Algorithm A1 The Metropolis-Hastings Markov chain Monte Carlo (MH-MCMC) algorithm

1:Input: Prior $p(\mathbf{\theta })$, stepsize $\gamma$, maximum number of iterations $N_{\mathrm{iter}}$. 2:Initialize:${\mathbf{\theta }}^{(0)}\sim p(\mathbf{\theta })$. 3: for $i=1,2,\dots ,N_{\mathrm{iter}}$ do 4:    Propose: ${\mathbf{\theta }}^{\mathrm{cand}}={\mathbf{\theta }}^{(i-1)}+\gamma \mathcal{N}(0,I).$ 5:    Acceptance Probability: 6:    $\alpha ({\mathbf{\theta }}^{\mathrm{cand}}|{\mathbf{\theta }}^{(i-1)})=\min \left\{1,\frac{p({\mathbf{\theta }}^{\mathrm{cand}}|D)}{p({\mathbf{\theta }}^{(i-1)}|D)}\right\}.$ 7:    Draw $u\sim \mathcal{U}(0,1)$. 8:    if $u<\alpha$ then 9:        Accept the proposal: ${\mathbf{\theta }}^{(i)}\leftarrow {\mathbf{\theta }}^{\mathrm{cand}}$. 10:    else 11:        Reject the proposoal: ${\mathbf{\theta }}^{(i)}\leftarrow {\mathbf{\theta }}^{(i-1)}$. 12:    end if 13: end for