A Gaussian mixture model with Soss.jl

soss-mixture.jl

### A Pluto.jl notebook ###
# v0.15.1

using Markdown
using InteractiveUtils

# ╔═╡ 6ca58e9a-0707-11ec-1a6c-bdb652993535
begin
	using Plots
	using StatsPlots
	using PGFPlotsX
	push!(PGFPlotsX.CUSTOM_PREAMBLE, raw"\usetikzlibrary{snakes, bayesnet}")
	import Distributions
	using MeasureTheory
	using Soss
	using SampleChainsDynamicHMC
	using FillArrays
	using MappedArrays
	using LinearAlgebra
	using TransformVariables
end

# ╔═╡ 59cb6cb4-3893-40a2-90c6-e09b735b8175
md"""
# Gaussian mixture
"""

# ╔═╡ dfeb4b28-f026-4f1f-be11-ceb6d1d6190c
plates = TikzDocument(raw"""
\begin{tikzpicture}
	
	\tikzstyle{selector}=[-latex, -|, snake=snake,segment amplitude=.4mm,segment length=2mm,line after snake=1mm, thick]
	
	\node[obs]                    (x)                   {$x_i$} ;
	\node[latent, above=of x]     (z)                   {$z_i$} ;
	\node[latent, above=of z]     (w)                   {$w$} ;	
	\node[const, above=of w]      (alpha)               {$\alpha$} ;
	\node[latent, right=2cm of x] (mu)                  {$\mu_k$} ;
	\node[const, right=2cm of mu, yshift=1cm] (mu0)     {$\mu_0 = 0$} ;
	\node[const, right=2cm of mu, yshift=-1cm] (sigma0) {$\sigma_0 = 1$} ; 
	
	\edge            {alpha}  {w} ;
	\edge            {w}      {z} ;
	\edge [selector] {z}      {x} ;
	\edge            {mu}     {x} ;
	\edge            {mu0}    {mu} ;
	\edge            {sigma0} {mu} ;
	
	\plate {zx} {(z)(x)} {$N$} ;
	\plate {mu} {(mu)}   {$K$} ;

\end{tikzpicture}
""")

# ╔═╡ ea113e33-0de9-4b44-8f30-952d4aacd152
# begin
# 	struct Product{
# 		S<:Distributions.ValueSupport,
# 		T<:Distributions.MultivariateDistribution{S},
# 		V<:AbstractVector{T},
# 	   } <: Distributions.MultivariateDistribution{S}
# 		v::V
# 		function Product(v::V) where
# 			V<:AbstractVector{T} where
# 			T<:Distributions.MultivariateDistribution{S} where
# 			S<:Distributions.ValueSupport
# 			return new{S, T, V}(v)
# 		end
# 	end

# 	Base.length(d::Product) = length(d.v)
# 	function Base.eltype(::Type{<:Product{S, T}}) where {S<:Distributions.ValueSupport,
# 												T<:Distributions.MultivariateDistribution{S}}
# 		return eltype(T)
# 	end

# 	function _rand!(rng::Distributions.AbstractRNG, d::Product, x::AbstractVector)
# 		broadcast!(dn->rand(rng, dn), x, d.v)
# 	end

# 	function _logpdf(d::Product, x::AbstractVector)
# 		sum(n -> logpdf(d.v[n], x[n]), 1:length(d))
# 	end
		
# 	function Distributions.rand(rng::Distributions.AbstractRNG, s::Product)
# 		_rand!(rng, s, Vector{Vector{eltype(s)}}(undef, length(s)))
# 	end
# end

# ╔═╡ 0874d9f4-05e5-475c-a48b-950feb7a503a
m = @model N begin
	# parameters
	σ0 ~ Lebesgue(ℝ)
	μ0 ~ Lebesgue(ℝ)
	α ~ Lebesgue(ℝ₊)
	K = 2
	# random variables
	μ ~ Normal(μ0, σ0) |> iid(K)
	w ~ Dists.Dirichlet(K, abs(α))
	
	# option 1
	# z ~ For(N) do _ Distributions.Categorical(w) end
	# x ~ For(z) do zi Distributions.MvNormal([μ[zi], μ[zi]], 1.) end
	
	# option 2
	# components = mappedarray(μ) do μk begin (Fill(μk, 2), 1.) end end
	# mixture = Dists.MixtureModel(Dists.MvNormal, components, w)
	# x ~ Product(fill(mixture, N))
	
