Lecture 8: Information Theory and Statistics
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1 Lecture 8: Information Theory and Statistics Part II: Hypothesis Testing and Estimation I-Hsiang Wang Department of Electrical Engineering National Taiwan University December 22, / 30 I-Hsiang Wang IT Lecture 8 Part II
2 1 Hypothesis Testing 2 / 30 I-Hsiang Wang IT Lecture 8 Part II
3 1 Hypothesis Testing 3 / 30 I-Hsiang Wang IT Lecture 8 Part II
4 Basic Setup We begin with the simplest setup binary hypothesis testing: 1 Two hypotheses regarding the observation X, indexed by θ {0, 1}: H 0 : X P 0 (Null Hypothesis, θ = 0) H 1 : X P 1 (Alternative Hypothesis, θ = 1) 2 Goal: design a decision making algorithm ϕ : X {0, 1}, x ˆθ, to choose one of the two hypotheses, based on the observed realization of X, so that a certain cost (or risk) is minimized. 3 A popular measure of the cost is based on probability of errors: Probability of false alarm (false positive; type I error): α ϕ P FA (ϕ) P {H 1 is chosen H 0 }. Probability of miss detection (false negative; type II error): β ϕ P MD (ϕ) P {H 0 is chosen H 1 }. 4 / 30 I-Hsiang Wang IT Lecture 8 Part II
5 Deterministic Testing Algorithm Decision Regions X Observation Space A 1 ( ) Acceptance Region of H 1. A 0 ( ) Acceptance Region of H 0. A test ϕ : X {0, 1} is equivalently characterized by its corresponding acceptance (decision) regions: ) { 1 A θ (ϕ) ϕ (ˆθ x X : ϕ (x) = ˆθ }, ˆθ = 0, 1. Hence, the two types of probability of error can be equivalently represented as α ϕ = β ϕ = x A 1 (ϕ) x A 0(ϕ) P 0 (x) = ϕ (x) P 0 (x), x X P 1 (x) = (1 ϕ (x)) P 1 (x). x X When the context is clear, we often drop the dependency on the test ϕ when dealing with acceptance regions Aˆθ. 5 / 30 I-Hsiang Wang IT Lecture 8 Part II
6 Likelihood Ratio Test Definition 1 (Likelihood Ratio Test) A (deteministic) likelihood ratio test (LRT) is a test ϕ τ, parametrized by constants τ > 0 (called threshold), defined as follows: { 1 if P 1 (x) > τp 0 (x) ϕ τ (x) = 0 if P 1 (x) τp 0 (x). For x supp P0, the likelihood ratio L (x) P 1(x) P 0(x). Hence, LRT is a thresholding algorithm on likelihood ratio L (x). Remark: For computational convenience, often one deals with log likelihood ratio (LLR) log (L(x)) = log (P 1 (x)) log (P 0 (x)). 6 / 30 I-Hsiang Wang IT Lecture 8 Part II
7 Trade-Off Between α (P FA ) and β (P MD ) Theorem 1 (Neyman-Pearson Lemma) For a likelihood ratio test ϕ τ and another deterministic test ϕ, α ϕ α ϕτ = β ϕ β ϕτ. pf: Observe x X, 0 (ϕ τ (x) ϕ (x)) (P 1 (x) τp 0 (x)), because if P 1 (x) τp 0 (x) > 0 = ϕ τ (x) = 1 = (ϕ τ (x) ϕ (x)) 0. if P 1 (x) τp 0 (x) 0 = ϕ τ (x) = 0 = (ϕ τ (x) ϕ (x)) 0. Summing over all x X, we get 0 (1 β ϕτ ) (1 β ϕ ) τ (α ϕτ α ϕ ) = (β ϕ β ϕτ ) + τ (α ϕ α ϕτ ). Since τ > 0, from above we conclude that α ϕ α ϕτ = β ϕ β ϕτ. 7 / 30 I-Hsiang Wang IT Lecture 8 Part II
