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128585
128585
05.03.2020 • 
Mathematics

Exponential family expectations: Let p(y|φ) = c(φ)h(y) exp{φt(y)} be an exponential family model. a) Take derivatives with respect to φ of both sides of the equation R p(y|φ) dy = 1 to show that E[t(Y )|φ] = −c 0 (φ)/c(φ). b) Let p(φ) ∝ c(φ) n0 e n0t0φ be the prior distribution for φ. Calculate dp(φ)/dφ and, using the fundamental theorem of calculus, discuss what must be true so that E[−c(φ)/c(φ)] = t0.

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