For a posterior distribution to be plausible in the Bayesian sense (Bayes' Plausible), it is said that:
$\mathbb{E}(\mu_{t+1} | \mu_t) = \mu_t$
where $\mu_t$ is the posterior distribution at time $t$, and consequently the prior at $t+1$ and $\mu_{t+1}$ is the posterior distribution at time $t+1$. So, for example $\mu_t(\omega)$ is the probability associated with the state being $\omega$ at time $t$. ( In other words, the belief).
Could someone explain why this should be the case? I think I understand why intuitively: on receiving no new informations, other than $\mu_t$, you can anticipate the new distribution to average to the current information you have.
But if this intuition is incorrect or incomplete, I'd really appreciate it if someone could point that out.