Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 207 / 3 / c / iv

Table of contents
- Solution iv

Solution

A Dirichlet process mixture model avoids fixing the number of occupied functions. Let

G_{0} = N_{n} (0, K)

be the finite-dimensional Gaussian process law on the common input grid and specify

G \sim DP (α, G_{0}), f_{i} ∣ G \sim iid G, y_{i} ∣ f_{i} \sim N_{n} (f_{i}, σ^{2} I_{n}) .

A draw from a Dirichlet process is almost surely discrete, so several

f_{i}

coincide and thereby form clusters. The number of occupied clusters is random and can grow with the data.

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