Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 326 / 2 / e

Table of contents
- Solution e

Solution

For observed

v \in R^{N}

, the Gaussian likelihood is

π (v ∣ u) = (2 π σ^{2})^{- N /2} exp [- \frac{1}{2 σ ^{2}} ∥ v - G (u) ∥_{R^{N}}^{2}] .

Thus the Bayesian inverse problem is to determine the posterior distribution of

U

given

V = v

. With

Φ (u; v) = \frac{1}{2 σ ^{2}} ∥ v - G (u) ∥^{2},

Bayes' formula gives

\frac{d μ ^{v}}{d μ _{0}} (u) = \frac{1}{Z ( v )} e^{- Φ (u; v)}, Z (v) = \int_{X} e^{- Φ (u; v)} d μ_{0} (u) .

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