Algorithm 3. KLD sampling algorithm

Input: ${\mathit{s}}_{\mathit{t}-\mathbf{1}}=\{\langle {\mathit{x}}_{\mathit{t}-\mathbf{1}}^{\left(\mathit{i}\right)},{\mathit{w}}_{\mathit{t}-\mathbf{1}}^{\left(\mathit{i}\right)}\rangle |\mathit{i}=\mathbf{1},\dots ,\mathit{n}\}$ , observations${\mathit{z}}_{\mathit{t}}$ , limits$\mathit{\epsilon }$ and$\mathit{\delta }$;

Output: ${\mathit{S}}_{\mathit{t}}$

1: ${\mathit{S}}_{\mathit{t}}≔\mathbf{\varnothing },\mathit{n}=\mathbf{0},\mathit{k}=\mathbf{0},\mathit{\alpha }=\mathbf{0}$    % initializing

2: do                % generating samples

3:  sampling from discrete distributions under the weight of known ${\mathit{s}}_{\mathit{t}-\mathbf{1}}$ , the sequence is$\mathit{j}\left(\mathit{n}\right)$

4:  sampling ${\mathit{x}}_{\mathit{t}}^{\left(\mathit{n}\right)}$ from$\mathbf{p}({\mathit{x}}_{\mathit{t}}|{\mathit{x}}_{\mathit{t}-\mathbf{1}})$ using ${\mathit{x}}_{\mathit{t}-\mathbf{1}}^{\left(\mathit{j}\left(\mathit{n}\right)\right)}$

5:  ${\mathit{w}}_{\mathit{t}}^{\left(\mathit{n}\right)}≔\mathit{p}({\mathit{z}}_{\mathit{t}}|{\mathit{x}}_{\mathit{t}}^{\left(\mathit{n}\right)})$       % calculate the importance weights

6:  $\mathit{\alpha }≔\mathit{\alpha }+{\mathit{w}}_{\mathit{t}}^{\left(\mathit{n}\right)}$          % update the normalization factor

7:  ${\mathit{S}}_{\mathit{t}}≔{\mathit{S}}_{\mathit{t}}{{\displaystyle \cup }}^{\text{}}\{\langle {\mathit{x}}_{\mathit{t}}^{\left(\mathit{n}\right)},{\mathit{w}}_{\mathit{t}}^{\left(\mathit{n}\right)}\rangle \}$    % insert the sample into the sample set

8:  if (${\mathit{x}}_{\mathit{t}}^{\left(\mathit{n}\right)}$ fall in $\mathit{b}\mathit{i}\mathit{n}\mathit{b}$ ) then    % update the number of $\mathit{b}\mathit{i}\mathit{n}\mathit{s}$

9:   $\mathit{k}≔\mathit{k}+\mathbf{1}$

10:   $\mathit{b}≔\mathit{n}\mathit{o}\mathit{n}-\mathit{e}\mathit{m}\mathit{p}\mathit{t}\mathit{y}$

11:  $\mathit{n}≔\mathit{n}+\mathbf{1}$          % update the number of generated samples

12: while ($\mathit{n}<\frac{\mathbf{1}}{\mathbf{2}\mathit{ϵ}}{\mathcal{X}}_{\mathit{k}-\mathbf{1},\mathbf{1}-\mathit{\delta }}^{\mathbf{2}}$)     % stop when come to the limits K-L with Equation (35)

13: for i: = 1, …, n do         % normalize importance weights

14:  ${\mathit{w}}_{\mathit{t}}^{\left(\mathit{i}\right)}={\mathit{w}}_{\mathit{t}}^{\left(\mathit{i}\right)}/\mathit{\alpha }$