I am bit puzzled on how we can interpret the posterior. Assume a coin which is 0.1 probable to be unfair. So our prior probability on the coin being unfair is 0.1, and being fair is 0.9. Also by unfair I mean, the probability of head is 2/3 instead of 1/2. Now, imagine a scenario where I toss this coin 10 times and I get 10 heads. So if I want to get the posterior, I do the following:
\begin{align} & P(\texttt{unfair}\mid\texttt{evidence}) = \frac{P(\texttt{evidence}\mid\texttt{unfair})\cdot P(\texttt{unfair})} {P(\texttt{evidence})} \\[8pt] = {} & \frac{P(\texttt{evidence}\mid\texttt{unfair})\cdot P(\texttt{unfair})} {P(\texttt{evidence}\mid\texttt{unfair})\cdot P(\texttt{unfair}) +P(\texttt{evidence}\mid\texttt{fair})\cdot P(\texttt{fair})} \\[8pt] = {} & \frac{(2/3)^{10}\cdot(0.1)}{(2/3)^{10}\cdot(0.1)+(1/2)^{10}\cdot(0.9)} = \frac{0.0017}{0.0017+0.0009} = \frac{0.0017}{0.0026} = 0.653 \end{align}
Now how can I interpret this posterior? It's technically saying that the probability of the coin being unfair given the evidence is higher than the prior belief on the coin being unfair ($0.653 > 0.1$), but is still below the prior belief on the coin being fair ($0.653 < 0.9$). So, the only conclusion is, the coin is still more likely to be fair. So, in order to change our belief about the coin we have "being unfair", we need more evidence (i.e. 100 heads in a row). Is it correct?
"It's technically saying that the probability of the coin being unfair given the evidence
is higher than the prior belief on the coin being unfair (0.653 > 0.1), but is still
below the prior belief on the coin being fair (0.653 < 0.9). So, the only conclusion is,"
we now belief that the coin is unfair rather than fair, but we are less confident in
that belief than we were before in our belief that the coin was fair rather than unfair.