Dual structure in the conjugate analysis of curved exponential families (Statistical Inference of Records and Related Statistics)
Toshio Ohnishi, Takemi Yanagimoto · Kyoto University Research Information Repository (Kyoto University) · 2005
Curved exponential families $\mathrm{a}\mathrm{d}$ mitting conjugate pliol densities are introduced $\mathrm{a}\alpha \mathrm{l}\mathrm{d}$ exploreaIntx 0- ducing extended versions of the mean and tlxe canonical I) arameters, we expand the conjugate analysis to these curved exponential families.Emphasis is put on dual structures-In fact, we derive the dual Pythagorean $1\mathrm{e}1\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{r}1\mathrm{b}\backslash \ddagger 1\mathrm{i}\mathrm{p}_{\mathrm{b}}$ with resPect to posterior risks, each of which makes it clear how the Bayes estimator do minates other estim $\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{b} eg$ .We also show that the conjugate prior density is the least infor mative.Key Words: closure under sampling, conjugacy, duality, least information, Legendre transfozntatiott, linearity, proper dispersion model.Pytllagolef.filrelationship, standgdized posterior mode Again, differentiating both sides of (A.5) with respect to $\eta_{k\backslash }$ we see thatThis implies the convexity of $\sqrt$ ) $(\eta)$ , which $\Gamma,0\mathrm{l}\mathrm{f}\mathrm{l}\mathrm{p}\mathrm{l}\mathrm{e}\mathrm{t}\mathrm{e}\mathrm{s}$ the proof of LPInlna 2.1.口