MIT15_097S12_lec15

# Second what if we assign the wrong prior we can

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

## 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.

Ask a homework question - tutors are online