Exercise 11.8 - Moments of a mixture of Gaussians

Answers

For question (a), the expectation of a mixture Gaussian distribution is:

𝔼 ( 𝐱 ) = ∫ 𝐱 ∑ k π k 𝒩 ( 𝐱 | μ k , Σ k ) d 𝐱 = ∑ k π k ( ∫ 𝐱 𝒩 ( 𝐱 | μ k , Σ k ) d 𝐱 ) = ∑ k π k μ k .

For question (b), recall that cov ( 𝐱 ) = 𝔼 ( 𝐱 𝐱 T ) − 𝔼 ( 𝐱 ) 𝔼 ( 𝐱 ) T . We have:

𝔼 ( 𝐱 𝐱 T ) = ∫ 𝐱 𝐱 T ∑ k π k 𝒩 ( 𝐱 | μ k , Σ k ) d 𝐱 = ∑ k π k ∫ 𝐱 𝐱 T 𝒩 ( 𝐱 | μ k , Σ k ) d 𝐱 ,

in which:

∫ 𝐱 𝐱 T 𝒩 ( 𝐱 | μ k , Σ k ) d 𝐱 = 𝔼 ( 𝐱 𝐱 T ) = cov ( 𝐱 ) + 𝔼 ( 𝐱 ) 𝔼 ( 𝐱 ) T = Σ k + μ k μ k T .

Therefore:

𝑐𝑜𝑣 ( 𝐱 ) = ∑ k π k ( Σ k + μ k μ k T ) − 𝔼 ( 𝐱 ) 𝔼 ( 𝐱 ) T .

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2021-03-24 13:42
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