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# hw2-2011 - the minimum value for H(X Y Z 4 Bottleneck...

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Unformatted text preview: the minimum value for H (X, Y, Z )? 4. Bottleneck. Suppose a (non-stationary) Markov chain starts in one of n states, necks down to k < n states, and then fans back to m > k states. Thus X1 → X2 → X3 , X1 ∈ {1, 2, . . . , n}, 2. The value .of ,a }, X ∈ {1, 2, . . . , m}, and p(x , x , x ) = p(x )p(x |x )p(x |x ). X2 ∈ {1, 2, . . k question. 3 1 2 3 1 21 32 Let X ∼ p(x), x = 1, 2, . . . , m. (a) Show that the ⊆University X We ask3 Urbana-Champaign We are given a set Sdependence ,of of 1Illinois, whether X by S and receive the answer {1, 2, . . . m}. and X is limited ∈ the bottleneck by proving that I (X1 ; X3 ) ≤ log k. ECE 563, Information Theory ,Fall 2011 if X that (b) Evaluate I (X1 ; X3 ) for k = 1, and1conclude∈ S no dependence can survive such Y= 0, if X ∈ S. a bottleneck. HW #2 Suppose Pr{X ∈ S } = α. Posted: 5. Find the decrease in uncertainty H (X September 13th, 2011 Conditional mutual information. ) − H (X |Y ). Due: September Consider a sequence with a given probability α 22nd, ,2011 . , X . Each Apparently any set S of n binary random variablesas good,as .anynother. n-sequence is X1 X2 . −(n−1) with an even number of 1’s has probability 2 and each n-sequence with an odd number of 1’s has probability 0. Find the mutual informations Problem 1: 3. Random questions. (X1 ; X , I (X2 ; X3 |X1 ), . . I ( pn−1 A question n− ) One wishes to Iidentify2 )a random object . X , ∼ X(x).; Xn |X1 , . . . , XQ 2∼. r (q ) is asked at random according to r (q ). This results in a deterministic answer A = A(x, q ) ∈ {a1 , a2 , . . .}. Suppose the object X and the question Q are independent. Then 6. I (X ; Q, A) is the uncertainty in X removed by the question-answer (Q, A). Fano’s inequality. Let Pr(X I (X ; Q, pi ) i= H1,A|Q.). , Interpret. p1 ≥ p2 ≥ p3 ≥ · · · ≥ pm . The minimal (a) Show = i) = A, = ( 2, . . m and let ˆ probability of error predictor of X is X = 1, with resulting probability of error Pe = (b) p . Maximize H (p) two i.i.d. questions Q1 , Q2 ∼ p (q= are to ﬁnd a bound answers Now suppose that subject to the constraint 1 − r ) P asked, eliciting on P in 1− 1 1 e e A of H . This is Fano’s two questions are less valuable than twice terms 1 and A2 . Show thatinequality in the absence of conditioning. the value of a single question in the sense that I (X ; Q1 , A1 , Q2 , A2 ) ≤ 2I (X ; Q1 , A1 ). Problem 2. 7. Random box size. 1 An n-dimensional rectangular box with sides X1 , X2 , X3 , . . . , Xn is to be constructed. The volume is Vn = n=1 Xi . The edge-length l of an n-cube with the same volume as i 1 the random box is l = Vn /n . Let X1 , X2 , . . . be i.i.d. uniform random variables over the interval [0,a]. 1 Find limn→∞ Vn /n , and compare to (EVn )1/n . Clearly the expected edge length does not capture the idea of the volume of the box. Problem 3. 10. Entropy of a disjoint mixture. Let X1 and X2 be discrete random variables drawn according to probability mass functions p1 (·) and p2 (·) over the respective alphabets X1 = {1, 2, . . . , m} and X2 = {m + 1, . . . , n}. Notice that these sets do not intersect. Let X= X1 , with probability α, X2 , with probability 1 − α. (a) Find H (X ) in terms of H (X1 ) and H (X2 ) and α. 2 (b) Maximize over α to show that 2H (X ) ≤ 2H (X1 ) + 2H (X2 ) and interpret using the notion that 2H (X ) is the eﬀective alphabet size. (c) Let X1 and X2 be uniformly distributed over their alphabets. What is the maximizing α and the associated H (X )? 11. Entropy of a random tree. Consider the following method of generating a random tree with n nodes. First expand the root node: X1 8. An AEP-like limit and the AEP. Problem 4. (a) p(x). Find ECE 563 Let X1 , X2 , . . . be i.i.d. drawn according to probability mass function Fall 2010 1 September 7, 2010 lim [p(X1 , X2 , . . . , Xn )] n . n→∞ HOMEWORK ASSIGNMENT 2 (b) Let Reading: .Sections 2.9-2.10 according to the following distribution: X1 , X2 , . . be drawn i.i.d. and Chapter 3 of Cover & Thomas Due Date: September 16, 2010 (in class) 2 2, 5 1. Problem 2.42 (Inequalities) on page 52Xi = 3, of text. 2 5 4, 1 2. Problem 2.32 (Fano) on page 50 of text. (Note: This is similar to Kevin’s example 5 that we talked about in class.) Find the limiting behavior of the product 3. Problem 2.39 (Entropy and pairwise independence) on page 52 of text. (X1 X2 · · · Xn )1/n . 4. Problem 3.1 (Markov’s Inequality and Chebyshev’s Inequality) on page 64 of text. 1 (c) Evaluate the limit of p(X1 , X2 , . . . , Xn ) n for the distribution in part b. 5. Problem 3.6 (AEP-like limit) on page 66 of text. 6. Problem 5. (AEP and source coding) on page 66 of text. Problem 3.7 9. AEP. 7. Let random variables ∼ 1 ,(X2y ). .We n be independent ∼ p(x), x ∈ of .the hypothesis that Let (Xi , Yi ) be i.i.d. X p x, , . . , X form the log likelihood ratio X Let Nx denote the number Y are independent symbol x in a giventhat X and1Yx2 , . . .dependent.empirical X and of occurences of a vs. the hypothesis sequence x , are , xn . The What is probabilityof the limit mass function is deﬁned by 1 p (X n )p (Y n ) log Nx n n ? pn (x) = p(X for x)∈ X ˆn , ,Y n (a) Show that ￿ 3 p(x1 , x2 , . . . , xn ) = p(x)Nx x ∈X and 1 p p − log p(x1 , x2 , . . . , xn ) = H (ˆn ) + D(ˆn ||p) n (b) For a given x1 , x2 , . . . , xn what is max p(x1 , x2 , . . . , xn ) p where the maximization is over all probability mass functions on X ? What probability mass function achieves this maximum likelihood? 8. In class we proved the following theorem: Theorem (AEP Converse): Let X1 , X2 , . . . be i.i.d. with entropy H . For any sequence of sets B (n) ∈ X n , if limn→∞ P(B (n) ) = 1, then lim inf n→∞ 1 log |B (n) | ≥ H n (X1 X2 · · · Xn )1/n . 1 (c) Evaluate the limit of p(X1 , X2 , . . . , Xn ) n for the distribution in part b. Problem 6. 9. AEP. Let (Xi , Yi ) be i.i.d. ∼ p(x, y ). We form the log likelihood ratio of the hypothesis that X and Y are independent vs. the hypothesis that X and Y are dependent. What is the limit of 1 p (X n )p (Y n ) log ? n p (X n , Y n ) 3 ...
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