Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 219 / 4 / a

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- Solution a

Solution

For one object, the probabilistic graphical model factorization is

p (x_{i}, y_{i}, ξ_{i}, η_{i} ∣ α, β, σ^{2}, μ, τ^{2}) = p (x_{i} ∣ ξ_{i}) p (y_{i} ∣ η_{i}) p (η_{i} ∣ ξ_{i}, α, β, σ^{2}) p (ξ_{i} ∣ μ, τ^{2}) .

Each factor is the normal distribution density specified by the model. This factorization displays the conditional independences of the latent variables

ξ_{i}, η_{i}

and the noisy observations

x_{i}, y_{i}

.

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