Poisson Models

Poisson models are appropriate for non-negative integer count data. Each cluster k has a rate parameter λ_k, and the likelihood for observation i in cluster k is:

y_i | λ_k ~ Poisson(λ_k)

For population models the rate is ρk × Pi, where P_i is a known exposure or population size:

y_i | ρ_k, P_i ~ Poisson(ρ_k × P_i)

All Poisson models use a conjugate Gamma prior on the rate, which allows analytical marginalisation.

PoissonClusterRates

Explicit cluster rates λ_k sampled via RJMCMC. Use this when you need posterior distributions over the cluster rates themselves. State holds a λ_dict::Dict{Vector{Int},Float64} mapping sorted customer index vectors to rates; priors are λ_a, λ_b for the Gamma(λ_a, λ_b) prior. See the API Reference for full type documentation.

Example:

data = CountData(y, D)
ddcrp = DDCRPParams(0.5, 1.0)
priors = PoissonClusterRatesPriors(1.0, 0.5)
opts = MCMCOptions(n_samples=3000)

samples, diag = mcmc(
    PoissonClusterRates(), data, ddcrp, priors, LogNormalMomentMatch(0.5);
    fixed_dim_proposal=WeightedMean(), opts=opts
)

PoissonClusterRatesMarg

Cluster rates λ_k are integrated out analytically. Inference uses conjugate Gibbs sampling, which is faster and typically mixes better than RJMCMC. State holds only c::Vector{Int}; no parameter dictionaries. See the API Reference for full type documentation.

Example:

data = CountData(y, D)
priors = PoissonClusterRatesMargPriors(1.0, 0.5)

samples = mcmc(PoissonClusterRatesMarg(), data, ddcrp, priors, ConjugateProposal(); opts=opts)

PoissonPopulationRates

Cluster rate multipliers ρ_k with per-observation population offsets P_i. The expected count for observation i is ρk × Pi. Supports missing observations via a BitVector mask on CountDataWithPopulation. See the API Reference for full type documentation.

Example:

data = CountDataWithPopulation(y, population, D)
priors = PoissonPopulationRatesPriors(1.0, 0.5)

samples, diag = mcmc(
    PoissonPopulationRates(), data, ddcrp, priors, LogNormalMomentMatch(0.5);
    fixed_dim_proposal=WeightedMean(), opts=opts
)

With missing data:

mask = BitVector([false, true, false, ...])   # true = missing
data = CountDataWithPopulation(y, population, D, mask)

Missing observations contribute only through the DDCRP prior, not the likelihood.

PoissonPopulationRatesMarg

Like PoissonPopulationRates but with cluster rate multipliers ρ_k marginalised out. Uses conjugate Gibbs sampling. See the API Reference for full type documentation.

Example:

data = CountDataWithPopulation(y, population, D)
priors = PoissonPopulationRatesMargPriors(1.0, 0.5)

samples = mcmc(PoissonPopulationRatesMarg(), data, ddcrp, priors, ConjugateProposal(); opts=opts)

Simulation

See simulate_poisson_data in the API Reference.