The marginal distribution of η is termed the prior distribution (prior in the sense of before the data have been taken into account), and must be specified by the user. The model provides us with the likelihood function P(Y|η), and hence the posterior distribution of η is determined (posterior in the sense of after the data have been taken into account). The posterior distribution is a description of the probabilities of possible values for η given the observed data and forms the basis for statistical inference. We may, for example, take the mean or the median of this distribution as our point estimate for η. Further, the interval between the 2.5th and the 97.5th percentile of the posterior distribution provides the so-called central 95% credibility region, which is analogous to a 95% confidence interval in the ML framework. For more on Bayesian statistics, the reader is referred to the introductions by Box and Tiao (1973) and Gelman et al. (2004).
