Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 219 / 3 / b

Table of contents
- Solution b

Solution

Let

c_{s} = c_{s} - E_{s}

and

M_{s} = M_{s} - R E_{s}

. With the stated improper hyperpriors, the unnormalized posterior density is

p ({E_{s}}, M_{0}, c_{0}, β, σ_{M}^{2}, σ_{c}^{2}, R, τ ∣ {d_{s}}) \propto 1_{{σ_{M}^{2}, σ_{c}^{2}, τ > 0}} s = 1 \prod N [\frac{1}{σ _{c}} exp {- \frac{( c _{s} - E _{s} - c _{0} ) ^{2}}{2 σ _{c}^{2}}} \frac{1}{σ _{M}} exp {- \frac{[ M _{s} - R E _{s} - M _{0} - β ( c _{s} - E _{s} ) ] ^{2}}{2 σ _{M}^{2}}} \frac{1}{τ} e^{- E_{s} / τ} 1_{{E_{s} \geq 0}}] .

Its Directed acyclic graph has, for each star, the arrows

(c_{0}, σ_{c}^{2}) \to c_{s}, (M_{0}, β, σ_{M}^{2}, c_{s}) \to M_{s}, τ \to E_{s},

followed by the deterministic observation arrows

(c_{s}, E_{s}) \to c_{s}, (M_{s}, E_{s}, R) \to M_{s} .

The latent variables

(c_{s}, M_{s}, E_{s})

are repeated inside a plate indexed by

s = 1, \dots, N

; all hyperparameters lie outside that plate.

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