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Machine Learning: a Probabilistic Perspective

Exercise 21.8 - Derivation of the structured mean field updates for FHMM

Answers

According to the conclusion from mean-field varitional methods, we resort to the moment matching:

E ( 𝐱 m ) = 𝔼 q ∕ m [ E ( p ¯ ( 𝐱 m ) ) ] ,

where the l.r.s. equals:

− ∑ t = 1 T 𝐱 t , m T ξ ~ t , m ,

and the r.h.s. is:

1 2 𝔼 [ ∑ t = 1 T ( 𝐲 t − ∑ l ≠ m M 𝐖 l 𝐱 t , m ) T Σ − 1 ( 𝐲 t − ∑ l ≠ m M 𝐖 l 𝐱 t , m ) ] + const .

From which we have dropped the irrevelant terms. Calibrating the coefficients for 𝐱 𝑡𝑚 yields (21.62)-(21.64).

Comparing the coefficient of x t , m , k (i.e. setting x t , m , k to 1) ends in:

𝜖 ~ t , m , k = W m T Σ − 1 ( 𝐲 t − ∑ l ≠ m W l 𝔼 [ 𝐱 t , l ] ) − 1 2 ( W m T Σ − 1 W m ) k , k

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2021-03-24 13:42
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