Posterior propriety
= Posterior propriety
A posterior kernel $L(\theta)\pi(\theta)$ defines a <posterior distribution> exactly when its integral is finite and nonzero. Proper full conditional densities do not suffice. For example, if a known positive measurement variance makes a marginal normal likelihood approach a strictly positive limit as a latent variance $v$ tends to zero, the log-flat prior $dv/v$ gives infinite posterior mass at that boundary. The same problem occurs for a latent exponential scale $\tau$ with $d\tau/\tau$ when the observed likelihood has a positive limit as the exponential contribution vanishes.