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3-data_compress - Harvard SEAS ES250 Information Theory...

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Harvard SEAS ES250 – Information Theory Gambling and Data Compression * 1 Gambling 1.1 Horse Race Definition The wealth relative S ( X ) = b ( X ) o ( X ) is the factor by which the gambler’s wealth grows if horse X wins the race, where b ( X ) is the fraction of the gambler’s wealth invested in horse X and o ( X ) is the corresponding odds. Definition The doubling rate of a horse race is W ( b , p ) = E [log S ( X )] = m X k =1 p k log b k o k Theorem Let the race outcomes X 1 , X 2 , · · · be i.i.d. p ( x ). Then the wealth of the gambler using betting strategy b grows exponentially at rate W ( b , p ); that is, S n . = 2 nW ( b , p ) Definition The optimum doubling rate W * ( p ) is the maximum doubling rate over all choices of the portfolio b : W * ( p ) = max b W ( b , p ) = max b : b i 0 , P i b i =1 m X i =1 p i log b i o i Theorem (proportional gambling is log-optimal) the optimal doubling rate is given by W * ( p ) = X p i log o i - H ( p ) and is achieved by the proportional gambling scheme b * = p . Theorem (Conservation theorem) For uniform fair odds, W * ( p ) + H ( p ) = log m Thus, the sum of the doubling rate and the entropy is a constant. If the gambler does not always bet all the money, then the optimum strategy may depend on the odds and will not necessarily have the simple form of proportional gambling. There are three cases: 1. Fair odds with respect to some distribution: 1 o i = 1. By betting b i = 1 o i , one achieves S ( X ) = 1, which is the same as keeping some cash aside. Proportional betting is optimal. 2. Superfair odds: 1 o i < 1. By choosing b i = c 1 o i , where c = 1 / 1 o i , one has S ( X ) = 1 / 1 o i > 1 with probability 1. In this case, the gambler will always want to bet all the money and the optimum strategy is again proportional betting. 3. Subfair odds: 1 o i > 1. Proportional gambling is no longer log-optimal. The gambler may want to bet only some of the money and keep the rest aside as cash, depending on the odds. * Based on Cover & Thomas, Chapter 5,6 1
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Harvard SEAS ES250 – Information Theory 1.2 Side Information and Entropy Rate Definition The increase Δ W is defined as: Δ W = W * ( X | Y ) - W * ( X ) , where W * ( X ) = max b ( x ) X x p ( x ) log b ( x ) o ( x ) W * ( X | Y ) = max b ( x | y ) X x,y p ( x, y ) log b ( x | y ) o ( x ) Theorem The increase Δ W in doubling rate due to side information Y for a horse race X is Δ W = I ( X ; Y ) .
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