= Empty component under an improper prior
If a model configuration leaves a parameter absent from its <likelihood function>, giving that parameter an independent flat <improper prior> makes the <posterior distribution> normalization infinite whenever the configuration has positive <prior distribution> mass. Integrate first over that parameter: a positive constant <likelihood function> factor is repeated over all of $\mathbb R$. This occurs for the second segment mean of a <Gaussian> <change-point detection> model when the proposed split is after the final observation. There is no <posterior distribution> or valid <Gibbs sampler> for that joint kernel. A proper <prior distribution> on the unused parameter is necessary if the no-change configuration is retained.
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