Gaussian change-point posterior with proper priors ($p(\theta ,\varphi ,v,m\mid x)\propto v^{-n/2}{e}^{-S_m/(2v)}p(\theta )p(\varphi )p(v)p(m)$)

Take independent proper normal prior distributions for the two segment means and a proper inverse-gamma distribution for the common variance. A uniform prior distribution over $m=1,\dots ,n$ may then retain the empty second segment. The posterior distribution is proper: its likelihood function is bounded above by $(2\pi {)}^{-n/2}{v}^{-n/2}$, whose integral against $\mathrm{IG}(a_0,b_0)$ is finite for $a_0,b_0>0$. At $m=n$ the second mean has its unchanged proper prior distribution as its full conditional distribution. The configuration's posterior probability now has the meaningful interpretation of no change inside the observed sample.

For independent means $\theta \sim N({\mu }_{\theta },t_{\theta })$ and $\varphi \sim N({\mu }_{\varphi },t_{\varphi })$, the first mean's conditional precision is $m/v+1/t_{\theta }$ and conditional precision-weighted mean is ${\sum }_{i\le m}{x}_i/v+{\mu }_{\theta }/t_{\theta }$; replace $m$ by $n-m$ for the second mean. With $v\sim \mathrm{IG}(a_0,b_0)$ its conditional law is $\mathrm{IG}(a_0+n/2,b_0+S_m/2)$. The discrete split has conditional probabilities proportional to $e^{-S_m/(2v)}$. These full conditional distributions give a Gibbs sampler including $m=n$. Averaging the sampled indicators of that configuration estimates its probability; averaging its conditional probabilities gives an estimate by Rao-Blackwellization.