C10. Gamma and inverse-Gamma kernels

Topic: Conjugate analysis for normal models · more questions on this topic
‹ Question 12

State the kernels and support of shape–rate \operatorname{Gamma}(a,b) and \operatorname{InvGamma}(a,b).

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Definitions/notation. A kernel omits only factors constant in the variable of interest; for t>0 and a,b>0, shape–rate \operatorname{Gamma}(a,b) has kernel t^{a-1}e^{-bt}, whereas \operatorname{InvGamma}(a,b) has kernel t^{-a-1}e^{-b/t}. The parameter b is the Gamma rate; under the stated inverse-Gamma convention, 1/T\sim\operatorname{Gamma}(a,b) when T\sim\operatorname{InvGamma}(a,b).