Lecture 17: Differential Entropy

Transcription

1 Lecture 17: Differential Entropy Differential entropy AEP for differential entropy Quantization Maximum differential entropy Estimation counterpart of Fano s inequality Dr. Yao Xie, ECE587, Information Theory, Duke University

2 From discrete to continuous world Dr. Yao Xie, ECE587, Information Theory, Duke University 1

3 Differential entropy defined for continuous random variable differential entropy: h(x) = S f(x) log f(x)dx S is the support of probability density function (PDF) sometimes denote as h(f) Dr. Yao Xie, ECE587, Information Theory, Duke University 2

4 Uniform distribution f(x) = 1/a, x [0, a] differential entropy: h(x) = a 0 1/a log(a)dx = log a bits for a < 1, log a < 0, differential entropy can be negative! (unlike the discrete world) interpretation: volume of support set is 2 h(x) = 2 log a = a > 0 Dr. Yao Xie, ECE587, Information Theory, Duke University 3

5 Normal distribution f(x) = 1 x2 e 2σ 2, x R 2πσ 2 Differential entropy: h(x) = 1 2 log 2πeσ2 bits Calculation: Dr. Yao Xie, ECE587, Information Theory, Duke University 4

6 Some properties h(x + c) = h(x) h(ax) = h(x) + log a h(ax) = h(x) + log det(a) Dr. Yao Xie, ECE587, Information Theory, Duke University 5

7 AEP for continuous random variable Discrete world: for a sequence of i.i.d. random variables p(x 1,..., X n ) 2 nh(x) Continuous world: for a sequence of i.i.d. random variables 1 n log f(x 1,..., X n ) h(x), in probability. Proof from weak law of large number Dr. Yao Xie, ECE587, Information Theory, Duke University 6

8 Size of typical set Discrete world: number of typical sequences A (n) ϵ 2 nh(x) Continuous world: volume of typical set Volume of set A: Vol(A) = A dx 1 dx n Dr. Yao Xie, ECE587, Information Theory, Duke University 7

9 p(a (n) ϵ ) > 1 ϵ for n sufficiently large Vol(A) 2 n(h(x)+ϵ) for n sufficiently large Vol(A) (1 ϵ)2 n(h(x) ϵ) for n sufficiently large Proofs: very similar to the discrete world 1 = f(x 1,, x n )dx 1 dx n A (n) ϵ f(x 1,, x n )dx 1 dx n 2 n(h(x)+ϵ) A (n) ϵ dx 1 dx n Dr. Yao Xie, ECE587, Information Theory, Duke University 8

10 Volume of smallest set that contains most of the probability is 2 nh(x) for n-dimensional space, this means that each dim has measure 2 h(x)! "#$%&! "#$%& Differential entropy is a the volume of the typical set Fisher information is a the surface area of the typical set Dr. Yao Xie, ECE587, Information Theory, Duke University 9

11 Mean value theorem (MVT) If a function f is continuous on the closed interval [a, b], and differentiable on (a, b), then there exists a point c (a, b) such that f (c) = f(b) f(a) b a Dr. Yao Xie, ECE587, Information Theory, Duke University 10

12 Relation of continuous entropy to discrete entropy Discretize a continuous pdf f(x), divide the range of X into bins of length MVT: exist a value x i within each bin such that f(x i ) = (i+1) i f(x)dx f(x) x Dr. Yao Xie, ECE587, Information Theory, Duke University 11

13 Define a random variable X {x 1,...} with probability mass function p i = (i+1) i f(x)dx = f(x i ) Entropy of X H(X ) = p i log p i = f(x i ) log f(x i ) log, If f(x) is Riemann integrable, H(X ) + log h(x), as 0 Dr. Yao Xie, ECE587, Information Theory, Duke University 12

14 Implication on quantization Let = 2 n, log = n In general, h(x) + n is the number of bits on average to describe a continuous variable X to n-bit accuracy e.g. if X is uniform on [0, 1/8], the first 3 bits must be zero. Hence to describe X to n bit accuracy we need n 3 bits. Agrees with h(x) = 3 if X N (0, 100), n + h(x) = n +.5 log(2πe 100) = n Dr. Yao Xie, ECE587, Information Theory, Duke University 13

15 Joint and conditional differential entropy Joint differential entropy h(x 1,..., X n ) = f(x 1,..., x n ) log f(x 1,..., x n )dx 1 dx n Conditional differential entropy h(x Y ) = f(x, y) log f(x, y)dxdy = h(x, Y ) h(y ) Dr. Yao Xie, ECE587, Information Theory, Duke University 14

16 Relative entropy and mutual information Relative entropy D(f q) = f log f g Not necessarily finite. Let 0 log(0/0) = 0 Mutual information I(X; Y ) = f(x, y) log f(x, y) f(x)f(y) dxdy I(X; Y ) 0, D(f q) 0 Same Venn diagram relationship as in the discrete world: chain rule, conditioning reduces entropy, union bound... Dr. Yao Xie, ECE587, Information Theory, Duke University 15

17 Entropy of multivariate Gaussian Theorem. Let X 1,..., X n N (µ, K), µ: mean vector, K: covariance matrix, then h(x 1, X 2,..., X n ) = 1 2 log(2πe)n K bits Proof: Dr. Yao Xie, ECE587, Information Theory, Duke University 16

18 Mutual information of multivariate Gaussian (X, Y ) N (0, K) K = [ ] σ 2 ρσ 2 ρσ 2 σ 2 h(x) = h(y ) = 1 2 log(2πeσ2 ) h(x, Y ) = 1 2 log(2πe)2 K h(x; Y ) = h(x) + h(y ) h(x; Y ) = 1 2 log(1 ρ2 ) ρ = ±1, perfectly correlated, mutual information is! Dr. Yao Xie, ECE587, Information Theory, Duke University 17

19 Multivariate Gaussian is maximum entropy distribution Theorem. Let X R n be random vector with zero mean and covariance matrix K. Then h(x) 1 2 log(2πe)n K Proof: Dr. Yao Xie, ECE587, Information Theory, Duke University 18

20 Estimation counterpart of Fano s inequality Random variable X, estimator ˆX E(X ˆX) 2 1 2πe e2h(x) equality iff X is Gaussian and ˆX is the mean of X corollary: given side information Y and estimator ˆX(Y ) E(X ˆX(Y )) 2 1 2πe e2h(x Y ) Dr. Yao Xie, ECE587, Information Theory, Duke University 19

21 Summary discrete random variable continuous random variable entropy differential entropy Many things similar: mutual information, AEP Some things are different in continuous world: h(x) can be negative, maximum entropy distribution is Gaussian. Dr. Yao Xie, ECE587, Information Theory, Duke University 20