= Gaussian–exponential hierarchical colour model
{c}
An observed astronomical colour is the sum of a normal intrinsic colour, a nonnegative dust contribution with an <exponential distribution>, and normal measurement noise. With latent $C_s\sim N(\mu,v)$, $E_s\sim\operatorname{Exp}(\text{mean }\tau)$ and $O_s\mid C_s,E_s\sim N(C_s+E_s,r_s)$, the <Gibbs sampler> conditionals for $C_s$ are normal and those for $E_s$ are <truncated normal distributions>. A flat prior on $\mu$ and inverse-gamma priors on $v,\tau$ give inverse-gamma conditional updates for the two scales. Log-flat priors on both scales instead make the joint posterior improper when all $r_s>0$, despite formally proper full conditionals at generic latent states: the marginal observed likelihood has a positive limit at either zero-scale boundary. Proper positive-scale priors restore a genuine joint posterior.
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