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May 2, 2022 16:23
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Turing example with updating belief after single coinflip
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| ### 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() | |
| # ╔═╡ 00000000-0000-0000-0000-000000000001 | |
| PLUTO_PROJECT_TOML_CONTENTS = """ | |
| [deps] | |
| Distributions = "31c24e10-a181-5473-b8eb-7969acd0382f" | |
| MCMCChains = "c7f686f2-ff18-58e9-bc7b-31028e88f75d" | |
| Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" | |
| PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8" | |
| Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c" | |
| StatsPlots = "f3b207a7-027a-5e70-b257-86293d7955fd" | |
| Turing = "fce5fe82-541a-59a6-adf8-730c64b5f9a0" | |
| [compat] | |
| Distributions = "~0.25.56" | |
| MCMCChains = "~5.2.0" | |
| Plots = "~1.28.1" | |
| PlutoUI = "~0.7.38" | |
| StatsPlots = "~0.14.33" | |
| Turing = "~0.21.1" | |
| """ | |
| # ╔═╡ 00000000-0000-0000-0000-000000000002 | |
| PLUTO_MANIFEST_TOML_CONTENTS = """ | |
| # This file is machine-generated - editing it directly is not advised | |
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| 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 | |
| # ╟─faf423b4-b9b5-4b66-a8e6-48128c4f2d43 | |
| # ╠═d69cfc8f-e48e-4e3e-9076-07ee07736e4e | |
| # ╠═0159ae15-ac81-4adc-8be5-b7ba9a640203 | |
| # ╠═314fec7b-827b-49d5-b9ec-df7ee87f7724 | |
| # ╟─9589511e-8f23-4611-9fe3-eefbd4d554aa | |
| # ╠═c2f70add-e0e6-44a2-ae59-7d8a667be1e2 | |
| # ╟─23f9c22e-6ea8-4602-9eff-661c0b3163c7 | |
| # ╠═62e88f44-2903-49d1-a6e6-9ec4877c4ecb | |
| # ╠═6dd570b1-0b3d-476e-a172-8686a29c1f0f | |
| # ╠═59ae1cb9-0dfc-4ae4-b8a0-c2428a5921cc | |
| # ╟─a7f72a2a-b3ab-4bbf-8e4d-34f6ba7f796a | |
| # ╟─c43efead-6daf-45c3-9c97-a4bfec45dd6d | |
| # ╠═7a8c4f4b-6741-4542-9ea3-2178916d9a8d | |
| # ╟─69fb63fd-9f74-4329-9663-e990f764bf42 | |
| # ╠═aca0d9e6-32be-4cd9-8252-624ec94b1879 | |
| # ╟─9d965e5c-b294-4777-9042-51b5db7ed7b0 | |
| # ╠═3afa6935-af80-41a8-b4c6-33441185cea2 | |
| # ╟─64210e98-d289-4a4a-ae7f-a5748358d5bf | |
| # ╠═3f4591c7-f7ec-4a65-bfc7-06415ffe4a01 | |
| # ╠═5014a882-a8ca-471c-8e5f-807e126c159a | |
| # ╟─97bc750b-ba7a-40d2-a6a4-b2ad6e2adf63 | |
| # ╠═2f0e9798-055b-478b-b65c-9004954802f9 | |
| # ╠═71fa56fb-a685-48fb-8789-57335ea51cf5 | |
| # ╠═16611fc5-c577-4fbf-961b-f3407dce63a5 | |
| # ╠═1999803a-45bf-4bef-8a6a-5941b3316aff | |
| # ╠═be9f297f-c24e-4ece-b1e0-93c42262d50b | |
| # ╠═521f0bf0-60c6-4291-867b-123e44d11d4f | |
| # ╠═d077b77d-d6ae-4ad1-a8a5-c235241fe190 | |
| # ╟─211033f2-587b-466d-869a-a7fe9c922ca6 | |
| # ╠═1b8ee8b9-e8d8-4b80-b610-fc9843d05b31 | |
| # ╠═bf119660-8e2b-4306-a762-e436808ce346 | |
| # ╟─00000000-0000-0000-0000-000000000001 | |
| # ╟─00000000-0000-0000-0000-000000000002 |
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