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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2008/iii/paper-47/6/a/solution

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Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 47 / 6 / a / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-14

The expectation-maximization algorithm alternates conditional averaging over missing data with maximization over the parameter. Start at a parameter value

θ^{(0)}

for which the observed likelihood function is positive. At iteration

t

, its E-step forms

Q (θ ∣ θ^{(t)}) = E_{θ^{(t)}} [lo g f (x, Z; θ) ∣ x],

using the current conditional law of

Z

but treating

θ

as the candidate parameter. Its M-step takes

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

Repeat until the parameter or observed log-likelihood stabilizes. An increase of

Q

implies an increase of the observed log-likelihood: if

q_{t} (z) = f (z ∣ x; θ^{(t)})

, then

ℓ (θ) - ℓ (θ^{(t)}) = Q (θ ∣ θ^{(t)}) - Q (θ^{(t)} ∣ θ^{(t)}) + D_{KL} (q_{t} ∥ f (\cdot ∣ x; θ)),

where Kullback-Leibler divergence is nonnegative. In general this monotonicity does not guarantee a global maximum; distinct initializations may approach different stationary points. The particular observed log-likelihood below is concave, which makes a stronger conclusion possible there.

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