8 (P MD ) (P MD ) (P FA ) 1 (P FA ) Question: What is the optimal trade-off curve? What is the optimal test achieving the curve? 8 / 30 I-Hsiang Wang IT Lecture 8 Part II
9 Randomized Testing Algorithm Randomized tests include deterministic tests as special cases. Definition 2 (Randomized Test) A randomized test decides ˆθ = 1 with probability ϕ (x) and ˆθ = 0 with probability 1 ϕ (x), where ϕ is a mapping ϕ : X [0, 1]. Note: A randomized test is characterized by ϕ, as in deterministic tests. Definition 3 (Randomized LRT) A randomized likelihood ratio test (LRT) is a test ϕ τ,γ, parametrized by cosntants τ > 0 and γ (0, 1), defined as follows: 1 if P 1 (x) > τp 0 (x) ϕ τ,γ (x) = γ if P 1 (x) = τp 0 (x). 0 if P 1 (x) < τp 0 (x) 9 / 30 I-Hsiang Wang IT Lecture 8 Part II
10 Randomized LRT Achieves the Optimal Trade-Off Consider the following optimization problem: Neyman-Pearson Problem minimize ϕ:x [0,1] subject to β ϕ α ϕ α Theorem 2 (Neyman-Pearson) A randomized LRT ϕ τ,γ with the parameters (τ, γ ) satisfying α = α ϕτ,γ, attains optimality for the Neyman-Pearson Problem. 10 / 30 I-Hsiang Wang IT Lecture 8 Part II
11 pf: First argue that for any α (0, 1), one can find (τ, γ ) such that α = α ϕτ,γ = ϕ τ,γ (x) P 0 (x) x X = P 0 (x) + γ P 0 (x) x: L(x)>τ x: L(x)=τ For any test ϕ, due to a similar argument as in Theorem 1, we have x X, (ϕ τ,γ (x) ϕ (x)) (P 1 (x) τ P 0 (x)) 0. Summing over all x X, similarly we get ( βϕ β ϕτ,γ ) + τ ( α ϕ α ϕτ,γ ) 0 Hence, for any feasible test ϕ with α ϕ α = α ϕτ,γ, its probability of type II error β ϕ β ϕτ,γ. 11 / 30 I-Hsiang Wang IT Lecture 8 Part II
12 Bayesian Setup Sometimes prior probabilities of the two hypotheses are known: π θ P {H θ is true}, θ = 0, 1, π 0 + π 1 = 1. In this sense, one can view the index Θ as a (binary) random variable with (prior) distribution P {Θ = θ} = π θ, for θ = 0, 1. With prior probabilities, it then makes sense to talk about the average probability of error for a test ϕ, or more generally, the average cost (risk): { P e (ϕ) π 0 α ϕ + π 1 β ϕ = E Θ,X [1 Θ ˆΘ }], ] R (ϕ) E Θ,X [r Θ, ˆΘ. The Bayesian hypothesis testing problem is to test the two hypotheses with knowledge of prior probabilities so that the average probability of error (or in general, a risk function) is minimized. 12 / 30 I-Hsiang Wang IT Lecture 8 Part II
13 Minimizing Bayes Risk Consider the following problem of minimizing Bayes risk. minimize ϕ:x [0,1] with known Bayesian Problem R (ϕ) E Θ,X [r Θ, ˆΘ (π 0, π 1 ) and r θ,ˆθ ] Theorem 3 (LRT is an Optimal Bayesian Test) Assume r 0,0 < r 0,1 and r 1,1 < r 1,0. A deterministic LRT ϕ τ threshold τ = (r 0,1 r 0,0 ) π 0 (r 1,0 r 1,1 ) π 1 attains optimality for the Bayesian Problem. with 13 / 30 I-Hsiang Wang IT Lecture 8 Part II
