Algorithm 2 Sequential Bayesian Predictive Inference Approximation with Uncertain Localization

At time $t\in {\mathbb{Z}}_{>0}$, do: 1:obtain new observations $y_t$ along with the probabilities for locations $\pi \left(q_t\right)$ 2:for$q_t^{\left(h\right)}\in \mathsf{\Omega }(t-m+1:t)$do 3: predict ${\mu }_{x_t|{\mathcal{D}}_{t-m+1:t}^{\left(h\right)}}$, ${\mathsf{\Sigma }}_{x_t|{\mathcal{D}}_{t-m+1:t}^{\left(h\right)}}$ and $\pi (y_t|{\mathcal{D}}_{t-m+1:t-1}^{\left(j\right)},q_t^{\left(k\right)})$ using Algorithm 1 4: compute $\pi \left(q_{t-m+1:t}^{\left(h\right)}|{\mathcal{R}}_{t-m+1:t}\right)$ by Lemma 5. 5:end for  6:compute ${\mu }_{x_t|{\mathcal{R}}_{t-m+1:t}}$ and ${\mathsf{\Sigma }}_{x_t|{\mathcal{R}}_{t-m+1:t}}$, using Theorem 3.   7: use following approximation to update estimations $${\mu }_{x_t|{\mathcal{R}}_{1:t}}\approx {\mu }_{x_t|{\mathcal{R}}_{t-m+1:t}},{\mathsf{\Sigma }}_{x_t|{\mathcal{R}}_{1:t}}\approx {\mathsf{\Sigma }}_{x_t|{\mathcal{R}}_{t-m+1:t}},$$ $${\mathcal{P}}_t\approx \{{\mu }_{x_t|{\mathcal{R}}_{t-m+1:t}},{\mathsf{\Sigma }}_{x_t|{\mathcal{R}}_{t-m+1:t}},\pi \left(\mathsf{\Theta }\right|{\mathcal{R}}_{t-m+1:t})\}.$$