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1

Since you know $P(D)$ and $P(D|H)$, you can compute $P(D)$ by the Law of Total Probability: \begin{align} P(D) = \int_{H^{*} \in \Theta}{P(D|H^{*})dP(H^{*})} \end{align} You usually don't need to compute this value in Bayesian computations, though.


2

This is known in the literature as the German tank problem. The discussion at Wikipedia is long and complex, but at the risk of greatly oversimplifying things, if you only see one taxi, numbered $m$, then the minimum variance unbiased estimator is $2m-1$.



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