b-466d-869a-a7fe9c922ca6
html"""<hr>"""

# ╔═╡ bf119660-8e2b-4306-a762-e436808ce346
TableOfContents()

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[[deps.SpecialFunctions]]
deps = ["ChainRulesCore", "IrrationalConstants", "LogExpFunctions", "OpenLibm_jll", "OpenSpecFun_jll"]
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[[deps.SplittablesBase]]
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[[deps.Static]]
deps = ["IfElse"]
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[[deps.StaticArrays]]
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[[deps.StatisticalTraits]]
deps = ["ScientificTypesBase"]
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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.StatsPlots]]
deps = ["AbstractFFTs", "Clustering", "DataStructures", "DataValues", "Distributions", "Interpolations", "KernelDensity", "LinearAlgebra", "MultivariateStats", "Observables", "Plots", "RecipesBase", "RecipesPipeline", "Reexport", "StatsBase", "TableOperations", "Tables", "Widgets"]
git-tree-sha1 = "4d9c69d65f1b270ad092de0abe13e859b8c55cad"
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[[deps.StructArrays]]
deps = ["Adapt", "DataAPI", "StaticArrays", "Tables"]
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[[deps.SuiteSparse]]
deps = ["Libdl", "LinearAlgebra", "Serialization", "SparseArrays"]
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[[deps.TOML]]
deps = ["Dates"]
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[[deps.TableOperations]]
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[[deps.TableTraits]]
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[[deps.Tables]]
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[[deps.Tar]]
deps = ["ArgTools", "SHA"]
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[[deps.TerminalLoggers]]
deps = ["LeftChildRightSiblingTrees", "Logging", "Markdown", "Printf", "ProgressLogging", "UUIDs"]
git-tree-sha1 = "62846a48a6cd70e63aa29944b8c4ef704360d72f"
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[[deps.Test]]
deps = ["InteractiveUtils", "Logging", "Random", "Serialization"]
uuid = "8dfed614-e22c-5e08-85e1-65c5234f0b40"

[[deps.Tracker]]
deps = ["Adapt", "DiffRules", "ForwardDiff", "LinearAlgebra", "LogExpFunctions", "MacroTools", "NNlib", "NaNMath", "Printf", "Random", "Requires", "SpecialFunctions", "Statistics"]
git-tree-sha1 = "0874c1b5de1b5529b776cfeca3ec0acfada97b1b"
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[[deps.Transducers]]
deps = ["Adapt", "ArgCheck", "BangBang", "Baselet", "CompositionsBase", "DefineSingletons", "Distributed", "InitialValues", "Logging", "Markdown", "MicroCollections", "Requires", "Setfield", "SplittablesBase", "Tables"]
git-tree-sha1 = "c76399a3bbe6f5a88faa33c8f8a65aa631d95013"
uuid = "28d57a85-8fef-5791-bfe6-a80928e7c999"
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[[deps.TreeViews]]
deps = ["Test"]
git-tree-sha1 = "8d0d7a3fe2f30d6a7f833a5f19f7c7a5b396eae6"
uuid = "a2a6695c-b41b-5b7d-aed9-dbfdeacea5d7"
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[[deps.Turing]]
deps = ["AbstractMCMC", "AdvancedHMC", "AdvancedMH", "AdvancedPS", "AdvancedVI", "BangBang", "Bijectors", "DataStructures", "Distributions", "DistributionsAD", "DocStringExtensions", "DynamicPPL", "EllipticalSliceSampling", "ForwardDiff", "Libtask", "LinearAlgebra", "MCMCChains", "NamedArrays", "Printf", "Random", "Reexport", "Requires", "SciMLBase", "SpecialFunctions", "Statistics", "StatsBase", "StatsFuns", "Tracker", "ZygoteRules"]
git-tree-sha1 = "ef0fdc72023c4480a9372f32db88cce68b186e8a"
uuid = "fce5fe82-541a-59a6-adf8-730c64b5f9a0"
version = "0.21.1"

[[deps.URIs]]
git-tree-sha1 = "97bbe755a53fe859669cd907f2d96aee8d2c1355"
uuid = "5c2747f8-b7ea-4ff2-ba2e-563bfd36b1d4"
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[[deps.UUIDs]]
deps = ["Random", "SHA"]
uuid = "cf7118a7-6976-5b1a-9a39-7adc72f591a4"

[[deps.UnPack]]
git-tree-sha1 = "387c1f73762231e86e0c9c5443ce3b4a0a9a0c2b"
uuid = "3a884ed6-31ef-47d7-9d2a-63182c4928ed"
version = "1.0.2"

[[deps.Unicode]]
uuid = "4ec0a83e-493e-50e2-b9ac-8f72acf5a8f5"

[[deps.UnicodeFun]]
deps = ["REPL"]
git-tree-sha1 = "53915e50200959667e78a92a418594b428dffddf"
uuid = "1cfade01-22cf-5700-b092-accc4b62d6e1"
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[[deps.Unzip]]
git-tree-sha1 = "34db80951901073501137bdbc3d5a8e7bbd06670"
uuid = "41fe7b60-77ed-43a1-b4f0-825fd5a5650d"
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[[deps.Wayland_jll]]
deps = ["Artifacts", "Expat_jll", "JLLWrappers", "Libdl", "Libffi_jll", "Pkg", "XML2_jll"]
git-tree-sha1 = "3e61f0b86f90dacb0bc0e73a0c5a83f6a8636e23"
uuid = "a2964d1f-97da-50d4-b82a-358c7fce9d89"
version = "1.19.0+0"

[[deps.Wayland_protocols_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4528479aa01ee1b3b4cd0e6faef0e04cf16466da"
uuid = "2381bf8a-dfd0-557d-9999-79630e7b1b91"
version = "1.25.0+0"

[[deps.Widgets]]
deps = ["Colors", "Dates", "Observables", "OrderedCollections"]
git-tree-sha1 = "505c31f585405fc375d99d02588f6ceaba791241"
uuid = "cc8bc4a8-27d6-5769-a93b-9d913e69aa62"
version = "0.6.5"

[[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"
version = "1.3.4+4"

[[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"
version = "1.2.12+1"

[[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"
version = "5.1.0+0"

[[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"
version = "1.41.0+1"

[[deps.p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"
version = "16.2.1+1"

[[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:
# ╟─c6030b7a-bfd7-4fba-82b2-16ac637d6cb8
# ╟─e3edcefe-9d05-46cd-9122-43fc6c32e2ad
# ╟─0bb322e1-c730-4e43-8c7f-aedfad29790a
# ╠═193f0cae-53a4-48d0-98ed-6eb94aad5712
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