COMP2610/COMP Information Theory

Transcription

1 COMP2610/COMP Information Theory Lecture 9: Probabilistic Inequalities Mark Reid and Aditya Menon Research School of Computer Science The Australian National University August 19th, 2014 Mark Reid and Aditya Menon (ANU) COMP2610/COMP Information Theory Semester 2 1 / 37

2 Last time Mutual information chain rule Jensen s inequality Information cannot hurt Data processing inequality Mark Reid and Aditya Menon (ANU) COMP2610/COMP Information Theory Semester 2 2 / 37

3 Review: Data-Processing Inequality Theorem if X Y Z then: I (X ; Y ) I (X ; Z) X is the state of the world, Y is the data gathered and Z is the processed data No clever manipulation of the data can improve the inferences that can be made from the data No processing of Y, deterministic or random, can increase the information that Y contains about X Mark Reid and Aditya Menon (ANU) COMP2610/COMP Information Theory Semester 2 3 / 37

4 This time Markov s inequality Chebyshev s inequality Law of large numbers Mark Reid and Aditya Menon (ANU) COMP2610/COMP Information Theory Semester 2 4 / 37

5 Outline 1 Properties of expectation and variance 2 Markov s inequality 3 Chebyshev s inequality 4 Law of large numbers 5 Wrapping Up Mark Reid and Aditya Menon (ANU) COMP2610/COMP Information Theory Semester 2 5 / 37

6 1 Properties of expectation and variance 2 Markov s inequality 3 Chebyshev s inequality 4 Law of large numbers 5 Wrapping Up Mark Reid and Aditya Menon (ANU) COMP2610/COMP Information Theory Semester 2 6 / 37

7 Expectation and Variance Let X be a random variable over X, with probability distribution p Expected value: E[X ] = x X x p(x). Variance: V[X ] = E[(X E[X ]) 2 ] = E[X 2 ] (E[X ]) 2. Standard deviation is V[X ] Mark Reid and Aditya Menon (ANU) COMP2610/COMP Information Theory Semester 2 7 / 37

8 Properties of expectation A key property of expectations is linearity: [ n ] n E X i = E [X i ]. i=1 i=1 This holds even if the variables are dependent! We have for any a R, E[aX ] = a E[X ]. Mark Reid and Aditya Menon (ANU) COMP2610/COMP Information Theory Semester 2 8 / 37

9 Properties of variance We have linearity of variance for independent random variables: [ n ] n V X i = V [X i ]. i=1 i=1 Does not hold if the variables are dependent We have for any a R, V[aX ] = a 2 V[X ]. Mark Reid and Aditya Menon (ANU) COMP2610/COMP Information Theory Semester 2 9 / 37

10 1 Properties of expectation and variance 2 Markov s inequality 3 Chebyshev s inequality 4 Law of large numbers 5 Wrapping Up Mark Reid and Aditya Menon (ANU) COMP2610/COMP Information Theory Semester 2 10 / 37

11 Markov s Inequality Motivation 1000 school students sit an examination The busy principal is only told that the average score is 40 out of 100 The principal wants to estimate the maximum number of students who scored more than 80 Would it make sense to ask about the minimum number of students? Mark Reid and Aditya Menon (ANU) COMP2610/COMP Information Theory Semester 2 11 / 37

12 Markov s Inequality Motivation Call x the number of students who score more than 80 We know: = 80x + S, where S is the total score of students scoring less than 80 Exam scores are nonnegative, so certainly S 0 Thus, 80x , or x 500. Can we formalise this more generally? Mark Reid and Aditya Menon (ANU) COMP2610/COMP Information Theory Semester 2 12 / 37

13 Markov s Inequality Theorem Let X be a nonnegative random variable. Then, for any λ > 0, p(x λ) E[X ] λ. Bounds probability of observing a large outcome Mark Reid and Aditya Menon (ANU) COMP2610/COMP Information Theory Semester 2 13 / 37

14 Markov s Inequality Alternate Statement Corollary Let X be a nonnegative random variable. Then, for any λ > 0, p(x λ E[X ]) 1 λ. Observations of nonnegative random variable unlikely to be much larger than expected value Mark Reid and Aditya Menon (ANU) COMP2610/COMP Information Theory Semester 2 14 / 37

15 Markov s Inequality Proof E[X ] = x p(x) x X = x p(x) + x p(x) x<λ x λ x λ x p(x) nonneg. of random variable x λ λ p(x) = λ p(x λ). Mark Reid and Aditya Menon (ANU) COMP2610/COMP Information Theory Semester 2 15 / 37

16 Markov s Inequality Illustration from Mark Reid and Aditya Menon (ANU) COMP2610/COMP Information Theory Semester 2 16 / 37

17 Markov s Inequality Illustration from Mark Reid and Aditya Menon (ANU) COMP2610/COMP Information Theory Semester 2 17 / 37

18 Markov s Inequality Illustration from Mark Reid and Aditya Menon (ANU) COMP2610/COMP Information Theory Semester 2 18 / 37

19 Markov s Inequality Illustration from Mark Reid and Aditya Menon (ANU) COMP2610/COMP Information Theory Semester 2 19 / 37

20 1 Properties of expectation and variance 2 Markov s inequality 3 Chebyshev s inequality 4 Law of large numbers 5 Wrapping Up Mark Reid and Aditya Menon (ANU) COMP2610/COMP Information Theory Semester 2 20 / 37

