ECE531 Screencast 11.4: Composite Neyman-Pearson Hypothesis Testing

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1 ECE531 Screencast 11.4: Composite Neyman-Pearson Hypothesis Testing D. Richard Brown III Worcester Polytechnic Institute Worcester Polytechnic Institute D. Richard Brown III 1 / 8

2 Basics Hypotheses H 0 and H 1 are associated with the subsets of states X 0 and X 1, where X 0 and X 1 form a partition on X. At least one hypothesis contains more than one state. Given a state x X 0, we can write the conditional false-positive probability as P fp,x (ρ) := Prob(ρ decides 1 state is x). Given a state x X 1, we can write the conditional false-negative probability as P fn,x (ρ) := Prob(ρ decides 0 state is x). or the conditional probability of detection as P D,x (ρ) := Prob(ρ decides 1 state is x) = 1 P fn,x (ρ). The binary composite N-P HT problem is then ρ NP = argmaxp D,x (ρ) for all x X 1 ρ subject to the constraint P fp,x (ρ) α for all x X 0. Worcester Polytechnic Institute D. Richard Brown III 2 / 8

3 Remarks We are trying to find one decision rule ρ : Y P M that maximizes the probability of detection for every state x X 1 subject to the constraints P fp,x (ρ) α for every x X 0. The decision rule can not be a function of x. If such a decision rule exists, it is called a uniformly most powerful (UMP) decision rule. UMP decision rules do not always exist. Worcester Polytechnic Institute D. Richard Brown III 3 / 8

4 Simple Null Hypothesis vs. Composite Alternative First let s consider a composite hypothesis testing problem where the null hypothesis H 0 is simple and the alternative hypothesis is composite, i.e. i.e. X 0 = x 0 X and X 1 = X\x 0. Pick two distinct states x 1 X\x 0 and x 2 X\x 0 and associate two new simple hypotheses H 1 x 1 and H 1 x 2 with these states. Consider the two simple binary hypothesis testing problems: H 0 vs. H 1 and H 0 vs. H 1. Let p j (y) denote the conditional density of the observations given that you started from state x j. The N-P lemma tells us the optimum decision rules are: 1 p 1 (y) > v p 0 (y) 1 p 2 (y) > v p 0 (y) ρ (y) = γ p 1 (y) = v p 0 (y) ρ (y) = γ p 2 (y) = v p 0 (y) 0 p 1 (y) < v p 0 (y) 0 p 2 (y) < v p 0 (y) The decision rule ρ cannot be more powerful than ρ for deciding between H 0 and H 1. Likewise, the decision rule ρ cannot be more powerful than ρ for deciding between H 0 and H 1. Worcester Polytechnic Institute D. Richard Brown III 4 / 8

5 Existence of UMP Decision Rule with Simple Null Hypoth. When might ρ and ρ have the same power for deciding between H 0 vs. H 1 and H 0 vs. H 1? Only when the critical region Y 1 = {y Y : ρ(y) decides H 1 } is the same (except possibly on a set of probability zero). In our example, we need {y Y : p 1 (y) > v p 0 (y)} = {y Y : p 2 (y) > v p 0 (y)} Actually, we need this to be true for all of the states x X 1. Theorem A UMP decision rule exists for the α-level binary composite HT problem H 0 x 0 versus H 1 X\x 0 = X 1 if and only if the critical region {y Y : ρ(y) decides H 1 } of the simple binary HT problem H 0 x 0 versus H 1 x 1 is the same for all x 1 X 1. Worcester Polytechnic Institute D. Richard Brown III 5 / 8

6 Example (part 1 of 2) Suppose we have the one-sided composite binary HT problem H 0 : x = µ 0 H 1 : x > µ 0 for X = [µ 0, ) and Y N(x,σ 2 ). Pick some µ 1 > µ 0 and associate this state with H 1. We already know how to find the α-level N-P decision rule for the simple binary problem H 0 : x = µ 0 H 1 : x = µ 1 The α-level N-P decision rule for H 0 versus H 1 is simply 1 y > σq 1 (α)+µ 0 ρ NP (y) = γ y = σq 1 (α)+µ 0 (1) 0 y < σq 1 (α)+µ 0 where γ is arbitrary since P fp = Prob(y > σq 1 (α)+µ 0 x = µ 0 ) = α. Worcester Polytechnic Institute D. Richard Brown III 6 / 8

7 Example (part 2 of 2) Note that the critical region {y Y : y > σq 1 (α)+µ 0 } does not depend on µ 1. Thus, by the theorem, (1) is a UMP decision rule for this binary composite HT problem. Also note that the probability of detection for this problem can be expressed as P D (ρ NP ) = Prob ( y > σq 1 ) (α)+µ 0 x = µ 1 ( σq 1 ) (α)+µ 0 µ 1 = Q σ does depend on µ 1, as you might expect. But there is no α-level decision rule that gives a better probability of detection (power) than ρ NP for any state x X 1. Worcester Polytechnic Institute D. Richard Brown III 7 / 8

8 Another Example What if we had this two-sided composite HT problem instead? for X = R and Y N(x,σ 2 ). H 0 : x = µ 0 H 1 : x µ 0 For µ 1 < µ 0, the most powerful critical region for an α-level decision rule is {y Y : y < µ 0 σq 1 (α)} Does not depend on µ 1. Unfortunately, this is not the same critical region as when µ 1 > µ 0. So, in fact, the critical region does depend on µ 1. We can conclude that no UMP decision rule exists for this binary composite HT problem. The theorem is handy for both finding UMP decision rules and also showing when UMP decision rules can t be found. Worcester Polytechnic Institute D. Richard Brown III 8 / 8

ECE531 Lecture 4b: Composite Hypothesis Testing

ECE531 Lecture 4b: Composite Hypothesis Testing D. Richard Brown III Worcester Polytechnic Institute 16-February-2011 Worcester Polytechnic Institute D. Richard Brown III 16-February-2011 1 / 44 Introduction