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Keeping the powers and exponentials of $v$ and $\tau$ in the joint kernel gives the two inverse-gamma distributions, using the printed shape–scale convention:Indeed their kernels are $v^{-(N/2+1)}\exp [-{\sum }_s(C_s-{\mu }_C{)}^2/(2v)]$ and ${\tau }^{-(N+1)}\exp [-{\sum }_s{E}_s/\tau ]$. With positive measurement variances and generic continuously drawn latent values, the two scale parameters are positive almost surely. Together with (a),(b), these are all $2N+3$ formal full conditional updates.

There is, however, no joint posterior probability distribution to sample under the stated priors. Integrate out the intrinsic $C_s$ first. For fixed ${\mu }_C,v>0$, the observational marginal density iswhere ${\varphi }_V$ is the centred normal density of variance $V$. Substitute $E=\tau u$. Dominated convergence givesThe convergence is uniform on any compact ${\mu }_C$ interval and any compact positive $v$ interval, since the normal densities are uniformly bounded and continuous there. Hence some such region has $L\ge c>0$ for all sufficiently small $\tau$, andIntegrating over those positive-measure intervals proves that the full normalizing constant is infinite. If all $r_s>0$, the $v\downarrow 0$ boundary also gives a positive marginal likelihood and a divergent $dv/v$ integral at fixed $\tau >0$. Thus $\boxed{\text{the stated joint posterior is improper}}$, even though its generic full conditionals are proper. Apparent stability of a simulated chain cannot establish a nonexistent stationary probability law.

A valid repair that preserves direct sampling is to replace the two log-flat priors by independent proper $v\sim \mathrm{IG}(a_v,b_v)$ and $\tau \sim \mathrm{IG}(a_{\tau },b_{\tau })$, with all four constants positive. The normal and truncated-normal updates remain as above, whileWith positive $r_s$, the flat ${\mu }_C$ prior is then harmless: after integrating it out, the Gaussian measurement likelihood is bounded uniformly over the latent shifts and the scales, and the remaining proper priors integrate to one. A proper prior on ${\mu }_C$ is also possible.

For this repaired model, initialize $v,\tau >0$, $E_s\ge 0$ and finite $C_s,{\mu }_C$. Repeatedly choose one of the $2N+3$ coordinates with fixed positive probabilities and redraw from its full conditional using the latest values. This Random-scan Gibbs sampler preserves the proper target and satisfies detailed balance: for each coordinate, both forward and reverse flows equal the marginal density of the remaining coordinates times the product of the two conditional densities. A systematic full sweep is another invariant scheme, though it need not itself be reversible.

Run several dispersed chains; inspect traces of both scales, ${\mu }_C$, representative latent values and the log posterior. Assess rank-normalized split $\hat{R}$, autocorrelation, and the effective sample size of a Markov chain, and extend runs until relevant Monte Carlo errors are sufficiently small. Scale–latent dependence can cause slow mixing, so diagnostics must include boundary exploration and alternative initializations. Keep post-warmup draws without automatic thinning. These are convergence checks for the repaired posterior; the printed model admits only the formal update algorithm, not a valid posterior-sampling interpretation.

Every move can be drawn from a full conditional distribution, producing a Gibbs sampler with acceptance probability one. Write $s_{x,i}^2={\sigma }_{x,i}^2$ and $s_{y,i}^2={\sigma }_{y,i}^2$. First update independentlyand thenLet $X$ have rows $(1,{\xi }_i)$ and $\eta =({\eta }_i)$. Update the linear regression coefficients jointly byand updateFinally, the flat positive variance priors give the following full conditionals, each an inverse-gamma distribution:A systematic sweep in the displayed order, using the newly sampled values immediately, defines the chain. The shapes are positive for $N>2$; full column rank of $X$ is also required.

The scaled inverse chi-squared distribution has densityIt is an inverse-gamma distribution under a change of parameters.