GammaPosterior

Gammaposterior, commonly referred to as the gamma posterior, is the posterior distribution of a positive rate parameter in Bayesian statistics when the prior distribution is a Gamma distribution. It arises from a conjugate prior relationship: if the data are modeled by a Poisson or Poisson-like likelihood, a Gamma prior yields a Gamma posterior, making analytic updating straightforward.

In the typical Poisson model, suppose N events are observed over an exposure time T, and the rate parameter is lambda. With a Gamma prior Gamma(α, β) expressed in shape α and rate β, the likelihood is proportional to lambda^N exp(-lambda T). The resulting posterior is Gamma(α+N, β+T). If a scale parameterization is used (scale θ = 1/β), the posterior parameters are adjusted accordingly. The posterior mean is E[lambda | data] = (α+N)/(β+T), and the posterior mode is (α+N-1)/(β+T) when α+N > 1.

The gamma posterior provides a convenient framework for uncertainty quantification and inference on the rate parameter. It also implies a gamma-Poisson (negative binomial) predictive distribution for future counts, integrating over the posterior uncertainty in lambda. This conjugacy and closed-form updating make the gamma posterior widely used in fields involving event counts, such as queueing theory, epidemiology, and reliability engineering.

Parameter choice for the prior (α, β) encodes prior beliefs about the rate; larger α and β shift and tighten the posterior accordingly. The gamma posterior is favored for its mathematical tractability and its natural interpretation as updating beliefs about a positive rate parameter with count data.