Introduction

In Chapter 20 we discussed how to guess the outcome of a binary random variable. We now extend the discussion to random variables that take on more than two—but still a finite—number of values. Statisticians call this problem “multi-hypothesis testing” to indicate that there may be more than two hypotheses. Rather than using H, we now denote the random variable whose outcome we wish to guess by M. (In Chapter 20 we used H for “hypothesis;” now we use M for “message.”) We denote the number of possible values that M can take by M and assume that M ≥ 2. (The case M = 2 corresponds to binary hypothesis testing.) As before the “labels” are not important and there is no loss in generality in assuming that M takes values in the setM= ﹛1, …, M﹜. (In the binary case we used the traditional labels 0 and 1 but now we prefer 1, 2, …, M.)

The Setup

A random variable M takes values in the set M = ﹛1, …, M﹜, where M ≥ 2, according to the prior

where

and where

We say that the prior is nondegenerate if

with the inequalities being strict, so M can take on any value in M with positive probability. We say that the prior is uniform if

The observation is a random vector Y taking values in Rd. We assume that for each the distribution of Y conditional on M = m has the density

where is a nonnegative Borel measurable function that integrates to one over Rd.

A guessing rule is a Borel measurable function from the space of possible observations Rd to the set of possible messages M. We think about as the guess we form after observing that Y = yobs. The error probability associated with the guessing rule is given by

Note that two sources of randomness determine whether we err or not: the realization of M and the generation of Y conditional on that realization. A guessing rule is said to be optimal if no other guessing rule achieves a lower probability of error.