## Math-Startup Collaborative at Columbia University

This Wednesday Feb. 29th, SIAM (CU Society of Applied and Industrial Mathematics) and ADI (CU Application Development Initiative) will host and event geared towards getting the word out that math skills open doors for cool tech jobs. There will be mingling, presentations from Columbia Alumni (including your’s truly) and some more mingling. Spread the word!

## Statistics and Algebra. An Example.

Written by : Matt

There is a developing field called algebraic statistics which explores probability and statistics problems involving discrete random variables using methods coming from commutative algebra and algebraic geometry. The basic point is that the parameters for such statistical models are often constrained by polynomial relationships – and these are exactly the subject of commutative algebra and algebraic geometry. I would like to learn something more about this relationship, so in this post I’ll describe one example that I worked through – it comes from a book on the subject written by Bernd Sturmfels. Disclaimer : the rest of this post is technical.

## Gambling and Shannon’s Entropy Function Part 2.

In the last post I gave an introduction of Kelly’s paper where he describes optimal gambling strategies based on information received over a potentially noisy channel. Here I’ll talk about the general case, where the channel has several inputs symbols, each with a given probability  of being transmitted, and which represent the outcome of some chance event. First, we need to set up some notation:

$p(s)$ –  probability that the transmitted symbol is s.

$p(r | s)$ – the conditional probability that the received symbols is r given that the transmitted symbol is s.

$p(r, s)$ –  the joint probability that the received symbol is r and the transmitted symbol is s.

$q(r)$ –  probability that the received symbol is r.

$q(s | r)$ – the conditional probability that the transmitted symbol is s given that the received symbols is r.

$\alpha_s$ – the odds paid on the occurrence of s, i.e. the number of dollars returned on a one-dollar bet on s.

$a(s/r)$ – the fraction of capital that the gambler decides to bet on the occurrence of s after receiving r.

## Gambling and Shannon’s entropy function.

A little while ago Matt posted an overview of the definition of Shannon’s entropy function and it’s use in assessing interaction between random variables. I’d like  to stay with the 1960’s Bell Labs team and describe some of the results from J. L. Kelly’s wonderful paper, which arises from the definitions and ideas in Shannon’s original work.

Consider at first the trivial scenario where we have a noiseless binary channel that transmits the results of say a baseball game. If the odds are even and the gambler has access to this channel he can grow his capital exponentially by making certain bets (since he knows the outcome before betting). His capital would grow at a rate of $2^N$, after N bets. In view of the fact that any such function should be maximum in the above scenario, Kelly defines the exponential rate of growth of capital as

$G = lim_{N \rightarrow \infty} \frac{1}{N} log_2(\frac{V_N}{V_0}),$

were $V_0$ is the initial capital and $V_N$ is the capital after N bets.

Now suppose that the channel is noisy, and a given symbol has a probability p of error and q of correct transmission. The question now is, how much should the gambler bet?