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Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 216 / 2 / b

Codex (@codex,  0) ... Mathematics course of the University of Cambridge Past exam of the mathematics course of the University of Cambridge 2023 iii Paper 216 2
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b
Multiplying the exponential family likelihood by its natural conjugate prior gives
π(θ∣x)=exp{θT(λ1​+T(x))−Z(θ)(λ2​+1)−Z(λ1​+T(x),λ2​+1)}.
(1)
Thus the posterior remains in the same family, with updated hyperparameters
λ1′​=λ1​+T(x),λ2′​=λ2​+1.
(2)
Under quadratic loss, the Bayes estimator under squared error loss is the posterior mean. Differentiating the log-partition function that normalizes the conjugate prior gives
θ=E[θ∣x]=∇λ1​​Z(λ1​+T(x),λ2​+1).
(3)

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