Transcription of Chapter 8 The exponential family: Basics
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Chapter 8 The exponential family: Basics
Chapter 8 The exponential family: BasicsIn this Chapter we extend the scope of our modeling toolbox to accommodate a variety ofadditional data types, including counts, time intervals and rates. We introduce the expo-nential family of distributions, a family that includes the Gaussian, binomial, multinomial,Poisson, gamma, von Mises and beta distributions, as well as many others. In this chapterwe focus on unconditional models and in the following Chapter we show how these ideas canbe carried over to the setting of conditional first blush this Chapter may appear to involve a large dose of mathematical detail, butappearances shouldn t deceive most of the detail involves working out examples that showhow the exponential family formalism relates to more familiar material.
2 CHAPTER 8. THE EXPONENTIAL FAMILY: BASICS where we see that the cumulant function can be viewed as the logarithm of a normalization factor.1 This shows that A(η) is not a degree of freedom in the specification of an exponential family density; it is determined once ν, T(x) and h(x) are determined.2 The set of parameters ηfor which the integral in Eq.
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