Gammapriori

Gammapriori is a term used in Bayesian statistics to denote the gamma prior distribution when modeling positive scale parameters such as Poisson rate parameters or precision terms.

In its standard form, a gamma prior for a rate λ takes shape α > 0 and rate β > 0, and is written as λ ~ Gamma(α, β). The density is proportional to λ^{α-1} e^{-β λ}, for λ > 0. The gamma prior is a conjugate prior for many Poisson-like models, which means the resulting posterior distribution remains in the gamma family under a Poisson likelihood.

Due to conjugacy with the Poisson likelihood, if data consists of counts x_i assumed Poisson with rate λ, the posterior for λ remains Gamma with updated parameters α_post = α + ∑ x_i and β_post = β + n, where n is the number of observations. This leads to closed-form expressions for posterior moments and convenient analytical updates.

Applications include modeling event rates in epidemiology, queueing theory, and reliability engineering, as well as various Bayesian machine learning contexts that require positive scale parameters. The gamma prior provides both interpretability and computational convenience, particularly in hierarchical and online updating settings.

Choice of hyperparameters reflects prior knowledge: α and β encode prior strength and the expected rate. Small values yield vague or weakly informative priors, while larger values exert stronger influence on the posterior. The gamma prior is also central to the gamma-Poisson mixture, where integrating out λ yields a Negative Binomial distribution for counts, linking Gamma priors to common count models.