statisticalrethinkin..

y ∼ Pois(λ)y ∼ Binom count eventslowratecountevents ∼ (µ, σ) 9.1. GENERALIZED LINEAR MODELS 269is(λ) count events ∼ (λ) count y ∼low Binom(n, probability p)low ratemany trials ∼ (λ) events ∼ large meanlow probability (λ, ) ∼ (µ, σ)many trials ∼ (, ) ∼ (λ)low probability ∼ (λ)many trials ∼ (λ, ) sum∼ (µ, σ) dgamma ∼ (, dnorm ) ∼ (λ) ∼ (µ, σ) ∼ (λ) ∼ (λ, ) ∼ (λ) ∼ (µ, σ) ∼ (, ) ∼ (λ) ∼ (λ, ) ∼ (λ) dexp ∼ (λ) ∼ (, ) count events count ∼ (λ, ) low rate events ∼ (, ) low probability dpois many trials dbinom FIGURE 9.1. Some of the exponential family distributions, their notation, and some of their relationships. Center: exponential distribution. Clockwise,from top-le: gamma, normal (Gaussian), binomial and Poisson dis- tributions. While there are other important distributions both in the family and outside it, you canusefully think of these distributions as the most common building blocks of probability modeling.Like the normal distribution (see Chapter 4), each of these exponential family distributionsarises (approximately or exactly) from natural processes. Not only do they arisefrom natural processes, but they are also related to one another through natural processesand the transformations that happen during measurement. ese are not ad hoc mathematicalconveniences. Instead they are natural distributions that arise through many processesin both nature and the laboratory. Just like Gaussian distributions populate the world, sotoo do exponential, gamma, Poisson, binomial, and others. Or the way I prefer to think of it,these distributions have strong informational “gravity” that attracts collections of measurestowards them. Just like a collection of sums from (almost) any distribution tends towards aGaussian distribution (Chapter 4), other ways of combining or transforming values lead toother distributions.Later, both in this chapter and the next, I’ll provide heuristic explanations of some processesthat give rise to these distributions, as well as how they relate to one another. For now,note the gray arrows in FIGURE 9.1. Each arrow is labeled with a process, whether it occursnaturally or through human measurement, that leads each distribution to generate another,either approximately or exactly. For example, as I’ll explain in more detail later in this chapter,summing values sampled from an exponential distribution (center) leads directly to the