chapter  7
Bayesian Modeling and Inference
ByGeorgiana Onicescu, Andrew B. Lawson
Pages 28

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

The development of Bayesian inference has as its kernel the data likelihood. The likelihood is the joint distribution of the data evaluated at the sample values. It can also be regarded as a function describing the dependence of a parameter or parameters on sample values. Hence, there can be two interpretations of this function. In Bayesian inference, it is this latter interpretation that is of prime importance. In fact, the likelihood principle, by which observations come into play through the likelihood function, and only through the likelihood function, is a fundamental part of the Bayesian paradigm (Bernardo and Smith, 1994, section 5.1.4). This implies that the information content of the data is entirely expressed by the likelihood function. Furthermore, the likelihood principle implies that any event that did not happen has no effect on an inference, since if an unrealized event does affect an inference, then there is some information not contained in the likelihood function.