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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.

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.

Discard burn-in from the MCMC output and retain the sampled $\Delta t$ coordinate. A normalized histogram or kernel density estimation of these draws approximates $p(\Delta t\mid D)$. Autocorrelation changes the Monte Carlo uncertainty, so uncertainty bands should use the chain's effective sample size rather than its raw length.