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. The real messageof this Chapter is the simplicity and elegance of exponentialfamily. Once the new ideas aremastered, it is often easier to work within the general exponential family framework thanwith specific The exponential familyGiven a measure , we define anexponential familyof probability distributions as thosedistributions whose density (relative to ) have the following general form:p(x| ) =h(x) exp{ TT(x) A( )}( )for a parameter vector , often referred to as thecanonical parameter, and for given func

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 chapter we focus on unconditional models and …

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  Opex, Exponential, Expo nential, Nential

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