ABSTRACT
The statistical methods in this book belong to a branch of statistics known as Bayesian statistics. An idealized Bayesian analysis has three steps. Step 1 is specifying a probabilistic model for known quantities in the data, unknown quantities of interest, and other unknown quantities needed in the model. Step 2 is inferring the unknown quantities from the observed data, also using probability distributions.
The joint probabilistic model specified in Step 1 can be decomposed into two terms. The first term is the probability distribution of the unknowns, and is referred to as the prior. The second term is the conditional probability distribution of the data, given a value for the unknowns, and is referred to as the likelihood. The inference step in a Bayesian analysis consists of deriving the conditional probability distribution of the unknowns, given the data, which is referred to as the posterior distribution.
Step 3 is model checking. We need to assess the quality of the inferences, by examining the agreement with the data and the substantive implications of the model, and trying alternative specifications. If problems are found, we return to Step 1, and reformulate our model.
