SLIDE 1
Divergence and Gambling Mathias Winther Madsen - - PowerPoint PPT Presentation
Divergence and Gambling Mathias Winther Madsen - - PowerPoint PPT Presentation
Divergence and Gambling Mathias Winther Madsen mathias.winther@gmail.com Institute for Logic, Language, and Computation University of Amsterdam March 2015 Divergence Definition If X is a random variable with probability densities p ( x ) ,
SLIDE 2
SLIDE 3
Divergence
x r a b p(x)
1/4 1/2 1/4
p-code 00 1 01 q(x)
1/2 1/4 1/4
q-code 10 11
rabarbararabaa ...
SLIDE 4
Divergence · rabarbararabaarbabra
5 10 15 20 25 30 35 40 Observation Surprisal
SLIDE 5
Divergence
Definition
The Kullback-Leibler divergence from p to q is D(p || q) = E
- log
1 q(X)
- − E
- log
1 p(X)
- =
- x
p(x) log p(x) q(x), where E[ · ] is the expectation with respect to p.
Properties of the KL divergence
- 1. Nonnegativity: D(p || q) ≥ 0;
- 2. Coincidence: D(p || q) = 0 if and only if p = q;
- 3. Asymmetry: Often, D(p || q) = D(q || p).
Solomon Kullback and Richard A. Leibler: “Information and Sufficiency,” Annals of Mathematical Statistics, 1951.
SLIDE 6
Divergence
0.5 1 1 2 0.3 1 1 2 x 1 2 p(x) .5 .5 x 1 2 p(x) .3 .7
SLIDE 7
Divergence
x 1 2 3 p(x)
1/4 1/4 1/2
SLIDE 8
Divergence a b
.38 1 .62 p(abaab . . . ) = p(a) · p(b | a) · p(a | b) · p(a | a) · · · q(abaab . . . ) = q(a) · q(b) · q(a) · q(a) · q(b) · · ·
SLIDE 9
Gambling
Problem
A horse race has three horses: x 1 2 3 p(x) .2 .3 .5
- (x)
4 4 2 b(x) ? ? ? Which capital distribution b maximizes the expected rate of return, E[ R ] = E [ o(X) b(X) ]?
SLIDE 10
Gambling
x 1 2 3 p(x) .2 .3 .5
- (x)
4 4 2 b(x) 1 0 2 4 6 8 10 20 40
SLIDE 11
Gambling
Daniel Bernoulli: “Specimen Theoriae Novae de Mensura Sortis,” Commentarii Academiae Scientiarum Imperialis Petropolitanae, 1738. John L. Kelly, Jr: “A New Interpretation of Information Rate,” Bell System Technical Journal, 1956.
SLIDE 12
Gambling
Definition
The doubling rate is the logarithm of the rate of return, W(X) = log R(X) = log o(X)b(X).
Proportional Gambling
The doubling rate attains its maximum at b∗ = p, regardless of the odds. x 1 2 3 p(x) .2 .3 .5
- (x)
4 4 2
SLIDE 13
Gambling
200 400 600 800 1,000 10−1 100 101 102 103 104 105 Rounds Capital
SLIDE 14