	# option 3
	# mixture = Dists.MixtureModel(Dists.Normal, μ, w)
	# x ~  Dists.MatrixReshaped(mixture |> iid(K*N), K, N)
	
	# option 4
	# https://github.com/cscherrer/Soss.jl/issues/296#issuecomment-907296790
	mixture = Dists.MixtureModel(Dists.Normal, μ, w)
	x ~ Dists.MatrixReshaped(Dists.Product(fill(mixture, K*N)), K, N)
	
end

# ╔═╡ 5d7c7b1c-171f-4695-bf9c-3836c53eda67
prior_model = predictive(m, :σ0, :μ0, :α)

# ╔═╡ 160b78c3-152c-4876-9b74-77b592c4ecbb
prior_data = rand(prior_model(N=30, σ0=1., μ0=0., α=1.))

# ╔═╡ 39f406bc-b49d-4746-a0b5-469d6a415963
begin
	# scatter(
	# 	[xi[1] for xi in prior_data.x], 
	# 	[xi[2] for xi in prior_data.x], 
	# 	legend = false, 
	# 	# color = prior_data.z,
	# 	title = "synthetic data"
	# )
	scatter(
		prior_data.x[1, :], 
		prior_data.x[2, :], 
		legend = false, 
		# color = prior_data.z,
		title = "synthetic data"
	)
	scatter!([prior_data.μ[1]], [prior_data.μ[1]], color=:yellow)
	scatter!([prior_data.μ[2]], [prior_data.μ[2]], color=:yellow)
end

# ╔═╡ 65ffda1c-47a6-46de-9f4a-a90a54302424
predx = predictive(m, :μ, :w)

# ╔═╡ c1015b69-fb26-45f2-adbb-4b5b48b4d795
data = rand(predx(N=30,  μ=[-3.5, 0.0], w=[0.5, 0.5]))

# ╔═╡ 9b1f9d4d-bc5b-4a4b-8957-c62c6c4be8dc
begin
	# scatter(
	# 	[xi[1] for xi in data.x], 
	# 	[xi[2] for xi in data.x], 
	# 	legend = false, 
	# 	# color = prior_data.z,
	# 	title = "synthetic data"
	# )
	scatter(
		data.x[1, :], 
		data.x[2, :], 
		legend = false, 
		# color = data.z,
		title = "synthetic data"
	)
	scatter!([-3.5], [-3.5], color=:yellow)
	scatter!([0.0], [0.0], color=:yellow)
end

# ╔═╡ c5a7b0db-1d65-4dd9-a4dc-38c73c0391b7
posterior = m(N=30)|(x=data.x,)

# ╔═╡ 1d4d27dd-f3ff-4a99-86ca-cdd825fd5d65
function Soss.xform(d::Dists.Dirichlet, _data::NamedTuple) 
	TransformVariables.UnitSimplex(length(d.alpha))
end

# ╔═╡ dda4ac2d-ef53-4276-80da-7394302b7d63
l(x) = logdensity(posterior, x)

# ╔═╡ b121795e-6904-45de-84fc-b1581e73b347
tr = xform(posterior)

# ╔═╡ d851b01f-7668-47db-939f-5cac63e2ab27
chain = newchain(3, dynamichmc(), l, tr)

# ╔═╡ e939f3b1-429e-48f1-a70d-64968aac7fc3
chain.μ

# ╔═╡ 3ebf7390-5a23-41a2-8fe1-8af2856b20ab
# fails with UndefVarError: sample not defined
sample(posterior, dynamichmc())

# ╔═╡ 00000000-0000-0000-0000-000000000001
PLUTO_PROJECT_TOML_CONTENTS = """
[deps]
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[compat]
Distributions = "~0.25.14"
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"""

# ╔═╡ 00000000-0000-0000-0000-000000000002
PLUTO_MANIFEST_TOML_CONTENTS = """
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[[Static]]
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git-tree-sha1 = "62701892d172a2fa41a1f829f66d2b0db94a9a63"
uuid = "aedffcd0-7271-4cad-89d0-dc628f76c6d3"
version = "0.3.0"

[[StaticArrays]]
deps = ["LinearAlgebra", "Random", "Statistics"]
git-tree-sha1 = "3240808c6d463ac46f1c1cd7638375cd22abbccb"
uuid = "90137ffa-7385-5640-81b9-e52037218182"
version = "1.2.12"

[[Statistics]]
deps = ["LinearAlgebra", "SparseArrays"]
uuid = "10745b16-79ce-11e8-11f9-7d13ad32a3b2"

[[StatsAPI]]
git-tree-sha1 = "1958272568dc176a1d881acb797beb909c785510"
uuid = "82ae8749-77ed-4fe6-ae5f-f523153014b0"
version = "1.0.0"

[[StatsBase]]
deps = ["DataAPI", "DataStructures", "LinearAlgebra", "Missings", "Printf", "Random", "SortingAlgorithms", "SparseArrays", "Statistics", "StatsAPI"]
git-tree-sha1 = "8cbbc098554648c84f79a463c9ff0fd277144b6c"
uuid = "2913bbd2-ae8a-5f71-8c99-4fb6c76f3a91"
version = "0.33.10"

