**Appendix A**

According to Equations (14), (15) and (42), we can obtain

$$\begin{split} r\_{p, \mathbb{K}k-1}^{(i)} &= r\_{k-1}^{(i)} \langle p\_{k-1}^{(i)}(\mathbf{x}\_{k-1}), p\_{S,k} \rangle \\ &= r\_{k-1}^{(i)} \left\langle \sum\_{m=1}^{\mathsf{u}\_{\mathbb{C}}} p\_{k-1}^{(i)}(\mathbf{x}\_{k-1}|\mathbf{c}^{m}, \mathsf{Z}^{k-1}) p\_{k-1}^{(i)}(\mathbf{c}^{m}|\mathsf{Z}^{k-1}), p\_{S,k} \right\rangle \\ &= r\_{k-1}^{(i)} \sum\_{m=1}^{\mathsf{u}\_{\mathbb{C}}} p\_{k-1}^{(i)}(\mathbf{c}^{m}|\mathsf{Z}^{k-1}) \Big\langle p\_{k-1}^{(i)}(\mathbf{x}\_{k-1}|\mathsf{c}^{m}, \mathsf{Z}^{k-1}), p\_{S,k} \rangle \end{split} \tag{A1}$$

*p* (*i*) *<sup>P</sup>*,*k*|*k*−1(*xk*|Z*k*−1) = ! *fk*|*k*−1(*xk*|*xk*−1),*<sup>p</sup>* (*i*) *<sup>k</sup>*−1(*xk*−1)*pS*,*<sup>k</sup>* " ! *p* (*i*) *<sup>k</sup>*−1(*xk*−1),*pS*,*<sup>k</sup>* " = . *f* k *<sup>k</sup>*|*k*−1(*xk*|*xk*−1), *n* \$c *m*=1 *p* (*i*) *<sup>k</sup>*−1(*xk*−1|*cm*,Z*k*−1)*<sup>p</sup>* (*i*) *<sup>k</sup>*−1(*cm*|Z*k*−1)*pS*,*<sup>k</sup>* / . *n* \$c *m*=1 *p* (*i*) *<sup>k</sup>*−1(*xk*−1|*cm*,Z*k*−1)*<sup>p</sup>* (*i*) *<sup>k</sup>*−1(*cm*|Z*k*−1),*pS*,*<sup>k</sup>* / = *n* \$c *m*=1 *p* (*i*) *<sup>k</sup>*−1(*cm*|Z*k*−1) ! *f* k *<sup>k</sup>*|*k*−1(*xk*|*xk*−1),*<sup>p</sup>* (*i*) *<sup>k</sup>*−1(*xk*−1|*cm*,Z*k*−1)*pS*,*<sup>k</sup>* " *n* \$c *m*=1 *p* (*i*) *<sup>k</sup>*−1(*cm*|Z*k*−1) ! *p* (*i*) *<sup>k</sup>*−1(*xk*−1|*cm*,Z*k*−1),*pS*,*<sup>k</sup>* " (A2)
