Ex. 8.3
Ex. 8.3
Justify the estimate (8.50), using the relationship
\[\begin{equation}
\text{Pr}(A) = \int \text{Pr}(A|B)d(\text{Pr}(B)).\non
\end{equation}\]
Soln. 8.3
From the relationship above, we have
\[\begin{eqnarray}
\widehat{\text{Pr}}_{U_k}(u) = \int \text{Pr}_{U_k|U_l: l\neq k}(u)d\text{Pr}_{U_l: l \neq k}.\non
\end{eqnarray}\]
The integral is thus estimated by a law-of-large-number way to be
\[\begin{equation}
\frac{1}{M-m+1}\sum_{t=m}^M\text{Pr}\left(u|U_l^{(t)}, l\neq k\right).\non
\end{equation}\]