MIT15_097S12_lec15

Second what if we assign the wrong prior we can

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Unformatted text preview: us, according to (17), the conjugate prior is � � p(θ) ∝ g (θ)η exp φ(θ)T ν θ = (1 − θ)η exp ν log 1−θ ν θ = (1 − θ)η 1−θ = θν (1 − θ)η−ν . 14 Reparameterizing by defining ν = α − 1 and η − ν = β − 1 gives us the Beta distribution that we expect. Example 2. (Normal Distribution is an Exponential Family) Con­ sider a normal distribution with known variance but unknown mean: p(yi |θ) = √ 1 exp − 1 (y − θ)2 . 2i 2σ 2πσ 2 �2 � Let f (yi ) = √21 2 exp −yi /2σ 2 , u(yi ) = yi /σ , φ(θ) = θ/σ , and g (θ) = � � πσ exp −θ2 /2σ 2 . Then, 12 1 1 yi exp − 2 θ2 exp 2θyi 2σ 2 2σ 2σ 2 2πσ 2 1 1 =√ exp − 2 (yi − θ)2 2σ 2πσ 2 = p(yi |θ). f (yi )g (θ) exp (φ(θ)u(yi )) = √ 1 exp − Thus the normal distribution with known variance is an exponential family. Example 3. (Exponential Distribution is an Exponential Family) Consider an exponential distribution: p(yi |θ) = θe−θyi . Let f (yi ) = 1...
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This note was uploaded on 03/24/2014 for the course MIT 15.097 taught by Professor Cynthiarudin during the Spring '12 term at MIT.

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