14 pf: R (ϕ) = r 0,0 π 0 P 0 (x) (1 ϕ (x)) + r 0,1 π 0 P 0 (x) ϕ (x) x X x X + r 1,0 π 1 P 1 (x) (1 ϕ (x)) + r 1,1 π 1 P 1 (x) ϕ (x) x X x X = r 0,0 π 0 + (r 0,1 r 0,0 ) π 0 P 0 (x) ϕ (x) x X + r 1,0 π 1 + (r 1,1 r 1,0 ) π 1 P 1 (x) ϕ (x) = x X x X ( ) {}}{ [ (r0,1 r 0,0 ) π 0 P 0 (x) (r 1,1 r 1,0 ) π 1 P 1 (x) ] ϕ (x) + r 0,0 π 0 + r 1,0 π 1. For each x X, we shall choose ϕ (x) [0, 1] such that ( ) is minimized. It is then obvious that we should choose { 1 if (r 0,1 r 0,0 ) π 0 P 0 (x) (r 1,1 r 1,0 ) π 1 P 1 (x) < 0 ϕ (x) = 0 if (r 0,1 r 0,0 ) π 0 P 0 (x) (r 1,1 r 1,0 ) π 1 P 1 (x) / 30 I-Hsiang Wang IT Lecture 8 Part II
15 Discussions For binary hypothesis testing problems, the likelihood ratio L (x) P1(x) P 0 (x) turns out to be a sufficient statistics. Moreover, a likelihood ratio test (LRT) is optimal both in the Bayesian and Neyman-Pearson settings. Extensions include M-ary hypothesis testing Minimax risk optimization (with unknown prior) Composite hypothesis testing, etc. Here we do not pursue these directions further. Instead, we would like to explore the asymptotic behavior of hypothesis testing, and the connection with information theoretic tools. 15 / 30 I-Hsiang Wang IT Lecture 8 Part II
16 1 Hypothesis Testing 16 / 30 I-Hsiang Wang IT Lecture 8 Part II
17 i.i.d. Observations So far we focus on the general setting where the observation space X can be arbitrary alphabets. In the following, we consider product space X n, length-n observation sequence X n drawn i.i.d. from one of the two distributions, and the two hypotheses are H 0 : X i i.i.d. P 0, i = 1, 2,..., n H 1 : X i i.i.d. P 1, i = 1, 2,..., n The corresponding probability of errors are denoted by α (n) P (n) FA P {H 1 is chosen H 0 } β (n) P (n) MD P {H 0 is chosen H 1 } Throughout the lecture we assume X = {a 1, a 2,..., a d } is a finite set. 17 / 30 I-Hsiang Wang IT Lecture 8 Part II
18 LRT under i.i.d. Observation (1) With i.i.d. observation, the likelihood ratio of a sequence x n X n is L (x n ) = n i=1 P 1 (x i ) P = 0(x i) a X ( ) N(a x n ) P1 (a) ( P 0(a) = P1 P a X 0(a)) nπ(a xn ), where N (a x n ) # of a s in x n and π (a x n ) 1 n N (a xn ) is the relative frequency of occurrence of symbol a in sequence x n. Note: From the above manipulation, we see that the collection of relative frequency of occurrence (as a X -dim probabilty vector), Π x n [ π (a 1 x n ) π (a 2 x n ) π (a d x n ) ] T, called the type of sequence x n, is a sufficient statistics for all the previously mentioned hypothesis testing problems. 18 / 30 I-Hsiang Wang IT Lecture 8 Part II
19 LRT under i.i.d. Observation (2) Let us further manipulate the LRT, by taking log likelihood ratio: L (x n ) τ n log (L (x n )) log τ n nπ (a x n ) log a X ( P1(a) P 0 (a) π (a x n ) log a X π (a x n ) log a X ) log τ n ) ( π(a x n ) P 0(a) ( π(a x n ) P 1 (a) ) 1 n log τ n D (Π x n P 0 ) D (Π x n P 1 ) 1 n log τ n 19 / 30 I-Hsiang Wang IT Lecture 8 Part II
20 Hypothesis Testing Probability Simplex Observation Space X P (X ) P1 A1 ( ) Acceptance Region of H1. (n) Acceptance F1 A0 ( ) P0 (n) Acceptance F0 Region of H0. Acceptance Region of H0. {xn Ai = decide Hi } 20 / 30 Region of H1. I-Hsiang Wang { (n) Πxn Fi IT Lecture 8 Part II } = decide Hi.
