Poisson approximation to Binomial

JuliaActuary is an ecosystem of packages that makes Julia the easiest language to get started for actuarial workflows.
modeling
statistics
experience-analysis

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:

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 analysis will show.

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:

using Turing
using CairoMakie
using StatsBase
using MCMCChains
using DataFrames
using ThreadsX
using Logging
using Markdown

include(joinpath(@__DIR__, "..", "..", "assets", "themes", "cassette_futurism.jl"))
set_theme!(cassette_futurism_theme())
Precompiling packages...
   4490.0 ms  ✓ QuartoNotebookWorkerMakieExt (serial)
  1 dependency successfully precompiled in 5 seconds
Precompiling packages...
   3518.3 ms  ✓ QuartoNotebookWorkerCairoMakieExt (serial)
  1 dependency successfully precompiled in 4 seconds

““”

Define the Models

@model function poisson(N,n_events) 
    q ~ Beta(1,1)
    
    n_events ~ Poisson(q*N)
end

@model function binom(N,n_events) 
    q ~ Beta(1,1)
    
    n_events ~ Binomial(N,q)
end
binom (generic function with 2 methods)

Simulation Parameters

trials = 30

# true probabilities
qs = [0.05,0.25,0.5,0.75,0.95]

# number of observations
Ns = [10,25,50,100,250,500,1000,5000]

# combine q and N into joint model points
model_points =  [(;q,N) for q in qs, N in Ns]
5×8 Matrix{@NamedTuple{q::Float64, N::Int64}}:
 (q = 0.05, N = 10)  (q = 0.05, N = 25)  …  (q = 0.05, N = 5000)
 (q = 0.25, N = 10)  (q = 0.25, N = 25)     (q = 0.25, N = 5000)
 (q = 0.5, N = 10)   (q = 0.5, N = 25)      (q = 0.5, N = 5000)
 (q = 0.75, N = 10)  (q = 0.75, N = 25)     (q = 0.75, N = 5000)
 (q = 0.95, N = 10)  (q = 0.95, N = 25)     (q = 0.95, N = 5000)

Sample from the Posterior

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:

Logging.disable_logging(Logging.Warn); #Disable warning logs to improve sampling time

bpchains = map(model_points) do mp
    
    ThreadsX.map(1:trials) do i
        claims = sum(rand() < mp.q for _ in 1:mp.N)
        bc = MCMCChains.Chains(sample(binom(mp.N,claims), NUTS(), 500))
        pc = MCMCChains.Chains(sample(poisson(mp.N,claims), NUTS(), 500))

        (;bc,pc)
    end
end
5×8 Matrix{Vector{@NamedTuple{bc::Chains{Float64, AxisArrays.AxisArray{Float64, 3, Array{Float64, 3}, Tuple{AxisArrays.Axis{:iter, UnitRange{Int64}}, AxisArrays.Axis{:var, Vector{Symbol}}, AxisArrays.Axis{:chain, UnitRange{Int64}}}}, Missing, @NamedTuple{parameters::Vector{Symbol}, internals::Vector{Symbol}}, @NamedTuple{varname_to_symbol::OrderedDict{AbstractPPL.VarName, Symbol}, start_time::Vector{Float64}, stop_time::Vector{Float64}}}, pc::Chains{Float64, AxisArrays.AxisArray{Float64, 3, Array{Float64, 3}, Tuple{AxisArrays.Axis{:iter, UnitRange{Int64}}, AxisArrays.Axis{:var, Vector{Symbol}}, AxisArrays.Axis{:chain, UnitRange{Int64}}}}, Missing, @NamedTuple{parameters::Vector{Symbol}, internals::Vector{Symbol}}, @NamedTuple{varname_to_symbol::OrderedDict{AbstractPPL.VarName, Symbol}, start_time::Vector{Float64}, stop_time::Vector{Float64}}}}}}:
 [(bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3}))  …  (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3}))]  …  [(bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3}))  …  (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3}))]
 [(bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3}))  …  (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3}))]     [(bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3}))  …  (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3}))]
 [(bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3}))  …  (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3}))]     [(bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3}))  …  (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3}))]
 [(bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3}))  …  (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3}))]     [(bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3}))  …  (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3}))]
 [(bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3}))  …  (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3}))]     [(bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3}))  …  (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3})), (bc = MCMC chain (500×15×1 Array{Float64, 3}), pc = MCMC chain (500×15×1 Array{Float64, 3}))]

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.

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:

function plot_band_under!(ax,plot,y,low,high,label="")
    function points(plot)
        # Modern Makie: positions on the Lines child are arg-1's Observable.
        # (Was plot.plots[2].converted[1][] pre-ComputePipeline.)
        pts = plot.plots[2][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′)
    typeof(xs′[filt]), typeof(fill(y,length(filt))), typeof(ys′[filt])
    b = band!(
        ax,
        Float64.(xs′[filt]), 
        Float64.(fill(y * 1.0, length(filt))), 
        Float64.(ys′[filt]),
        color=(plot.color[], 0.9),transparency=true,shading = false) # 0.25 alpha
    translate!(b,0,0,5)
end


let
    f = Figure(size=(1100, 720))
    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]...)
            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