Let's say we have a sequence of repeated coin tosses of a coin with unknown bias $p$. The resulting observations are $x = (x_1, \dots, )$. The posterior distribution in this scenario is $$P(p|x) = \text{Beta}(a_0 + \text{#heads}, b_0 + \text{#tails})$$ where $\text{Beta}(a_0,b_0)$ is the prespecified prior distribution.
I am wondering what mathematical tools we have to describe the evolution of the posterior as more data is added. Specifically, is there a way to model the continuum limit of the posterior using differential equations?
This seems like a natural question since we already have a description of the continuum limit for the coin flips, namely random walks on a line.
Roughly, I know the posterior gets "thinner" as more data is added, and its mean approaches the true bias, $p$. This reminds me of the Fokker-Planck equation used for describing the distribution of a randomly moving particle, except the opposite (instead of spreading out over time, the posterior loses entropy).