21 Probability Simplex P (X ) P 1 F (n) 1 P F (n) 0 P 0 By Sanov s Theorem, we know that α (n) = P n 0 ( F (n) 1 ) ( 2 nd(p P 0 ), β (n) = P n 1 F (n) 0 ) 2 nd(p P 1 ). 21 / 30 I-Hsiang Wang IT Lecture 8 Part II
22 Asymptotic Behaviors 1 Neyman-Pearson: β (n, ε) min ϕ n :X n [0,1] β (n) ϕ n, subject to α (n) ϕ n ε. It turns out that for all ε (0, 1), { lim 1 n n log β (n, ε) } = D (P 0 P 1 ) { 2 Bayesian: P e (n) min π 0 α (n) ϕ n:x n ϕ [0,1] n + π 1 β (n) ϕ n }. It turns out that { lim 1 n n log P e (n) } = D (P λ P 0 ) = D (P λ P 1 ) where P λ (a) (P 0(a)) λ (P 1 (a)) 1 λ x X (P 0(x)) λ (P 1 (x)) 1 λ, a X, and λ (0, 1) such that D (P λ P 0 ) = D (P λ P 1 ) 22 / 30 I-Hsiang Wang IT Lecture 8 Part II
23 in Neyman-Pearson Setup Theorem 4 (Chernoff-Stein) { For all ε (0, 1), lim 1 n n log β (n, ε) } = D (P 0 P 1 ). pf: We shall prove the achievability and the converese part separately. Achievability: construct a sequence of tests {ϕ n } with α (n) ϕ n ε for { } n sufficiently large, such that lim inf 1 n n log β(n) ϕ n D (P 0 P 1 ). Converse: for any sequence of tests {ϕ n } with α (n) ϕ n ε for n { } sufficiently large, show that lim sup 1 n log β(n) ϕ n D (P 0 P 1 ). n We use method of types to prove both the achievability and the converse. Alternatively, Chapter 11.8 of Cover&Thomas[1] uses a kind of weak typicality to prove the theorem. 23 / 30 I-Hsiang Wang IT Lecture 8 Part II
24 Achievability: Consider a deterministic test ϕ n (x n ) = 1 {D (Π x n P 0 ) δ n }, δ n 1 n ( log 1 ε + d log(n + 1)). In other words, it determines H 1 if D (Π x n P 0 ) δ n, and H 0 otherwise. Check the probability of Type I error : By Prop. 4 in Part I, we have α (n) ϕ n ( = P i.i.d. {D (Π X n P 0 ) δ n } 2 n Xi P 0 where (a) is due to our construction. δ n d log(n+1) n ) (a) = ε, Analyze the probability of Type II error : ((b) is due to Prop. 3 in Part I) β (n) ϕ n = P n 1 (T n (Q)) (b) Q P n : D(Q P 0 )<δ n P n 2 nd n, where D n Q P n : D(Q P 0 )<δ n 2 nd(q P 1) min {D (Q P 1 )}. Q P n : D(Q P 0 )<δ n Since lim n δ n = 0, we have lim n D n = D (P 0 P 1 ), and achievability is done. 24 / 30 I-Hsiang Wang IT Lecture 8 Part II
25 Converse: We prove the converse for deterministic tests. Extension to randomized tests is left as en exercise (HW6). Let A (n) i {x n ϕ n (x n ) = i}, the acceptance region of H i, for i = 0, 1. Let B (n) {x n D (Π x n P 0) < ε n}, ε n 2d log(n+1). By Prop. 4, we have n ) P n 0 (B (n) = 1 P i.i.d. {D (Π X n P 0 ) ε n } Xi P 0 ( 1 2 n ε n d log(n+1) ) n = 1 2 d log(n+1) 1 as n. ) ( ) Hence, for sufficiently large n, both P n 0 (B (n) and P n 0 A (n) 0 > 1 ε, and ( ) ) ( ) ( ) P n 0 B (n) A (n) 0 = P n 0 (B (n) + P n 0 A (n) 0 P n 0 B (n) A (n) 0 > 2 (1 ε) 1 = 1 2ε. Note B (n) = T n (Q). Hence Q n P n, D (Q n P 0) < ε n Q P n: D(Q P 0 )<ε n such that ( ) P n 0 T n (Q n ) A (n) 0 > (1 2ε) P n 0 (T n (Q n )). (1) 25 / 30 I-Hsiang Wang IT Lecture 8 Part II
