A Bayesian analysis of tree-structured statistical decision problems

Abstract

This paper concerns the Bayesian analysis of statistical experiments having tree-like structures made up of sequences of independent subexperiments or trials. Members of this class of problems are treated as compound Bernoulli experiments constructed of linear sequences of independent generalized Bernoulli trials having unknown outcome probabilities. Replication instances of these experiments are assumed to follow a multinomial-like sampling scheme capable of generating complete and/or partial observations. From the viewpoint of the theory of statistical distributions, the preposterior analysis of this class of experiments yields a new family of discrete multivariate distributions — the hyper-compound multinomial distribution. The properties of this family of distributions are presented as well as an illustrative numerical example concerning a Bayesian analysis of a simple tree-structured decision model for a 30-day medical prognosis, i.e., either early death or survival, for patients who have suffered an acute myocardial infarction (heart attack).