The 'pathology' of the Natural Conjugate Prior Density in the Regression Model

In a Bayesian analysis of the linear regression model, one may have prior information on the error variance. If one incorporates this kind of information in a natural conjugate prior density, under certain conditions the posterior mean of the coefficients on which one is informative is equal to a constrained least squares estimator. The value of the posterior covariance matrix is also studied. We discuss and illustrate how to avoid getting posterior results too close to the “pathological” results summarized above.