Let's consider a coin with bias $r$ in $]0,1[$.
We conduct $n$ coin flip experiments. At each step, $P(H)=r$ and $P(T)=1-r$ where $H$ stands for heads and $T$ stands for tail.
Let's start with an arbitrary function $P_0(r)$ defined on $[0, 1]$ with $\int_0^1 P_0(x) dx\ = 1$. After each experiment $X_n$ we apply Bayes theorem to calculate the posterior probability function $P_{n+1}(r)$ from $P_n(r)$. For instance if $x=H$ then $P_{n+1}(r)=\frac{r P_n(r)}{\int_0^1 x P_n(x) dx}$.
My question is: does $P_n(r)$ converge (to a normal distribution)? Does this depend on $P_0(r)$?