[[StatsFuns]]
deps = ["IrrationalConstants", "LogExpFunctions", "Reexport", "Rmath", "SpecialFunctions"]
git-tree-sha1 = "20d1bb720b9b27636280f751746ba4abb465f19d"
uuid = "4c63d2b9-4356-54db-8cca-17b64c39e42c"
version = "0.9.9"

[[StatsPlots]]
deps = ["Clustering", "DataStructures", "DataValues", "Distributions", "Interpolations", "KernelDensity", "LinearAlgebra", "MultivariateStats", "Observables", "Plots", "RecipesBase", "RecipesPipeline", "Reexport", "StatsBase", "TableOperations", "Tables", "Widgets"]
git-tree-sha1 = "e7d1e79232310bd654c7cef46465c537562af4fe"
uuid = "f3b207a7-027a-5e70-b257-86293d7955fd"
version = "0.14.26"

[[StructArrays]]
deps = ["Adapt", "DataAPI", "Tables"]
git-tree-sha1 = "44b3afd37b17422a62aea25f04c1f7e09ce6b07f"
uuid = "09ab397b-f2b6-538f-b94a-2f83cf4a842a"
version = "0.5.1"

[[SuiteSparse]]
deps = ["Libdl", "LinearAlgebra", "Serialization", "SparseArrays"]
uuid = "4607b0f0-06f3-5cda-b6b1-a6196a1729e9"

[[SymbolicCodegen]]
deps = ["DataStructures", "MLStyle", "MacroTools", "RuntimeGeneratedFunctions", "SymbolicUtils"]
git-tree-sha1 = "cf1ff3370c3bd1ce8e43c6355bc17c5a28720e72"
uuid = "fc9b0551-4f1b-42e1-8440-e8a535f5a551"
version = "0.2.2"

[[SymbolicUtils]]
deps = ["AbstractAlgebra", "AbstractTrees", "Combinatorics", "ConstructionBase", "DataStructures", "IfElse", "LabelledArrays", "LinearAlgebra", "NaNMath", "Setfield", "SparseArrays", "SpecialFunctions", "StaticArrays", "TimerOutputs"]
git-tree-sha1 = "e0094258633c2d7c00fa61140c2f21c592603d8c"
uuid = "d1185830-fcd6-423d-90d6-eec64667417b"
version = "0.11.0"

[[TOML]]
deps = ["Dates"]
uuid = "fa267f1f-6049-4f14-aa54-33bafae1ed76"

[[TableOperations]]
deps = ["SentinelArrays", "Tables", "Test"]
git-tree-sha1 = "019acfd5a4a6c5f0f38de69f2ff7ed527f1881da"
uuid = "ab02a1b2-a7df-11e8-156e-fb1833f50b87"
version = "1.1.0"

[[TableTraits]]
deps = ["IteratorInterfaceExtensions"]
git-tree-sha1 = "c06b2f539df1c6efa794486abfb6ed2022561a39"
uuid = "3783bdb8-4a98-5b6b-af9a-565f29a5fe9c"
version = "1.0.1"

[[Tables]]
deps = ["DataAPI", "DataValueInterfaces", "IteratorInterfaceExtensions", "LinearAlgebra", "TableTraits", "Test"]
git-tree-sha1 = "d0c690d37c73aeb5ca063056283fde5585a41710"
uuid = "bd369af6-aec1-5ad0-b16a-f7cc5008161c"
version = "1.5.0"

[[Tar]]
deps = ["ArgTools", "SHA"]
uuid = "a4e569a6-e804-4fa4-b0f3-eef7a1d5b13e"

[[Test]]
deps = ["InteractiveUtils", "Logging", "Random", "Serialization"]
uuid = "8dfed614-e22c-5e08-85e1-65c5234f0b40"

[[TimeZones]]
deps = ["Dates", "Future", "LazyArtifacts", "Mocking", "Pkg", "Printf", "RecipesBase", "Serialization", "Unicode"]
git-tree-sha1 = "6c9040665b2da00d30143261aea22c7427aada1c"
uuid = "f269a46b-ccf7-5d73-abea-4c690281aa53"
version = "1.5.7"

[[TimerOutputs]]
deps = ["ExprTools", "Printf"]
git-tree-sha1 = "209a8326c4f955e2442c07b56029e88bb48299c7"
uuid = "a759f4b9-e2f1-59dc-863e-4aeb61b1ea8f"
version = "0.5.12"

[[Trajectories]]
deps = ["RecipesBase", "Tables"]
git-tree-sha1 = "9c7a662752d8b5dd43afd56384738590a58a4cdc"
uuid = "2c80a279-213e-54d7-a557-e9a14725db56"
version = "0.2.2"

