The Reverend Bayes and tennis probability, Part II

For our Big Bayesian Experiment, let's take an example from 2005, which is the tennis-data spreadsheet I currently have open. Roger Federer* took on Andre Agassi in the semi-final of the Dubai Duty Free men's tournament on Feb. 26th. A little over a month earlier, they played in the quarter-finals of the Australian Open, which Federer won handily, 6-3, 6-4, 6-4.

I'm going to take a very simple model of tennis, one which could almost certainly be improved by taking into account strength of serve and the number of service breaks in each set - unfortunately, I have neither the data, the programming skill nor the patience to put that kind of model into effect for a short blog article. Anyway, contrary to all good sense, I'll assume that each game has an identical probability of each player winning, and that parameter carries forward to the next match.

And to begin with, let's also assume that we know nothing about Federer and Agassi - the chance of Federer being a Scottish no-hoper and having no chance of getting near Agassi (p_F=0, p_A=1) is the same as that of the roles being reversed (p_F=1, p_A=0) - all values of p_F are equally likely. This is our prior distribution. We're looking for a posterior distribution of p_F - how likely each value of the variable is.

There's some number-crunching to be done here. What we're going to do is examine every (well, almost) possible value of p_F, find the probability for each of Federer winning the match with that score, and examine the distribution that comes out. Some examples:


.pF. P(6-3) P(6-4) P(6-3, 6-4, 6-4)
0.40 0.0493 0.0666 0.00022
0.50 0.1091 0.1228 0.00165
0.60 0.1670 0.1505 0.00378
0.70 0.1780 0.1204 0.00258


The highest of these is 0.60 (in fact, the most likely value for p_F is the number of games he won - 18 - over the number played - 29 - or about 0.62). The number-crunching (combined with NeoOffice) give a nice graph of the likelihoods, which I reproduce here:




You can see that the peak of the PDF (I won't go into the terminology here, it's getting long) is indeed around 0.62 (0.6207, to be precise). But we're interested in a confidence range, for which we read off of the CDF the p_F values for P at various levels. For instance, we're 95% certain that p_F lies between the 0.025 level and the 0.975 level - i.e. between 0.4385 and 0.7734. We're 50% sure that it lies between the 0.25 and 0.75 levels - so between 0.5550 and 0.6734.

How does this help us? Well, it narrows down p_F substantially - remember, we didn't have a clue what it was before. Now we can say with some certainty where it isn't - it's unlikely to be more than 0.7734 or less than 0.4385, eliminating almost two-thirds of the possible values at a stroke. It's as likely as not to be in the range [0.5550, 0.6734]. Can we translate this into a match probability for the next game? Of course. Again, it's a number-crunching exercise, but we get the following for our key values of p_F:


..pF.. P(Set) P(2-0) P(2-1) P(Win)
0.4385 0.3307 0.1093 0.1464 0.2557
0.5550 0.6522 0.4253 0.2959 0.7217
0.6207 0.8075 0.6250 0.2510 0.8760
0.6734 0.8977 0.8058 0.1649 0.9707
0.7734 0.9825 0.9653 0.0338 0.9991


This tells us: the most likely probability of Federer beating Agassi in two sets is 87.6% (1.14), that it's as likely as not to be between 72.17% (1.39) and 97.07% (1.03), and 95% certain to be between 25.57 (3.91) and 0.9991 (1.001). In the event, the best available odds were 1.35 with Expekt - and depending on how much confidence-in-value we wanted, we would take those and reap the benefit of Federer's 6-3, 6-1 demolition job.

This system is, I have to stress, very basic and currently only works if the two players met in the recent past. In the next instalment, I might have a shot at a link function so we can rate players who haven't met recently - but have played a common opponent.

* easy to spell; difficult to stop spelling