26 Key Observation : Note that the probability of each sequence in the same type class is the same, under any product distribution. Hence, (1) is equivalent to T n (Q n ) A (n) > (1 2ε) T n (Q n ), ( ) which implies P n 1 T n (Q n) A (n) 0 > (1 2ε) P n 1 (T n (Q n)). 0 Hence, for sufficiently large n, Q n P n with D (Q n P 0 ) < ε n such that ( ) ( ) P n 1 A (n) 0 P n 1 T n (Q n) A (n) 0 > (1 2ε) P n 1 (T n (Q n)) where (c) is due to Prop. 3. (c) (1 2ε) P n 1 2 nd(q n P 1 ), Finally, as lim εn = 0, we have lim D (Qn P1) = D (P0 P1), and the n n converse proof is done. 26 / 30 I-Hsiang Wang IT Lecture 8 Part II
27 in Bayesian Setup Theorem 5 (Chernoff) where { lim 1 n n log P e (n) } = D (P λ P 0 ) = D (P λ P 1 ) ( P λ (a) = max log λ [0,1] (P 0(x)) λ (P 1(x)) 1 λ x X }{{} Chernoff Information CI (P 0, P 1 ) (P0(a))λ (P 1(a)) 1 λ x X (P 0 (x)) λ (P 1 (x)) 1 λ, a X, and λ (0, 1) such that D (P λ P 0 ) = D (P λ P 1 ). Note: The optimal Bayesian test (for minimizing P e ) is the maximum a posterier (MAP) test: ϕ MAP (x n ) = 1 {π 1 P n 1 (xn ) π 0 P n 0 (xn )}. 1 ) 27 / 30 I-Hsiang Wang IT Lecture 8 Part II
28 pf: The proof is based on application of large deviation in analyzing the optimal test, MAP: ϕ MAP (x n ) = 1 {π 1 P 1 (x n ) π 0 P 0 (x n )}. Analysis of error probabilities of MAP test : ( ) ( α (n) = P n 0, β (n) = P n 1 where F (n) 1 F (n) 0 ), { F (n) 1 Q P (X ) D (Q P 0) D (Q P 1) 1 log π 0 n { F (n) 0 Q P (X ) D (Q P 0 ) D (Q P 1 ) 1 log π 0 n : By Sanov s Theorem, we have π 1 } π 1 } } } lim { 1n log n α(n) = min D (Q P 0), lim { 1n log Q F 1 n β(n) = min D (Q P 1), Q F 0 where F 1 {Q P (X ) D (Q P 0 ) D (Q P 1 ) 0} and F 0 {Q P (X ) D (Q P 0) D (Q P 1) 0}.,. 28 / 30 I-Hsiang Wang IT Lecture 8 Part II
29 Exponents : Characterizing the two exponents is equivalent to solving the two (convex) optimization problems: min D (Q P 0 ) Q F 1 min D (Q P 1 ) Q F 0 minimize (Q 1,...,Q d ) d l=1 Q l log Q l P 0 (a l ) minimize (Q 1,...,Q d ) d l=1 Q l log Q l P 1 (a l ) subject to d l=1 Q l log P 1(a l ) P 0 (a l ) 0 Q l 0, l = 1,..., d d l=1 Q l = 1 subject to d l=1 Q l log P 1(a l ) P 0 (a l ) 0 Q l 0, l = 1,..., d d l=1 Q l = 1 It turns out that both problems have a common optimal solution P λ (a) = (P 0(a)) λ (P 1 (a)) 1 λ x X (P 0 (x)) λ (P 1 (x)) 1 λ, a X, with λ [0, 1] such that D (P λ P 0) = D (P λ P 1). 29 / 30 I-Hsiang Wang IT Lecture 8 Part II
30 Hence, both types of error probabilities have the same exponent, and so does the average error probability. This completes the proof of the first part. Chernoff Information : To show that ( CI (P 0, P 1 ) max log λ [0,1] simply observe that a X D (P λ P 0 ) = D (P λ P 1 ) 1 (P 0 (x)) λ (P 1 (x)) 1 λ x X (P 0 (a)) λ (P 1 (a)) 1 λ (log P 0 (a) log P 1 (a)) = 0 ) = D (P λ P 0 ), 1 D (P λ P 0) = D (P λ P 1) = log (P 0 (x)) λ (P 1 (x)) 1 λ. Proof complete. x X 30 / 30 I-Hsiang Wang IT Lecture 8 Part II
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