[[TransformVariables]]
deps = ["ArgCheck", "DocStringExtensions", "ForwardDiff", "LinearAlgebra", "Pkg", "Random", "UnPack"]
git-tree-sha1 = "9433efc8545a53a9a34de0cdb9316f9982a9f290"
uuid = "84d833dd-6860-57f9-a1a7-6da5db126cff"
version = "0.4.1"

[[Tricks]]
git-tree-sha1 = "8280b6d0096e88b77a84f843fa18620d3b20e052"
uuid = "410a4b4d-49e4-4fbc-ab6d-cb71b17b3775"
version = "0.1.4"

[[TupleVectors]]
deps = ["ArraysOfArrays", "ElasticArrays", "GeneralizedGenerated", "MacroTools", "NestedTuples", "Requires", "StatsBase", "StructArrays"]
git-tree-sha1 = "d1fac3c97b1aabdecae445256aae9eb00bb71e51"
uuid = "615932cf-77b6-4358-adcd-5b7eba981d7e"
version = "0.1.2"

[[URIs]]
git-tree-sha1 = "97bbe755a53fe859669cd907f2d96aee8d2c1355"
uuid = "5c2747f8-b7ea-4ff2-ba2e-563bfd36b1d4"
version = "1.3.0"

[[UUIDs]]
deps = ["Random", "SHA"]
uuid = "cf7118a7-6976-5b1a-9a39-7adc72f591a4"

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

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

[[UnsafeArrays]]
git-tree-sha1 = "038cd6ae292c857e6f91be52b81236607627aacd"
uuid = "c4a57d5a-5b31-53a6-b365-19f8c011fbd6"
version = "1.0.3"

[[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"

[[Wayland_protocols_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll"]
git-tree-sha1 = "2839f1c1296940218e35df0bbb220f2a79686670"
uuid = "2381bf8a-dfd0-557d-9999-79630e7b1b91"
version = "1.18.0+4"

[[Widgets]]
deps = ["Colors", "Dates", "Observables", "OrderedCollections"]
git-tree-sha1 = "eae2fbbc34a79ffd57fb4c972b08ce50b8f6a00d"
uuid = "cc8bc4a8-27d6-5769-a93b-9d913e69aa62"
version = "0.6.3"

[[WoodburyMatrices]]
deps = ["LinearAlgebra", "SparseArrays"]
git-tree-sha1 = "59e2ad8fd1591ea019a5259bd012d7aee15f995c"
uuid = "efce3f68-66dc-5838-9240-27a6d6f5f9b6"
version = "0.5.3"

[[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"

[[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"

[[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"

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

[[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"

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

[[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"

[[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"

[[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"

[[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"

[[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"

[[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"

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

[[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"

[[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"

[[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"

[[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"

[[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"

[[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"

[[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"

[[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"

[[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"

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

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

[[Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "cc4bf3fdde8b7e3e9fa0351bdeedba1cf3b7f6e6"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.0+0"

[[ZygoteRules]]
deps = ["MacroTools"]
git-tree-sha1 = "9e7a1e8ca60b742e508a315c17eef5211e7fbfd7"
uuid = "700de1a5-db45-46bc-99cf-38207098b444"
version = "0.2.1"

[[libass_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "acc685bcf777b2202a904cdcb49ad34c2fa1880c"
uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
version = "0.14.0+4"

[[libfdk_aac_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "7a5780a0d9c6864184b3a2eeeb833a0c871f00ab"
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
version = "0.1.6+4"

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

[[libvorbis_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"]
git-tree-sha1 = "c45f4e40e7aafe9d086379e5578947ec8b95a8fb"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.7+0"

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

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

[[x264_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "d713c1ce4deac133e3334ee12f4adff07f81778f"
uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
version = "2020.7.14+2"

[[x265_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "487da2f8f2f0c8ee0e83f39d13037d6bbf0a45ab"
uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
version = "3.0.0+3"

[[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:
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# ╠═6ca58e9a-0707-11ec-1a6c-bdb652993535
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# ╠═0874d9f4-05e5-475c-a48b-950feb7a503a
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# ╠═39f406bc-b49d-4746-a0b5-469d6a415963
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# ╠═c1015b69-fb26-45f2-adbb-4b5b48b4d795
# ╠═9b1f9d4d-bc5b-4a4b-8957-c62c6c4be8dc
# ╠═c5a7b0db-1d65-4dd9-a4dc-38c73c0391b7
# ╠═1d4d27dd-f3ff-4a99-86ca-cdd825fd5d65
# ╠═dda4ac2d-ef53-4276-80da-7394302b7d63
# ╠═b121795e-6904-45de-84fc-b1581e73b347
# ╟─d851b01f-7668-47db-939f-5cac63e2ab27
# ╠═e939f3b1-429e-48f1-a70d-64968aac7fc3
# ╠═3ebf7390-5a23-41a2-8fe1-8af2856b20ab
# ╟─00000000-0000-0000-0000-000000000001
# ╟─00000000-0000-0000-0000-000000000002