Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 216 / 4 / a

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- Solution a

Solution

Let

θ = (β, π)

and regard

X_{U}

as the latent variable. At iteration

t

, the E-step forms

Q (θ ∣ θ^{(t)}) = E_{θ^{(t)}} [lo g p (Y, X_{U}, X_{O}, θ) ∣ Y, X_{O}] .

The M-step updates

θ^{(t + 1)} \in ar g max_{θ} Q (θ ∣ θ^{(t)}) .

This is the expectation-maximization algorithm for the posterior objective: including

lo g p (θ)

in the complete-data log density makes the maximizer a maximum a posteriori estimate rather than a maximum-likelihood estimate.

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