Posteriors, conjugacy, and exponential families for completely random measures

Understand

We demonstrate how to calculate posteriors for general CRM-based priors and likelihoods for Bayesian nonparametric models.

• We further show how to represent Bayesian nonparametric priors as a sequence of finite draws using a size-biasing approach---and how to represent full Bayesian nonparametric models via finite marginals.
• Motivated by conjugate priors based on exponential family representations of likelihoods, we introduce a notion of exponential families for CRMs, which we call exponential CRMs.
• This construction allows us to specify automatic Bayesian nonparametric conjugate priors for exponential CRM likelihoods.