Pseudo-prior

A pseudo-prior is a proper density assigned to a parameter that is inactive under a particular model. Integrating it out leaves that model's likelihood and marginal evidence unchanged. Its choice affects movement between model states, but not the intended marginal posterior over models.

A pseudo-prior can make model parameter vectors have equal dimensions, allowing identity-matched RJ-MCMC moves. If active and inactive coordinates share the same prior density in the two augmented targets, these factors cancel in model-switch ratios. Within-model updates must preserve the augmented target, including the inactive coordinate's pseudo-prior law.

Compare independent Poisson means with one shared mean under proper shape-rate gamma distribution priors. Adding the second mean as an inactive parameter under the shared-mean model gives a simple identity model switch. With equal model priors and identical active/inactive priors, its ratio from separate to shared means is $e^{{\gamma }_2-{\gamma }_1}({\gamma }_1/{\gamma }_2{)}^{x_2}$. Poisson-gamma conjugacy supplies exact within-model updates, and integrating the gamma kernels supplies an independent Bayes factor benchmark.