= Improper posterior from a log-uniform random-effect scale prior
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 $\int_0^\varepsilon d\lambda/\lambda=\infty$. 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.
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