Improper prior

An improper prior is a nonnegative prior kernel with infinite total mass. It can still produce a proper posterior after multiplication by the likelihood and normalization.

The measure $d\sigma /\sigma$ is invariant under multiplication by every positive constant. In the logarithmic coordinate it is Lebesgue measure. It cannot be normalized over $(0,\infty )$, although a bounded truncation is a log-uniform distribution.

A random-effect model with positive marginal likelihood at scale zero cannot yield a proper posterior distribution under a prior proportional to the reciprocal scale near zero. On a compact set of the other parameters, continuity and positivity give a positive likelihood lower bound, while ${\int }_0^{\epsilon }d\lambda /\lambda =\infty$. Individual full conditional distributions can still be proper, so their existence alone does not justify a Gibbs sampler with a probability distribution as its joint target.

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.