Algorithm 1 Sequential Bayesian Predictive Inference

Initialization: 1:initialize $F_s$   2:for $\theta \in \mathsf{\Theta }$, initialize $Q_{\theta }$, and compute $Q_{\theta }^{-1}$ At time $t\in {\mathbb{Z}}_{>0}$, do: 1:obtain new observations $y_t$ at current locations $q_t$   2:find the map ${\mathsf{\Gamma }}_{q_t}$ from $q_t$ to spacial sites $\mathcal{S}$, and compute radial basis values $F_{q_t}$ in $q_t$. 3:for$\theta \in \mathsf{\Theta }$do 4: predict ${\mu }_{x_t|\theta ,{\mathcal{D}}_{1:t-1}}$ and $Q_{x_t|\theta ,{\mathcal{D}}_{1:t-1}}$ using measurements up to time $t-1$, given by Equation (21). 5: compute ${\mu }_{x_t|\theta ,{\mathcal{D}}_{1:t}}$ and $Q_{x_t|\theta ,{\mathcal{D}}_{1:t}}$ given by Equation (24). 6: compute ${\mu }_{y_t|\theta ,{\mathcal{D}}_{1:t-1},q_t}$ and ${\mathsf{\Sigma }}_{y_t|\theta ,{\mathcal{D}}_{1:t-1},q_t}$ by Equation (23). 7:calculate $\pi \left(\theta \right|{\mathcal{D}}_{1:t})$ given by Equation (22). 8:end for 9:compute the predictive mean and variance using Equation (27).