### A Pluto.jl notebook ### # v0.19.11 using Markdown using InteractiveUtils # ╔═╡ 283eda68-3d69-4b2d-a722-a684fe6b7275 begin using Turing using CairoMakie using StatsBase using MCMCChains using DataFrames using ThreadsX using Logging using Markdown end # ╔═╡ 8432d88b-58cd-48fc-9cb1-0bba406c394a # hideall md"""# Poisson approximation to Binomial This notebook explores using the Poissson approximation to the Binomial. This can be useful for a number of reasons: 1. The Binomial probability mass formula becomes more unwieldy for large ``N`` faster than it does for the Poisson. 2. In actuarial, and other contexts, ``N`` can be a non-integer value (think partial period exposures) and rounding up or down could materially distort the posterior estimate if ``N`` is small. We will look at the approximation across a range of parameters ``q`` (the probabilty a binomial event occurs) and ``N`` (the number of chances for the event to occur, or "exposures"). Under certain conditions, the Poisson distribution can approximate the Binomial, where the average number of events, ``λ=N*q``. What are those conditions? [According to Wikipedia](https://en.wikipedia.org/wiki/Binomial_distribution#Poisson_approximation): > According to two rules of thumb, this approximation is good if n ≥ 20 and p ≤ 0.05, or if n ≥ 100 and np ≤ 10. It's not really that simple, and arguably a bit restrictive as this notebook will show. """ # ╔═╡ 159061c3-7149-44d5-a446-bd73d441b648 md" ## Define the Models" # ╔═╡ f875e0a0-84ec-4dbe-a869-bb5903d32727 @model function poisson(N,n_events) q ~ Beta(1,1) n_events ~ Poisson(q*N) end # ╔═╡ b80727b8-96a5-44b8-aed4-230b54c93889 @model function binom(N,n_events) q ~ Beta(1,1) n_events ~ Binomial(N,q) end # ╔═╡ 9d6f631e-b294-40d2-8bcf-846aa31f8eb5 md" ## Simulation Parameters" # ╔═╡ 3e7cf5a7-74c5-4c08-972f-9247c2ffa766 trials = 30 # ╔═╡ 15b90365-97de-49a4-9393-3922731cf0d3 # hideall Markdown.parse("""## Approach We will use Julia and Turing.jl to simulate the posterior distribution. Our prior for the parameter \$q\$ will be \${Uniform} = Beta(1,1)\$. We will sample from the posterior using the No-U-Turn (NUTS) sampler, and aggregate the results of the chains over $(trials) trials of simulated outcomes for the given \$q\$ and \$N\$. This is overkill for a toy problem where we could just model the parameters themselves, but it demonstrates using Bayesian MCMC techniques in a simple, exploratory fashion. We begin by importing the relevant packages: """) # ╔═╡ 0b846cdb-1444-433e-b846-f0eeb6cc3ad2 qs = [0.05,0.25,0.5,0.75,0.95] # ╔═╡ a96caa3b-0f0a-4efc-93ce-9c5f94205f67 Ns = [10,25,50,100,250,500,1000,5000] # ╔═╡ 3cb42e1f-8484-448b-a678-c959a89faeed model_points = [(;q,N) for q in qs, N in Ns] # ╔═╡ f5bc4ab6-c1fd-42bd-9827-427faad4b1f3 md" ## Sample from the Posterior " # ╔═╡ bc44706c-3ecf-43e5-a794-eb6987743e69 Logging.disable_logging(Logging.Warn); #Disable warning logs to improve sampling time # ╔═╡ f600049f-1165-4420-8237-7485764fa338 md" This is a collection of samples (chains) from Markov chains that are sampled in proportion to the posterior density. If this is new to you, I highly recommend the book *Statistical Rethinking*:" # ╔═╡ 216032cf-b8ed-4bbd-9892-32ed58ac877e @time bpchains = map(model_points) do mp ThreadsX.map(1:trials) do i claims = sum(rand() < mp.q for _ in 1:mp.N) bc = sample(binom(mp.N,claims), NUTS(), 500) pc = sample(poisson(mp.N,claims), NUTS(), 500) (;bc,pc) end end # ╔═╡ 426b3cb1-e160-4800-b0a5-9cfe4c6a13e0 # hideall Markdown.parse(""" ## Results The results indicate that the Poisson is a good fit when \$q\$ is small, where "small" depends on N, but in general it seems to provide a good fit in less restrictive cases than the "rule of thumb" quoted below. E.g. go ahead and use the Poisson approximation when you've got enough expsoures even if \$q\$ is well above \$0.10\$. The Poisson approximation also isn't terrible when \$N\$ is as low as 10 as long as \$q\$ is very small (e.g. \$<0.05\$ ). The fit remains poor when \$q >> 0.5\$ and when \$N\$ is small. """) # ╔═╡ 70b82a0d-f8d5-4c4d-9991-40693603567c md"### Visualization This visualization shows the aggregated posterior distribution across all of the trials and model points. The darker shaded band indicates the middle 50% of the posterior:" # ╔═╡ 4e555af6-c486-4d68-b960-0d78df3bbb82 function plot_band_under!