Posterior coin selection from repeated tosses ($P(B\mid H^m)=\pi p_B^m/[(1-\pi )p_A^m+\pi p_B^m]$)

If a hidden coin is selected once with prior probability $\pi$ of coin $B$, repeated tosses are independent conditional on that coin. Bayes' theorem weights the prior by each all-heads likelihood. They need not be independent without conditioning on the chosen coin.