## 3 Model and Methods
### Posterior Inference  
Bayesian inference is performed by summarizing the posterior distribution in (12), which is analytically intractable. We take a sampling-based MCMC approach for posterior inference using a block-wise random-walk Metropolis-Hastings (RW-MH) algorithm; see [61; 62] for details. The basic idea of MH algorithms is to build a Markov chain with the target posterior distribution as its stationary distribution. The implementation of a MH sampler is based on the choice of suitable proposal distributions for the parameters, from which candidates for the Markov chain are sampled.  
We use block-wise sampling by grouping the parameters of interest, i.e., \(R_{\text{th},i}\), \(\mathbf{\theta}_{R_{\text{th}}}\) and \(I_{f,i}\), into blocks, and we use Gaussian proposal distributions in each block, centred around the previous value of the chain. The variance parameter of the proposal distributions is tuned to achieve block-wise acceptance rates (i.e., proportion of candidate values that is accepted during MCMC sampling) between 20% and 40%, which proves well for random-walk proposals; cf. [24; 63; 64]. The MCMC sampling scheme is implemented in R [65]. Algorithm 1 outlines the sampling steps to generate samples from the hierarchical posterior distribution.  
Given the MCMC samples from the posterior distribution, i.e., \(\mathbf{\[PERSON]}_{i}^{(m)}\sim p(\mathbf{\[PERSON]}_{i}|y_{i})\), for \(m=1,\ldots,M\), samples from the posterior predictive distribution can be obtained as  
\[y_{i}^{*(m)}=\Delta T_{0_{i}}^{*(m)}\sim p(y_{i}|\mathbf{\[PERSON]}_{i}^{(m)},\mathbf{z} _{i}) \tag{14}\]  
based on draws from the sampling distribution specified in (6).  
**1**: Given the current state \(m=1,\ldots,M\) of the Markov chain for the parameters \(\mathbf{\[PERSON]}_{k}^{(m)}\) of the \(k\)th block, \(k=1,\ldots,3\), a candidate is sampled from the proposal distribution  
\[\mathbf{\[PERSON]}_{k}^{(m+1)}\sim q\Big{(}\mathbf{\[PERSON]}_{k}^{(m+1)}|\mathbf{\[PERSON]}_{k}^{(m )},\mathbf{\[PERSON]}_{\smallsetminus k}^{(m)}\Big{)}\]  
given by a Gaussian distribution centred around the current value \(\mathbf{\[PERSON]}_{k}^{(m)}\) and the variance \(\epsilon^{2}\mathbf{I}_{d}\), where \(\epsilon\) is a tuning parameter, and \(\mathbf{I}_{d}\) is the \(d\)-dimensional identity matrix of dimension \(d\) of the parameter vector of the respective block. The parameter vector \(\mathbf{\[PERSON]}_{\smallsetminus k}^{(m)}\) denotes all parameters except for those summarized in block \(\mathbf{\[PERSON]}_{k}^{(m)}\).  
**2**: The proposed state is accepted if  
\[U_{j}\leq\alpha\Big{(}\mathbf{\[PERSON]}_{k}^{(m+1)}|\mathbf{\[PERSON]}_{k}^{(m)}\Big{)}\]  
with acceptance probability \(\alpha(\cdot)\), which simplifies to  
\[\alpha\Big{(}\mathbf{\[PERSON]}_{k}^{(m+1)}|\mathbf{\[PERSON]}_{k}^{(m)}\Big{)}=\min\left( 1,\frac{p(\mathbf{\[PERSON]}_{k}^{(m+1)},\mathbf{\[PERSON]}_{\smallsetminus k}^{(m)}|\mathbf{y} )}{p(\mathbf{\[PERSON]}_{k}^{(m)},\mathbf{\[PERSON]}_{\smallsetminus k}^{(m)}|\mathbf{y})}\right)\]  
and \(U_{j}\) is a random draw from \(\mathcal{U}_{[0,1]}\).  
Otherwise, the proposed value is rejected, i.e., \(\mathbf{\[PERSON]}_{k}^{(m+1)}=\mathbf{\[PERSON]}_{k}^{(m)}\).  
**Algorithm 1** RW-MH algorithm for posterior sampling.