ece830_fall11_lecture10.pdf

Q q 1 p f a r a 2 s 2 σ 2 where we term snr a 2 s 2

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= Q Q - 1 ( P F A ) - r A 2 || s || 2 σ 2 ! where we term SNR = A 2 || s || 2 2 . If s = A sin( 2 π 10 ) , ‘ = 1 , · · · , 100 , P F A = 10 - 2 1.3 Two-sided Tests To see how special the UMP condition is, consider the following simple generalization of the testing problems above. H 0 : x ∼ N (0 , 1) H 1 : x ∼ N ( μ, 1) , μ 6 = 0 The log-likelihood ratio statistic is log Λ( x ) = - ( x - μ ) 2 2 + x 2 2 = μx - μ 2 / 2 and the log-LRT has the form μ x - μ 2 / 2 H 1 H 0 γ 0 . We can move the term μ 2 / 2 to the other side and absorb it into the threshold, but this leaves us with a test of the form μ x H 1 H 0 γ , and since μ is unknown (and not necessarily positive) the test is uncomputable. How can we proceed? Look at two densities in Figure 4. Intuitively the test | x | H 1 H 0 γ seems reasonable. This is called the Wald Test .
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Lecture 10: Composite Hypothesis Testing 6 Figure 3 The P F A of the Wald test can be seen from Figure 5. P F A = 2 Q ( γ ) γ = Q - 1 ( P F A / 2) P D = Z γ 1 2 π e - ( x - μ ) 2 / 2 dx + Z - γ -∞ 1 2 π e - ( x - μ ) 2 / 2 dx y = x - μ = Z γ - μ N (0 , 1) dy + Z - γ - μ -∞ N (0 , 1) dy = Q ( γ - μ ) + Q ( γ + μ ) . The P D depends on μ , which is unknown. Figure 4 1.4 Two “Derivations” of the Wald Test Generalized Likelihood Ratio Test (GLRT) Model μ as a deterministic, but unknown, parameter. Estimate μ from the data and plug the estimate into the LRT. Under H 1 the distribution is X ∼ N ( μ, 1), so a natural estimate for μ is b μ = x , the observation itself. The plugging this into the likelihood ratio yields b Λ( x ) = p ( x | b μ ) p ( x | 0) = exp( - ( x - b μ ) 2 / 2) exp( - x 2 / 2) = e x 2 / 2 .
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Lecture 10: Composite Hypothesis Testing 7 This is the generalized likelihod ratio. In effect, this compares the best fitting model in the composite hypothesis H 1 with the model H 0 . Taking the log yields the test log b Λ( x ) = x 2 H 1 H 0 γ , which is equivalent to the Wald test. Bayes Factor Model μ as an independent random parameter with prior probability distribution p ( μ ). The alternative hypothesis is that μ 6 = 0, and with no other prior information it is natural to take p ( μ ) to be symmetric about the origin. In particular, the prior probability distirbution μ ∼ N (0 , σ 2 ) is symmetric and models a prior belief that smaller values of μ are more probable that larger values. The Gaussian form is also convenient to use with the Gaussian likelihood. The Bayes Factor is the ratio Λ BF ( x ) = R p ( x | μ ) p ( μ ) p ( x | 0) .
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