Presenting: Lihu Berman Hypothesis Testing Presenting: Lihu Berman

Agenda Basic concepts Neyman-Pearson lemma UMP Invariance CFAR

Basic concepts X is a random vector with distribution is a parameter, belonging to the parameter space disjoint covering of the parameter space denotes the hypothesis that Binary test of hypotheses: vs. M-ary test: vs.

Basic concepts (cont.) If then is said to be a simple hypothesis Otherwise, it is said to be a composite hypothesis Example: vs. simple vs. composite hypotheses RADAR – Is there a target or not ? Physical model – is the coin we are tossing fair or not ? A two-sided test: the alternative lies on both sides of A one-sided test (for scalar ): vs.

Basic concepts (cont.) Introduce the test function for binary test of hypotheses: is a disjoint covering of the measurement space If the measurement is in the Acceptance region – is accepted If it is in the Rejection region – is rejected, and is accepted.

Basic concepts (cont.) Probability of False-Alarm (a.k.a. Size): simple: composite: i.e. the worst case Detection probability (a.k.a. Power): simple: composite:

Basic concepts (cont.) Receiving Operating Characteristics (ROC): Chance line The best test of size has the largest among all tests of that size

The Neyman-Pearson lemma and let denote the density function of X, then: Is the most powerful test of size for testing which of the two simple hypotheses is in force, for some Let:

The Neyman-Pearson (cont.) Proof: Let denote any test satisfying: Obviously:

The Neyman-Pearson (cont.) Note 1: If then the most powerful test is: Note 2: Introduce the likelihood function: Then the most powerful test can be written as:

The Neyman-Pearson (cont.) Note 3: Choosing the threshold k. Denote by the probability density of the likelihood function under , then: Note 4: If is not continuous (i.e. ) Then the previous equation might not work! In that case, use the test: Toss a coin, and choose if heads up

Binary comm. in AWGN Source Mapper ‘1’ = Enemy spotted. ‘0’ = All is clear. Prior probabilities unknown !

Binary comm. in AWGN Natural logarithm is monotone, enabling the use of Log-Likelihood

Binary comm. (cont.)

Binary comm. (cont.) Assume equal energies: and define

Binary comm. (cont.)

Binary comm. (cont.)

Binary comm. (cont.)

UMP Tests The Neyman-Pearson lemma holds for simple hypotheses. Uniformly Most Powerful tests generalize to composite hypotheses A test is UMP of size , if for any other test , we have:

UMP Tests (cont.) Consider scalar R.Vs whose PDFs are parameterized by scalar If the likelihood-ratio is monotone non-decreasing in x for any pair Karlin-Rubin Theorem (for UMP one-sided tests): Is the UMP test of size for testing , then the test:

UMP Tests (cont.) Proof: begin with fixed values By the Neyman-Pearson lemma, the most powerful test of size for testing is: As likelihood is monotone, we may replace it with the threshold test

UMP Tests (cont.) The test is independent of , so the argument holds for every making the most powerful test of size for testing the composite alternative vs. the simple hypothesis Consider now the power function At . For any because is more powerful than the test A similar argument holds for any

UMP Tests (cont.) Thus, we conclude that is non-decreasing Consequently, is also a test whose size satisfies Finally, no test with size can have power , as it would contradict Neyman-Pearson, in

A note on sufficiency The statistic T(x) is sufficient for if and only if No other statistic which can be calculated from the same sample provides any additional information as to the value of the parameter Fisher-Neyman factorization theorem: The statistic T(x) is sufficient for if and only if One can write the likelihood-ratio in terms of the sufficient statistic

UMP Tests (cont.) UMP one-sided tests exist for a host of problems ! Theorem: the one-parameter exponential family of distributions with density: has a monotone likelihood ratio in the sufficient statistic provided that is non-decreasing Proof:

UMP Tests (cont.) Example:

UMP Tests (cont.) Therefore, the test is the Uniformly Most Powerful test of size for testing

Invariance Revisit the binary communication example, but with a slight change. Source Mapper So what?! Let us continue with the log-likelihood as before… Oops

Invariance (cont.) Intuitively: search for a statistic that is invariant to the nuisance parameter Project the measurement on the subspace orthogonal to the disturbance! Optimal signals ?

Invariance (formal discussion) Let G denote a group of transformations. X has probability distribution:

Invariance (cont.) Revisit the previous example (AWGN channel with unknown bias) The measurement is distributed as

Invariance (cont.) organizes the measurements x into equivalent classes where:

Invariance (cont.)

Invariance (cont.) Let us show that is indeed a maximal invariant statistic

Invariance (another example) Consider the group of transformations: The hypothesis test problem is invariant to G

Invariance (another example) What statistic is invariant to the scale of S ? The angle between the measurement and the signal-subspace (or the subspace orthogonal to it: ) In fact, Z is a maximal invariant statistic to a broader group G’, that includes also rotation in the subspace. G’ is specifically appropriate for channels that introduce rotation in as well as gain

Invariance (UMPI & summary) Invariance may be used to compress measurements into statistics of low dimensionality, that satisfy invariance conditions. It is often possible to find a UMP test within the class of invariant tests. Steps when applying invariance: 1. Find a meaningful group of transformations, for which the hypothesis testing problem is invariant. 2. Find a maximal invariant statistic M, and construct a likelihood ratio test. 3. If M has a monotone likelihood ratio, then the test is UMPI for testing one sided hypotheses of the form Note: Sufficiency principals may facilitate this process.

CFAR (introductory example) Project the measurement on the signal space. A UMP test ! The False-Alarm Rate is Constant. Thus: CFAR

CFAR (cont.) m depends now on the unknown . Test is useless. Certainly not CFAR Redraw the problem as: Utilize Invariance !!

CFAR (cont.) As before: Change slightly: independent

CFAR (cont.) The distribution of is completely characterized under even though is unknown !!! Thus, we can set a threshold in the test: in order to obtain CFAR ! Furthermore, as the likelihood ratio for non-central t is monotone, this test is UMPI for testing in the distribution when is unknown !

CFAR (cont.) The actual probability of detection depends on the actual value of the SNR

Summary Basic concepts Neyman-Pearson lemma UMP Invariance CFAR