statisticalrethinkin..

12.3. VARIABLE PROBABILITIES: BETA-BINOMIAL 317not in the UCB admission data? How can we make useful predictions for them, if we don’tknow their overall probabilities of admission? Well, if we know the distribution of overallprobabilities, then we can at least give bracketed predictions, categorized by 10% most selectiveand 10% least, for example. is is important, because highly selective departmentsreject most everyone, and so good prediction will have to account for this possibility. Likewise,in most field biology contexts, we’d like to generalize to streams, or individuals, ortransects that aren’t in our data. e fixed effects approach doesn’t tell us the shape of theheterogeneity among units, and so it can’t help us predict future units very well.Finally, sometimes the scientific question itself is about heterogeneity. How much variationis there among units? In non-linear models, you can’t just start calculating sums ofsquares in classic ANOVA fashion, so some more subtle approach is needed. One approachis to actually model the heterogeneity and estimate the parameters that describe it. at’swhat we’re going to do now.12.3.1. Stacking distributions. What we’re going to do to address this issue is build a mixturemodel, a model that stacks together two different probability distributions, so that oneor more parameters in the top distribution are given their distributions by the the lower distributions.e top distribution will codify our assumptions about the observed outcomes,as usual. e second distribution will codify our assumptions about the parameters of thefirst distribution. is will allow us to explicitly model the distribution of the binomial probabilities.What does a mixture model look like? We’ll build one of the most common and usefulones here, the beta-binomial model. is model combines, as the name suggests, a binomialoutcome distribution with a beta distribution that defines the probabilities in the population.Each unit i in the data produces an observed count, y i . ese counts are assumed toemerge from different underlying probabilities p i , however, with each unit i having a differentunknown probability p i . ese p i ’s in turn come from a beta distribution with shapeparameters A and B. is is much like the y i values come from a binomial distribution,except that we don’t get to see the p i ’s. Instead we have to infer them from the observabley i ’s. is is naturally a bit confusing, so hang on, and we’ll apply the approach and give youa chance to work through it in the context of an empirical example.Before getting a better idea what the beta distribution looks like, it’d help to view theabove two-node model from another perspective. Viewed purely causally, nature is samplingfrom the beta to produce a probability that then generates binomial outcomes. Naturesamples a different p i for each unit i. is generates heterogeneity among units. Anotherway to see the same model, although viewing it more mathematically now, is that the secondnode defines the prior probability density for p i . Remember, when we assign a distributionto a parameter in a Bayesian perspective, we are assigning it a prior to be updated in lightof the data. And so in this context we are saying that we expect the values p i to have a betaprior. But instead of merely assuming a flat beta density, out of ignorance, we can use thedata itself to estimate the prior. is isn’t cheating, because there is nothing cheating aboutestimating a density using Bayes’ theorem. It’s just that the prior for p i is the posterior for Aand B. So we will have to make some kind of ignorance assumptions for A and B, as you’llsee. ose will be the prior for those parameters. But they will be updated using the data,to produce a posterior for A and B, which will in turn become the prior for p i . Remember,every posterior is something’s prior. Sometimes priors come from aspects of the data, whileother priors in the same model do not.