The Arbitrary Relation between Probability of Error and Measurement Subset

Abstract

Let P_e (S) denote the Bayes risk (probability of error) in testing two equally likely hypotheses H ₀ versus H ₁ using measurements in S, a subset of the set of possible measurements.

The possible values of P_e (S) as a function of S are considered. It has been shown that all orderings on P_e (S), satisfying the natural monotonicity constraint S′ ⊂ S ⇒ P_e (S′) ≥ P_e (S) can occur. We show that no other restrictions exist on the numbers P_e (S), 0 < P_e (S) ≤ ½, thus extending the known result from the achievability of orderings to the achievability of numerically specified sequences. Thus nonexhaustive (suboptimal) measurement-selection algorithms can be arbitrarily bad.

Key Words:

• Hypothesis testing
• Measurement selection
• Probability of error
• Choice of regression variables
• Mahalanobis distance