When is the posterior distribution equal to the prior? So I have heard that if the prior distribution is in the subexponential class, applying Bayes rule does not change the belief. I have been trying to find an example of this but I am unable to do so. I was wondering if somebody around here has an example. This seems similar to the martingale property but I can't quite make the link...
 A: For a given realisation, $x_0$, of $X\sim f(x|\theta)$, it is possible that $f(x_0|\theta)$ is a constant function of $\theta$ in which case the posterior is equal to the prior for this observation. However, if the posterior is almost surely equal to the prior, in the sense of almost every realisation of $X$, this implies that the density of $X$ does not depend on $\theta$.
I would welcome the reference to the subexponential distributions as there is no mathematical reason for a heavy or very heavy tail distribution to fail to update the prior. For instance, here is the comparison of prior (dashes) and posterior (full) when the prior is a Laplace (double-exponential) distribution and the likelihood is a Cauchy with observation 3.14:

The prior has clearly moved when becoming the posterior and the tail behaviour changed. A likelihood flatter than the Cauchy like a $t$ distribution with $10^{-1}$ or $10^{-2}$ degrees of freedom would exhibit the same pattern, albeit keeping the same mode for prior and posterior (as mentioned in my book). 
