 Solution

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.