Let $P(\theta|x_1)$ denote the posterior probability of $\Theta = \theta$ given the data $X = x_1$. Suppose now you observe new data $X = x_2$, does it make sense the concept of equality in distribution, $$ P(\theta|X = x_1) = P(\theta|X = x_2) \quad \text{for all $\theta$},$$
in this context? references welcome if this makes any sense.