Ex. 8.3

Ex. 8.3

Justify the estimate (8.50), using the relationship

$$\text{Pr}(A)=\int \text{Pr}(A|B)d(\text{Pr}(B)).$$

Soln. 8.3

From the relationship above, we have

$$\begin{array}{r}{\widehat{\text{Pr}}}_{U_k}(u)=\int {\text{Pr}}_{U_k|U_l:l\ne k}(u)d{\text{Pr}}_{U_l:l\ne k}.\end{array}$$

The integral is thus estimated by a law-of-large-number way to be

$$\frac{1}{M-m+1}\sum _{t=m}^M\text{Pr}(u|U_l^{(t)},l\ne k).$$