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Lecture 9 Entropy (Section 3.1, a bit of 3.2)

Lecture 9 Entropy (Section 3.1, a bit of 3.2). Theory of Information. Information Obtained from a Source Symbol. Source: {“has two kidneys”, “has one kidney”} Three source symbols are emitted, one carrying information about Bob, one about Jane, and one about Bill: Bob: Has two kidneys

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Lecture 9 Entropy (Section 3.1, a bit of 3.2)

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  1. Lecture 9Entropy(Section 3.1, a bit of 3.2) Theory of Information

  2. Information Obtained from a Source Symbol Source: {“has two kidneys”, “has one kidney”} Three source symbols are emitted, one carrying information about Bob, one about Jane, and one about Bill: Bob: Has two kidneys Jane: Has two kidneys Bill: Has one kidney Which piece of information was most informative? Why?

  3. Information Obtained from a Source Symbol Conclusion: The lower the probability of a source symbol, the more information it contains. I(s) --- the information carried by source symbol s. Unsuccessful attempt: Why not simply stipulate I(s)=1/P(s)? Not a good idea. Source alphabet: {M, F} (={Male, Female}) Total info: A: Male 1 unit of info 1 B: Male 1 unit of info 2 C: Female 1 unit of info 3 D: Male 1 unit of info 4 … However, 1/P({M,M,F,M}) is 1/16 rather than 1/4

  4. Information Obtained from a Source Symbol Successful attempt: Let us stipulate I(s)=lg(1/P(s)) Good idea. “Bit” (binary unit) – measure of information. Source alphabet: {M, F} I(M)=I(F)=lg(1/0.5)=1. Total info: A: Male 1 bit 1 bit B: Male 1 bit 2 bits C: Female 1 bit 3 bits D: Male 1 bit 4 bits … lg(1/P({M,M,F,M})= lg(1/(1/16))=lg16=4

  5. Information Obtained from a Source Symbol DEFINITION The information I(p) (or I(s)) obtained from a source symbol s with probability of occurrence p>0 is given by I(p) (or I(s)) =lg(1/p) Example: {a,b,c} P(a)=1/8, P(b)=1/32, P(c)=27/32 I(a)=? I(b)=? I(c)=?

  6. Definition of Entropy DEFINITION Let S=(S,P) be a source, with probability distribution P={p1,…,pq}. The average information H(S) obtained from a single sample (symbol) from S, also called the ENTROPY of the source, is H(S) = p1I(p1)+…+pqI(pq) = p1lg(1/p1)+…+pqlg(1/pq). Intuition: Entropy = degree of uncertainty about what symbol a given symbol (emitted from a black box) would be. Example 1. S=({s},1). H(S)= Example 2. S=({a,b}, 0.5, 0.5) H(S)= Example 3. S=({a,b}, 99%, 1%) H(S)= Example 4. S=({a,b,c,d}, ¼, ¼, ¼, ¼ ) H(S)= Example 5. S=({a,b,c}, ¼, ¼ , ½) H(S)=

  7. The Entropy of 2-symbol Sources Consider a source S=({a,b), P). The graph for H(S), showing the entropy of S depending on P(a): 1 ½ 1

  8. An Important Property of Entropy THEOREM 3.2.3 For a source S of size q, the entropy H(S) satisfies 0 H(S) lg q. Furthermore, H(S)=lg q (is the biggest) if and only if all source symbols are equally likely to occur, and H(S)=0 if and only if one of the symbols has probability 1 of occurring.

  9. Homework • 9.1. Exercises 1,2,5,6 of Section 3.1. • 9.2. The definitions of I(s) (the information obtained from a source • symbol) and H(S) (entropy). • 9.3. Give a probability distribution for the source alphabet • {a,b,c,d,e,f,g,h} that yields: • The lowest possible entropy. What does that entropy equal to? • b) The highest possible entropy. What does that entropy equal to?

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