We now return to the question of proposing a prior, based first upon the assumption of prior independence, p( ?, ?) = p( ?) p( ?), such that

Using the likelihood function that follows from the Gauss Markov conditions and the assumption of normally-distributed errors, we have

How do we propose priors p( ?) and p( ?)?

Let ? _MLE = ? _LS be the maximum likelihood estimate of ?, i.e., that maximizing the likelihood by minimizing S( ?). Let us now write S( ?) as

so that the posterior becomes

We now define the sample variance s ²:

In the supplemental material in the accompanying website, we show that if the Gauss Markov conditions (8.54) hold, s ² provides an unbiased estimate for ? ². That is, if we redo the set of experiments many times and average the computed s ² in each, E[ s ²] = ? ².

Using the definition of the sample variance, the posterior becomes

We now define a likelihood function for ? given s,

and a conditional likelihood function for ? given y and ?,

The posterior density then partitions into two contributions,

We now consider each contribution independently to search for priors that seem most satisfying, in terms of being reproducible by different analysts.