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

The Reverend Bayes and tennis probability

The Reverend Thomas Bayes was pretty much your archetypal dour, 18th century English Nonconformist minister. Except that he was a probability whizz, and gave us a law which has annoyed amateur statisticians for the better part of three centuries. It goes something like this: the probability of one thing happening given that another happens, is the probability of both happening divided by the probability of the other thing: P(A|B) = P(A & B)/P(B)

A typical example would be "Given that my girlfriend has two cats, at least one of which is female, what is the probability of her having two girl cats?" The probability of two cats both being female is (in our idealised world) one in four, or 25%. The probability of at least one female cat in two is 3/4 or 75% (FF, FM, MF all include a girl, MM doesn't). So the probability of a second girl given a first girl is (25%)/(75%) or 1/3 (33.33%) - higher than the 1/4 given no information.

It is natural that Bayes's Law should remind me of tennis, since I have spent a lot of time looking at both twisting my head from one side to the other. How can we use previous results to determine unknown probabilities? And how well do we know them?

We will need to broach the difficult subject of probability distributions. The easiest way (for me at least) to visualise a probability distribution is as a graph. The graph outlines a region of unit area* - the x axis is an outcome, often a continuum from 0 to 1 but not forcibly; the y axis is a mystical variable called probability density. You drop a pin onto the graph so that the point is equally likely to land anywhere in the region, and read off the value on the x-axis. You can see that the peaks in the graph correspond to more likely outcomes.



Let's look first at the PDF of the sum of three rolled dice. It peaks in the middle, around 10 and 11 - meaning that you're much more likely to roll 11 than 3. This makes sense - you have many more ways to roll 11 (27, I think) than to roll 3 (just one).

Some important distributions include the uniform distribution, which is a level straight line (all outcomes equally likely) and the normal distribution, which is a bell curve - outcomes near the mean are much likelier than ones far away.

What we're going to try in the next article is create a very simple tennis model and find how our knowledge about one game affects our knowledge of the next. Our strategy is going to be as follows.

We start with a uniform prior distribution of p, a variable that describes how likely one of the players is to win a single game. We then take each possible value of p (from 0 to 1) and see what the probability of the result of a known game would be, given that value of p. Out of that we get a likelihood graph showing what values of p are more likely than others - which can be converted into a PDF if we multiply by a constant. Given the PDF, we can determine confidence limits for our value of p - we want to be, say, 75% sure it's no lower than a given value so we can evaluate the odds on offer for the next game.

* at least for continuous PDFs. For discrete ones, the sum of the probabilities is one, which amounts to the same thing in the limit.

[This article and the next were originally one article. But I realised afterwards that I'd lost the thread somewhere and needed to explain things a bit better.]