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I've been considering how sports with binary outcomes might be modelled e.g. the probability of a tennis player winning a point on serve.

In text books the usual Bayesian approach uses the beta-binomial. The updating rule is equivalent to starting with some pseudo observations for successes and failures and adding on new successes and failures as they occur, from which you calculate your new probability of winning that particular point.

I've read an alternative is the binomial logistic-normal where the log of the odds (logit) approaches a normal distribution. What would my Bayesian updating rule be here for each new success or fail? I assume the odds get multiplied by a different factor each point (adding to the logit)?

I don't yet have the ability in statistics to understand GLMs or GLMMs properly but I'm still curious to know the updating rule as I could plot graphs for p and the logit, calculate moments, and get some intuition for how it compares to Beta-Binomal as n gets bigger.

On page 164 of 'The Mathematics of tennis' http://www.strategicgames.com.au/book.pdf the authors use an updating rule which is the following:

"For player i the proportion of initial serving statistics $X_i$ is combined with actual serving statistics $Y_i$ to give updated serving statistics $Z_i$ at any point within the match."

"n represents the total number of points played and c is a constant."

$Z_i = e^{\!-n/c}X_i + (1−e^{\!-n/c} )Y_i$

Is this binomial logistic-normal or something different? (I emailed the one contactable author but he never got back to me.)

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    $\begingroup$ I think the problem is that there isn't an analytically tractable answer to your question. Beta Binomial is nice because you have an analytical updating formula, and can just plug in the numbers. Move away from that and you need to use some computer based iterative or simulation based approach. MCMC is the simulation tool that really opened up problems like this, but you might not need to do anything quite so complicated. $\endgroup$
    – Paul Hewson
    Commented Dec 18, 2019 at 15:19
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    $\begingroup$ By "there isn't an analytically tractable answer" I think you're asking a good question! $\endgroup$
    – Paul Hewson
    Commented Dec 18, 2019 at 15:19
  • $\begingroup$ Thank you Paul. Due to my lack of Statistical sophistication that I alluded to I approached this problem from different angles. I thought the answer might be analytically tractable and lie with the multiplicative central theorem. I thought understanding the more ubiquitous log-normal distribution might help me understand the logit-normal (but I could be completely wrong). Options traders assume stock returns are log-normally distributed and price them with a binomial tree. I found a paper on the log-normal galton board. The kelly criterion used by gamblers optimises log growth and.. $\endgroup$
    – Batfink
    Commented Dec 18, 2019 at 16:24
  • $\begingroup$ involves entropy. The derivative of the binary entropy function is the negative of the logit function. The log of the ratio of two binomial proportions is approximately normally distributed. I assumed the taylor series would be involved. I thought I might get my answer by understanding these? I was hoping to reverse engineer the parameters from the betting markets but sounds like that might be very difficult. $\endgroup$
    – Batfink
    Commented Dec 18, 2019 at 16:35
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    $\begingroup$ Hi, everything you say sounds reasonable to me. I think the reasons it took 100s of years for anyone to do Bayesian statistics was because there were no closed form solutions to "real" problems like yours. Not funny doing thousands of iterations if you are inverting matrices by hand. I'm sure once you start looking at a computationally intensive method to solve the problem you find it liberating. At6 the moment my favourite MCMC toy is called STAN mc-stan.org but that might be massive overkill for what you want. $\endgroup$
    – Paul Hewson
    Commented Dec 19, 2019 at 10:20

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