## ABSTRACT

The key to statistical inference is to understand the random mechanism by which a given set of data (denoted by y ) is generated. Better understanding of the random mechanism, which generates the data, leads to better explanation of the differences observed and potentially better prediction of what one might observe in the future. For example, in item response theory (IRT) applications, the data, y , might be a matrix of item responses of a sample of P examinees to a set of I items. From this set of item responses, we might want to rank order the examinees or determine the extent to which each examinee has acquired a certain skill; we may also be interested in predicting how likely a given student will be to correctly answer the next question, for example, in a computerized adaptive testing environment. Although classical or frequentist statistical approaches (e.g., maximum likelihood, confidence intervals, p values) are commonly used in operational testing and the most widely available methods available, Bayesian statistical methods are recently becoming more readily available and increasingly utilized in psychometrics and IRT.