= Gaussian change-point posterior with proper priors
{c}
{title2=$p(\theta,\phi,v,m\mid x)\propto v^{-n/2}e^{-S_m/(2v)}p(\theta)p(\phi)p(v)p(m)$}
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 $m=1,\ldots,n$ 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 $\operatorname{IG}(a_0,b_0)$ 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.
Back to article page
