statisticalrethinkin..

318 12. MONSTERS AND MIXTURESDensity0.6 0.8 1.0 1.2 1.4alpha = 1, beta = 1Density0.0 0.5 1.0 1.5alpha = 3, beta = 30.0 0.2 0.4 0.6 0.8 1.00.0 0.2 0.4 0.6 0.8 1.0p ip ialpha = 4, beta = 1alpha = 0.9, beta = 0.8Density0 1 2 3 4Density1.0 1.2 1.4 1.6 1.80.0 0.2 0.4 0.6 0.8 1.00.0 0.2 0.4 0.6 0.8 1.0p ip iFIGURE 12.2. e beta distribution defines the probabilities of differentprobabilities, p i . e horizontal axis in each plot is the range of binomialprobabilities from which to draw samples. It is a parameter, but now witha different value for each case i. e vertical axis is the likelihood (density)of each probability, according to a beta distribution with shape defined bythe A and B values shown at the top of each plot.12.3.2. Beta distribution. You haven’t seen a beta distribution before, unless it was outsidethis book, so it’s worth saying a little about it. e BETA DISTRIBUTION is a probability densityfor probabilities themselves, continuous values between zero and one. It is a member ofthe exponential family, just like the Gaussian and Poisson and binomial. But I didn’t mentionit before, because it isn’t so oen used as a top-level distribution. 110 Instead, it becomesvaluable when we start stacking and mixing probability distributions.e beta distribution is very flexible, and its shape is defined by two parameters, A andB. Sadly, the conventional names for these parameters are α and β, and that’s how theyappear in most books. But since α and β are the same symbols oen used to build linearmodels, I’m going to use the “fancy” capital letter names for them. Hopefully this will reduceany confusion for now. As you’ll see a little later, we’re going to re-parameterize the betadistribution anyway.Let’s aim to understand how the distribution can take different shapes. You can see someof its characterist shapes, and how they correspond to the values of A and B, in FIGURE 12.2.