2.13 Conjugate joint prior for the normal distribution

2.13.1 The form of the conjugate prior

In Section 2.12, we considered a reference prior for a normal distribution with both parameters unknown, whereas in this section we shall consider a conjugate prior for this situation. It is, in fact, rather difficult to determine which member of the conjugate family to use when substantial prior information is available, and hence in practice the reference prior is often used in the hope that the likelihood dominates the prior. It is also the case that the manipulations necessary to deal with the conjugate prior are a bit involved, although the end results are, of course, similar to those when we use a reference prior, with some of the parameters altered slightly. Part of the problem is the unavoidable notational complexity. Further, the notation is not agreed among the different writers on the subject. A new notation is introduced below.

We first recall that the likelihood is

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Now suppose that your prior distribution of is (a multiple of) an inverse chi-squared on  degrees of freedom. It may be convenient to think of as l0–1, so that your prior knowledge about is in some sense worth l

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