A Introduction
The frequentist statistics in the previous chapter dealt with countable, identifiable items and probably seemed reasonably familiar even if you don't do statistical calculations on a daily basis.
Statistics is not always so straightforward, though. Getting a good grip on what “a 30% chance of rain tomorrow” means, for instance, is trickier than deducing your chances of rolling a seven with two dice. For one thing, the chance of rain is a “single-event probability.”1 You can't sample from a large set of identical tomorrows, so calculating the exact odds of rain is out of the question; obviously, the frequency interpretation of probability does not apply. Moreover, just what the “%” refers to—the reference class—is often unclear: Are we talking about the percentage of time that it will rain tomorrow, the area of the local region that will get rained on, or something else?The statistical methods that can handle issues like “the chance of rain” are part of a large and complex class known today as Bayesian statistics after its inventor, the English clergyman Thomas Bayes. Despite Bayes' discovery of the primary rule, or theorem, of his method, his work only appeared posthumously after a friend put Bayes' notes together and published them. This branch of statistics was independently discovered and elaborated by the French mathematician, Pierre- Simon Laplace, whose contribution was notable enough that some writers feel that we should be talking about “Laplacian” instead of “Bayesian” statistics. Nevertheless, a name change is unlikely at this point.
My main goal here is to consider how we might put Bayesian reasoning together with conventional hypothesis testing scientific methods. I'll go into the fundamentals of Bayesian statistics only as much as necessary to reach the goal. Because it is usually omitted from the science curriculum, this chapter may at first look dauntingly dense and equation-filled, but you'll soon see that what looks like several equations is really just one, shape-shifting its way through the chapter; if you stick with it, your intuition about what's going on will improve. Readers with a solid background in the basics of Bayesian methods may want to skip ahead to Section 6.D, where we'll zoom out to consider how to integrate these methods with hypothesis-based thinking in ways that are rarely discussed, even by Bayesians.
6.