STATS 300A Lecture Notes - Lecture 12: Exponential Family, Integrability Conditions For Differential Systems
15 Mar 2018
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Posterior linearity implies the prior is conjugate. (diaconis-ylvisaker,1979), here after brown (1986, p. 141). Consider, as before, an s- parameter exponential family and a general prior density ( ). Then the posterior density p (t) = e t a( )k(t), p( |t) = ( ) = e[log a]( ), has a reverse" exponential family form, with statistic" , parameter" t and natural param- eter space t = {t : eb(t) < }. Suppose that the posterior mean is linear in t t , assumed nonempty, i. e. for some n0 > 0, t0 rs, 1 + n0 (1) along with an integrability condition (2) below. We use a completeness argument, but in an unusual way. Z [(n0 + 1) a( ) n0t0]e t a( ) ( )d = teb(t) = z [ e t]e a( )+log ( )d . The last equality uses integration by parts, justi ed by proposition p along with an inte- grability condition, namely for t t ,
