Sequential Hypothesis Testing

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Problem 9.19 from Gallager R.G. "Stochastic Processes: Theory for Applications" [Gal13]More problems in Chapter 9 "Random walks, large deviations, and martingales"

(a)(b)

Consider the following binary hypothesis testing problem: there are two hypotheses $X=0, X=1$ , and, conditional on each hypothesis, IID observations $Y_{1}, \ldots , Y_{n}$ are drawn with conditional density $\mathsf{f}_{Y|X}(y \mid 0)$ or $\mathsf{f}_{Y|X}(y \mid 1)$ respectively.

(a)

Conditional on $X = 0$ for the hypothesis testing problem, consider the rv s $Z_{i} = \ln \left[\mathsf{f}(y_{i} \mid X=1)/\mathsf{f}(y_{i} \mid X=0)\right]$ . Show that $r^{*}$ , the positive solution to $\mathsf{g}(r) = 1$ , where $\mathsf{g}(r) = \mathbb {E}\left[\exp (rZ_{i})\right]$ , is given by $r^{*} = 1$ .

(b)

Assuming that $Y$ is a discrete rv (under each hypothesis), show that the tilted rv $Z_{r}$ with $r = 1$ has the PMF $\mathsf{p}_{Y}(y \mid X=1)$ .

References

[Gal13] Gallager R.G. Stochastic Processes: Theory for Applications. Cambridge University Press, 2013.