Exercise 9.2 - The MVN is in the exponential family

Answers

The sufficient statistics for 𝐱 is:

ϕ ( 𝐱 ) = ( x 1 , x 2 , ⋯ , x M , x 1 x 1 , ⋯ , x M x M ) T ,

including altogether M + M 2 components.

Expand the information form for a Gaussian, the linear terms within the exponential are:

− 1 2 ∑ m , m ′ Λ m , m ′ x m x m ′ + ∑ m x m ∑ m ′ Λ m , m ′ μ m ′ ,

hence the parameters can be collected by:

𝜃 = ( ∑ m ′ Λ 1 , m ′ μ m ′ , ⋯ , ∑ m ′ Λ M , m ′ μ m ′ , Λ 1 , 1 , ⋯ , Λ M , M ) .

𝜃 contains exactly the same information as μ , Σ . Finally, we have:

A ( 𝜃 ) = 1 2 μ T Λ μ + M 2 ln ⁡ ( 2 π ) + 1 2 ln ⁡ | Σ | .

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2021-03-24 13:42
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