The Bayesian Estimation of a Linear Functional Relationship

Summary The purpose of this note is twofold. First, to demonstrate by means of an example that a statistical approach based solely on the likelihood function can be misleading, whereas the Bayesian method yields sensible results. Second, to develop the Bayesian method and apply the results to the problem of estimating a linear functional relationship. In Section 1 we discuss the likelihood approach. Section 2 concerns the Bayesian method, which is applied to the functional relationship situation in Section 3: it is shown that the asymptotic behaviour of the distributions exhibits a novel form. In Section 4 the posterior distributions derived in Section 3 are considered in more detail: in particular, a standard error for the estimate of slope is provided. Section 5 contains various comments on the method, and, in particular, indicates other fields of application.