# Beta-binomial model

1. Aug 18, 2012

### ParoXsitiC

I asked a question here: http://math.stackexchange.com/questions/183483/rating-system-incorporating-experience and I'd like to understand how I do the beta-binomial model, but he is taking a bit long to answer so I am hoping to get an answer.

One option is to use something like the [beta-binomial model][1].

The general idea is that players' true *success rates* (or winning frequency) come from an underlying distribution (e.g. a beta distribution). As a player plays more games and you get actual information on wins and losses, the prior information from the beta distribution is combined with the wins/losses information which is expected to follow a binomial distribution (based on the true success rate $p$), and a posterior estimate of the success rate made as
$$\hat{p}=\frac{n_{\text{wins}}+\nu\rho}{n_{\text{games}}+\nu}$$
where the beta distribution essentially has the effect of a prior information equivalent to $\nu$ games with $\rho$ success rate.

The advantage of this method is that for playes with few games, the estimated success rate is *shrinked* towards the population mean; extreme success rates due to highly uncertain success rate estimates for playes with few games are avoided.

Due to the similarity with Bayesian methods, this type of approach is often referred to as empirical Bayes. However, the parameters $\nu$ and $\rho$ used to specify the beta distribution are estimated using traditional frequentist methods (moment or maximum likelihood estimates).

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I looked the wikipedia on it here: http://en.wikipedia.org/wiki/Beta_binomial

and found an example calculating the alpha and beta using sample methods - Given this data set:

Males 0 1 2 3 4 5 6 7 8 9 10 11 12

They came up with the 1st sample moment as : 6.23
2nd sample moment as 42.31

I searched for a while how this calculation was actually done, and finally came to this wikipedia page: http://en.wikipedia.org/wiki/Beta_distribution#Parameter_estimation

Which says the 1st sample moment and 2nd sample moment are just the sample mean and sample variance,

which I found to be 6 and 15.166667 - different than their 6.23 and 42.31.

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I just want find a good rating system that involves win and losses and need a laymen explanation on how it's done. I also asked the question here: http://www.rugatu.com/questions/2845/understanding-statistics-beta-binomial-model but he also failed to follow up and make sure I did it right :\

2. Aug 18, 2012

### Stephen Tashi

That article says the variance is the second moment centered about the the mean, so perhaps the $m_2$ in the other article is the second moment centered about 0 , which is just the mean of the squares of the data. I leave the arithmetic of testing that theory to you!

The Wikipedia article on the beta-binomial is ambiguous about the meaning of $m_2$ given that the Wikipedia article on moments refers only to "moments about" some value. I'll put a comment on the talk page of the article and see if there is a response.