21 Chebyshev s Inequality Motivation Markov s inequality only uses the mean of the distribution What about the spread of the distribution (variance)? Mark Reid and Aditya Menon (ANU) COMP2610/COMP Information Theory Semester 2 21 / 37

22 Chebyshev s Inequality Theorem Let X be a random variable with E[X ] <. Then, for any λ > 0, p( X E[X ] λ) V[X ] λ 2. Bounds the probability of observing an unexpected outcome Do not require non negativity Two-sided bound Mark Reid and Aditya Menon (ANU) COMP2610/COMP Information Theory Semester 2 22 / 37

23 Chebyshev s Inequality Alternate Statement Corollary Let X be a random variable with E[X ] <. Then, for any λ > 0, p( X E[X ] λ V[X ]) 1 λ 2. Observations are unlikely to occur several standard deviations away from the mean Mark Reid and Aditya Menon (ANU) COMP2610/COMP Information Theory Semester 2 23 / 37

24 Chebyshev s Inequality Proof Define Y = (X E[X ]) 2. Then, by Markov s inequality, for any ν > 0, But, Also, Thus, setting λ = ν, p(y ν) E[Y ] ν. E[Y ] = V[X ]. Y ν X E[X ] ν. p( X E[X ] λ) V[X ] λ 2. Mark Reid and Aditya Menon (ANU) COMP2610/COMP Information Theory Semester 2 24 / 37

25 Chebyshev s Inequality Illustration For a binomial with N trials and success probability θ, we have e.g. p( X Nθ 2Nθ(1 θ)) 1 2. Mark Reid and Aditya Menon (ANU) COMP2610/COMP Information Theory Semester 2 25 / 37

26 Chebyshev s Inequality Example Suppose we have a coin with bias θ, i.e. p(x = 1) = θ Say we flip the coin n times, and observe x 1,..., x n {0, 1} We use the maximum likelihood estimator of θ: ˆθ n = x x n n Estimate how large n should be such that 1% probability of a 5% error p( ˆθ n θ 0.05) 0.01? Mark Reid and Aditya Menon (ANU) COMP2610/COMP Information Theory Semester 2 26 / 37

27 Chebyshev s Inequality Example Observe that E[ˆθ n ] = V[ˆθ n ] = n i=1 E[x i] = θ n n i=1 V[x i] θ(1 θ) n 2 =. n Thus, applying Chebyshev s inequality to ˆθ n, p( ˆθ n θ > 0.05) We are guaranteed this is less than 0.01 if When θ = 0.5, n 10, 000! n θ(1 θ) (0.05) 2 (0.01). θ(1 θ) (0.05) 2 n. Mark Reid and Aditya Menon (ANU) COMP2610/COMP Information Theory Semester 2 27 / 37

28 1 Properties of expectation and variance 2 Markov s inequality 3 Chebyshev s inequality 4 Law of large numbers 5 Wrapping Up Mark Reid and Aditya Menon (ANU) COMP2610/COMP Information Theory Semester 2 28 / 37

29 Independent and Identically Distributed Let X 1,..., X n be random variables such that: Each X i is independent of X j The distribution of X i is the same as that of X j Then, we say that X 1,..., X n are independent and identically distributed (or iid) Example: For n independent flips of an unbiased coin, X 1,..., X n are iid from Bernoulli ( ) 1 2 Mark Reid and Aditya Menon (ANU) COMP2610/COMP Information Theory Semester 2 29 / 37

30 Law of Large Numbers Theorem Let X 1,..., X n be a sequence of iid random variables, with and V[X i ] <. Define E[X i ] = µ Then, for any ɛ > 0, X n = X X n. n lim p( X n µ > ɛ) = 0. n Given enough trials, the empirical success frequency will be close to the expected value Mark Reid and Aditya Menon (ANU) COMP2610/COMP Information Theory Semester 2 30 / 37

31 Law of Large Numbers Proof Since X i s are identically distributed, E[ X n ] = µ. Since the X i s are independent, [ ] X X n V[ X n ] = V n = V [X X n ] n 2 = nσ2 n 2 = σ2 n. Mark Reid and Aditya Menon (ANU) COMP2610/COMP Information Theory Semester 2 31 / 37

32 Law of Large Numbers Proof Applying Chebyshev s inequality to X n, p( X n µ ɛ) V[ X n ] ɛ 2 = σ2 nɛ 2. As n, the right hand side 0. Thus, p( X n µ < ɛ) 1. Mark Reid and Aditya Menon (ANU) COMP2610/COMP Information Theory Semester 2 32 / 37

33 Law of Large Numbers Illustration N = 1000 trials with Bernoulli random variable with parameter Mark Reid and Aditya Menon (ANU) COMP2610/COMP Information Theory Semester 2 33 / 37

34 Law of Large Numbers Illustration N = trials with Bernoulli random variable with parameter x 10 Mark Reid and Aditya Menon (ANU) COMP2610/COMP Information Theory Semester 2 34 / 37

35 1 Properties of expectation and variance 2 Markov s inequality 3 Chebyshev s inequality 4 Law of large numbers 5 Wrapping Up Mark Reid and Aditya Menon (ANU) COMP2610/COMP Information Theory Semester 2 35 / 37

36 Summary & Conclusions Markov s inequality Chebyshev s inequality Law of large numbers Mark Reid and Aditya Menon (ANU) COMP2610/COMP Information Theory Semester 2 36 / 37

37 Next time Ensembles and sequences Typical sets Approximation Equipartition (AEP) Mark Reid and Aditya Menon (ANU) COMP2610/COMP Information Theory Semester 2 37 / 37