ABSTRACT
This chapter discusses inference and model checking.
In most cases, the posterior distribution is diffcult to work with using algebra alone. Bayesian statisticians use Markov chain Monte Carlo techniques to generate a large sample of draws from the posterior distribution, and use the sample to calculate summary measures of the posterior distribution, such as point estimates and credible intervals. We show that it is easy to derive summary measures for two types of new unknown quantities: (1) quantities that are completely determined by unknown quantities in our posterior distribution; (2) quantities that depend on unknown quantities in our posterior distribution, but are not determined completely by them.
The Bayesian approach can naturally deal with missing data. The missing values are treated as unknown quantities, and are included in the joint probabilistic model and the posterior distribution along with other unknown quantities. The Bayesian approach can also unify estimation and forecasting, by treating forecasting as a type of estimation with missing data.
We introduce two techniques for Bayesian model checking: heldback data and replicate data. We discuss in an optional section how simulations can be used to assess a model's performance over a range of different datasets.
