College Math Teaching

April 5, 2019

Bayesian Inference: what is it about? A basketball example.

Let’s start with an example from sports: basketball free throws. At a certain times in a game, a player is awarded a free throw, where the player stands 15 feet away from the basket and is allowed to shoot to make a basket, which is worth 1 point. In the NBA, a player will take 2 or 3 shots; the rules are slightly different for college basketball.

Each player will have a “free throw percentage” which is the number of made shots divided by the number of attempts. For NBA players, the league average is .672 with a variance of .0074.

Now suppose you want to determine how well a player will do, given, say, a sample of the player’s data? Under classical (aka “frequentist” ) statistics, one looks at how well the player has done, calculates the percentage (p ) and then determines a confidence interval for said p : using the normal approximation to the binomial distribution, this works out to \hat{p} \pm z_{\frac{\alpha}{2}} \sqrt{n}\sqrt{p(1-p)}

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Yes, I know..for someone who has played a long time, one has career statistics ..so imagine one is trying to extrapolate for a new player with limited data.

That seems straightforward enough. But what if one samples the player’s shooting during an unusually good or unusually bad streak? Example: former NBA star Larry Bird once made 71 straight free throws…if that were the sample, \hat{p} = 1 with variance zero! Needless to say that trend is highly unlikely to continue.

Classical frequentist statistics doesn’t offer a way out but Bayesian Statistics does.

This is a good introduction:

But here is a simple, “rough and ready” introduction. Bayesian statistics uses not only the observed sample, but a proposed distribution for the parameter of interest (in this case, p, the probability of making a free throw). The proposed distribution is called a prior distribution or just prior. That is often labeled g(p)

Since we are dealing with what amounts to 71 Bernoulli trials where p = .672 so the distribution of each random variable describing the outcome of each individual shot has probability mass fuction p^{y_i}(1-p)^{1-y_i} where y_i = 1 for a make and y_i = 0 for a miss.

Our goal is to calculate what is known as a posterior distribution (or just posterior) which describes g after updating with the data; we’ll call that g^*(p) .

How we go about it: use the principles of joint distributions, likelihood functions and marginal distributions to calculate g^*(p|y_1, y_2...,y_n) = \frac{L(y_1, y_2, ..y_n|p)g(p)}{\int^{\infty}_{-\infty}L(y_1, y_2, ..y_n|p)g(p)dp}

The denominator “integrates out” p to turn that into a marginal; remember that the y_i are set to the observed values. In our case, all are 1 with n = 71 .

What works well is to use the beta distribution for the prior. Note: the pdf is \frac{\Gamma (a+b)}{\Gamma(a) \Gamma(b)} x^{a-1}(1-x)^{b-1} and if one uses p = x , this works very well. Now because the mean will be \mu = \frac{a}{a+b} and \sigma^2 = \frac{ab}{(a+b)^2(a+b+1)} given the required mean and variance, one can work out a, b algebraically.

Now look at the numerator which consists of the product of a likelihood function and a density function: up to constant k , if we set \sum^n_{i=1} y_i = y we get k p^{y+a-1}(1-p)^{n-y+b-1}
The denominator: same thing, but p gets integrated out and the constant k cancels; basically the denominator is what makes the fraction into a density function.

So, in effect, we have kp^{y+a-1}(1-p)^{n-y+b-1} which is just a beta distribution with new a^* =y+a, b^* =n-y + b .

So, I will spare you the calculation except to say that that the NBA prior with \mu = .672, \sigma^2 =.0074 leads to a = 19.355, b= 9.447

Now the update: a^* = 71+19.355 = 90.355, b^* = 9.447 .

What does this look like? (I used this calculator)

That is the prior. Now for the posterior:

Yes, shifted to the right..very narrow as well. The information has changed..but we avoid the absurd contention that p = 1 with a confidence interval of zero width.

We can now calculate a “credible interval” of, say, 90 percent, to see where p most likely lies: use the cumulative density function to find this out:

And note that P(p < .85) = .042, P(p < .95) = .958 \rightarrow P(.85 < p < .95) = .916 . In fact, Bird’s lifetime free throw shooting percentage is .882, which is well within this 91.6 percent credible interval, based on sampling from this one freakish streak.

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