(ax,plot,y,low,high,label="") function points(plot) pts = plot.plots[2].converted[1][] [p[1] for p in pts], [p[2] for p in pts] end xs′, ys′ = points(plot) filt = findall(x-> (x ≥ low) && (x ≤ high),xs′) b = band!(ax,xs′[filt], fill(y,length(filt)), ys′[filt], color=(plot.color, 0.9),transparency=true,shading = false,) # 0.25 alpha translate!(b,0,0,5) end # ╔═╡ b963dfd0-8849-4d1a-9aea-646b339db394 let f = Figure(resolution=(1280,960)) a = Any # outer variable to set as axis for to grab legend for (i,N) in enumerate(reverse(Ns)) ax = Axis(f[i+1,1:10], xticks=qs, ylabel="N=$N" ) a = ax xlims!(0,1) model_idx = findall(x->x.N == N,model_points) for n in model_idx c = bpchains[n] q = model_points[n].q y = model_points[n].N bpoints = vcat([x.bc["q"][:] for x in c]...) ppoints = vcat([x.pc["q"][:] for x in c]...) @show bpoints bqtls = quantile(bpoints,[.25,.75]) pqtls = quantile(ppoints,[.25,.75]) j = density!( bpoints, linewidth=10, strokewidth = 1, strokecolor = (:grey30,0.6), label="Binomial", color=(:red,.25), ) plot_band_under!(ax,j,0,bqtls[1],bqtls[2],"Binomial") j = density!( ppoints, color=(:blue,.25), linewidth=10, strokewidth = 1, strokecolor = (:grey30,0.6), label="Poisson", ) plot_band_under!(ax,j,0,pqtls[1],pqtls[2],"Poisson") hideydecorations!(ax,label=false) end scatter!( qs, zeros(length(qs)), marker = :vline, markersize=20, color=:grey30, label="actual value" ) end Legend(f[1,5],a,unique=true,orientation=:horizontal) f end # ╔═╡ ea621f6d-4bc2-4dfe-bf50-7715a2983e1e # hidealll html""" """ # ╔═╡ 00000000-0000-0000-0000-000000000001 PLUTO_PROJECT_TOML_CONTENTS = """ [deps] CairoMakie = "13f3f980-e62b-5c42-98c6-ff1f3baf88f0" DataFrames = "a93c6f00-e57d-5684-b7b6-d8193f3e46c0" Logging = "56ddb016-857b-54e1-b83d-db4d58db5568" MCMCChains = "c7f686f2-ff18-58e9-bc7b-31028e88f75d" Markdown = 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version = "0.2.3+0" [[deps.libaom_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] git-tree-sha1 = "3a2ea60308f0996d26f1e5354e10c24e9ef905d4" uuid = "a4ae2306-e953-59d6-aa16-d00cac43593b" version = "3.4.0+0" [[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 = "5.1.1+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.libsixel_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] git-tree-sha1 = "78736dab31ae7a53540a6b752efc61f77b304c5b" uuid = "075b6546-f08a-558a-be8f-8157d0f608a5" version = "1.8.6+1" [[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 = "1.48.0+0" [[deps.p7zip_jll]] deps = ["Artifacts", "Libdl"] uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0" version = "17.4.0+0" [[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" """ # ╔═╡ Cell order: # ╟─8432d88b-58cd-48fc-9cb1-0bba406c394a # ╟─15b90365-97de-49a4-9393-3922731cf0d3 # ╠═283eda68-3d69-4b2d-a722-a684fe6b7275 # ╟─159061c3-7149-44d5-a446-bd73d441b648 # ╠═f875e0a0-84ec-4dbe-a869-bb5903d32727 # ╠═b80727b8-96a5-44b8-aed4-230b54c93889 # ╟─9d6f631e-b294-40d2-8bcf-846aa31f8eb5 # ╠═3e7cf5a7-74c5-4c08-972f-9247c2ffa766 # ╠═0b846cdb-1444-433e-b846-f0eeb6cc3ad2 # ╠═a96caa3b-0f0a-4efc-93ce-9c5f94205f67 # ╠═3cb42e1f-8484-448b-a678-c959a89faeed # ╟─f5bc4ab6-c1fd-42bd-9827-427faad4b1f3 # ╠═bc44706c-3ecf-43e5-a794-eb6987743e69 # ╟─f600049f-1165-4420-8237-7485764fa338 # ╠═216032cf-b8ed-4bbd-9892-32ed58ac877e # ╟─426b3cb1-e160-4800-b0a5-9cfe4c6a13e0 # ╟─70b82a0d-f8d5-4c4d-9991-40693603567c # ╠═b963dfd0-8849-4d1a-9aea-646b339db394 # ╠═4e555af6-c486-4d68-b960-0d78df3bbb82 # ╠═ea621f6d-4bc2-4dfe-bf50-7715a2983e1e # ╟─00000000-0000-0000-0000-000000000001 # ╟─00000000-0000-0000-0000-000000000002