Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 219 / 4

One simple algorithm is importance sampling from the prior. Draw $({\theta }_j,{\mathbf{\alpha }}_j)$ independently from $\pi (\theta )\pi (\mathbf{\alpha })$ and assign weight $w_j=L_A({\theta }_j,{\mathbf{\alpha }}_j)$. Thenestimates the Bayesian model evidence, and the normalized weights $w_j/{\sum }_k{w}_k$ represent the posterior. A weighted histogram or kernel density estimation of the ${\theta }_j$ estimates the marginal ${\mathcal{P}}_A(\theta )$; discarding ${\mathbf{\alpha }}_j$ performs the marginalization.

Independence of the experiments giveswhereThe marginal posterior is obtained by integrating the displayed joint posterior over both nuisance-parameter vectors.

Define the prior-predictive nuisance integralsThenFor each individual analysis, Bayes theorem also gives

Estimate each one-dimensional marginal posterior from its individual experiment's weighted samples, for example by weighted kernel density estimation. Combining that density estimate with the individual evidence estimate givesTheir product with $\pi (\theta )$ can be normalized on the one-dimensional $\theta$ space, avoiding all joint nuisance-parameter sampling.

Compute the joint evidence by one-dimensional numerical integration,Equivalently, the individual outputs giveQuadrature or one-dimensional importance sampling evaluates this integral without entering the $(\theta ,\mathbf{\alpha },\mathbf{\beta })$ space.