## 06 July, 2015

### Bayesian Credible Intervals for a Batting Average

(This post is a followup to my previous post on bayesian inference)

I've posted a fair amount about confidence intervals for various quantities. All of the ones I've posted so far - central limit based theorem intervals, Wald theory intervals, and likelihood intervals - are based on a frequentist understanding of probability - that is to say, probability is defined as the limiting result of the proportion of times an event (say, a hit) happens as the number of trials (say, at-bats) goes to infinity.

The term "95% confidence" refers to the construction of the interval itself - that is, if we were to calculate a 95% confidence interval for the true batting average $\theta$ for each of our millions and millions of trials, then 95% of them will contain the true batting average $\theta$.

Statisticians tend to avoid making probability statements about confidence intervals. The statement that "There is a 95% chance that $\theta$ is in the interval." is incorrect because $\theta$ is conceptualized as a fixed quantity. Furthermore, the statement "There is a 95% chance that the interval contains $\theta$." is awkward because once an interval has been calculated, there's no more randomness anymore - it either contains $\theta$, or it doesn't. This is why statisticians prefer to use confidence rather than probability to describe intervals.

But what if you are working in a Bayesian framework? The end result of a Bayesian analysis is a distribution $p(\theta | x)$ that represents the distribution of believe in $\theta$ after the data has been accounted for - so it makes perfect sense to write, for example, $P(0.250 \le \theta \le 0.300)$.

All the code used to generate the images in this article may be found on my github.

## Credible Intervals

Instead of confidence intervals, Bayesian statisticians instead calculate  "Credible" intervals - these are intervals from the distribution of $p(\theta | x)$ that contain the desired amount of probability. Say, for example, that $P(0.250 \le \theta \le 0.300) = 0.95$ - then the interval $(0.250, 0.300)$ would be a 95% credible interval for $\theta$.

The main issue with this method is that there is more than one way to get a 95% credible interval given $p(\theta | x)$ - technically, any interval $(L, U)$ with $P(L \le \theta \le U) = 0.95$ is a valid 95% credible interval. Statisticians have several ways to determine which one to use, but I'm going to show you one that can be easily done with most computer software.

## Baseball Example

In the previous post, I explained how to use the beta-binomial model to get a posterior distribution for a batting average using a beta-binomial model. Let's take observer A, who for the batter with $15$ hits in $n = 50$ at-bats had a prior distribution of belief given by a beta distribution with parameters $\alpha = 1$ and $\beta = 1$

And a posterior distribution of belief given by a beta distribution with parameters $\alpha' = 16$ and $\beta' = 36$

To get a credible interval, we can take quantiles from the beta distribution. A quantile is a value $Q_{p}$ for a distribution so that $P(X \le Q_{p}) = p$. To get a 95% credible interval, we can take $Q_{0.025}$ as the lower boundary of the interval and $Q_{0.975}$ as the upper boundary of the interval (since 97.5% - 2.5% = 95%) so that the interval contains the middle 95% of the probability.

Since these do not have nice formulas to calculate by hand, it's easiest to use computer software to get the - any good statistical software should be able to give quantiles for common distributions. In the program $R$, the command to do this is

> qbeta(c(.025,.975),16,36)
[1] 0.1911040 0.4382887

So for observer A, a 95% credible interval for $\theta$ is given by $(0.191, 0.438)$. With quantiles, the posterior belief for observer A looks like

The area under the curve between the two vertical lines is 0.95 - and so the values of the vertical lines give the 95% credible interval.

What about observer B? Observer B used a beta distribution as their prior with $\alpha = 53$ and $\beta = 147$, for a posterior distribution that is beta with $\alpha' = 68$ and $\beta' = 182$. Quantiles from observer B's posterior distribution are

> qbeta(c(.025,.975),68,182)
[1] 0.2187438 0.3287111

So observer B's 95% credible interval for $\theta$ is $(0.219, 0.329)$. Note that observer B's interval is much more realistic - as baseball fans, we know that a $\theta = 0.400$ batting average is very, very unlikely - so good prior information can lead to improved inference.