Gamma Models

Gamma models are appropriate for positive continuous data. Each cluster k has a shape parameter α_k and a rate parameter β_k, and the likelihood for observation i in cluster k is:

y_i | α_k, β_k ~ Gamma(α_k, β_k)

Currently one Gamma model is implemented: GammaClusterShapeMarg, which marginalises out the rate parameters β_k analytically and samples the shape parameters α_k via Metropolis–Hastings.

GammaClusterShapeMarg

The rate β_k is integrated out analytically using a conjugate Gamma prior, while the shape α_k is sampled via Metropolis–Hastings with a log-normal random walk proposal. The proposal standard deviation for α_k is controlled by prop_sds[:α] in MCMCOptions. See the API Reference for full type documentation.

Example:

using DistanceDependentCRP, Random
Random.seed!(7)

sim = simulate_gamma_data(60, [2.0, 8.0], [1.0, 1.0]; α=0.3, scale=1.5)
data = ContinuousData(sim.y, sim.D)

ddcrp = DDCRPParams(0.3, 1.5)
priors = GammaClusterShapeMargPriors(α_a=2.0, α_b=1.0, β_a=2.0, β_b=1.0)
opts = MCMCOptions(n_samples=5000, prop_sds=Dict(:α => 0.3))

# GammaClusterShapeMarg uses RJMCMC (shape α is explicit)
samples, diag = mcmc(
    GammaClusterShapeMarg(), data, ddcrp, priors, LogNormalMomentMatch(0.3);
    fixed_dim_proposal=WeightedMean(), opts=opts
)

println(acceptance_rates(diag))

Tuning tips:

• The MH proposal SD for α_k is set via prop_sds=Dict(:α => σ). Values around 0.2–0.5 work well for moderate-sized clusters.
• If acceptance rates are too low, reduce prop_sds[:α]; if too high (> 0.8), increase it.
• GammaClusterShapeMarg always uses RJMCMC because the shape parameter is not conjugately marginalised. Use PriorProposal(), LogNormalMomentMatch(σ), or FixedDistributionProposal(d) as birth proposals.

Simulation

See simulate_gamma_data in the API Reference.