Bayesian analysis for controlled branching processes

Abstract

A controlled branching process is a stochastic model that is well suited to describing the probabilistic evolution of populations in which, for various reasons of an environmental, social, or other nature, there is a mechanism that establishes the number of progenitors who take part in each generation. For this model, a Bayesian analysis is described, considering a non-parametric offspring distribution and control distributions belonging to the power series family that depend on a single parameter termed the control parameter. Inferences on the offspring distribution, on the offspring mean, and on the control parameter (or on its parametrization as the migration parameter) are considered within two sampling schemes: first, the classical branching theory scheme based on the observation of the entire family tree; and second, the more realistic situation in which only the generation-by-generation population size is observed. In this latter case, the Dirichlet process and the Gibbs sampler are used to estimate the posterior density of the main parameters of interest. Inference on posterior predictive distributions for as-yet unobserved generations is also considered. Monte Carlo sampling based and analytical approximations are discussed. The results are illustrated with simulated data.