Algorithm 1.

Dynamic program for inference in the HNS model.

${\alpha }^{\left(1\right)}(j,k)=\{\begin{array}{cc}{\psi }_j^{\left(1\right)}\left(1\right)\hfill & \text{if}j=k\hfill \\ {\alpha }^{\left(1\right)}(j,k-1){\psi }_k^{\left(1\right)}\left(0\right)\hfill & k>j\hfill \end{array}\phantom{\}}$

${\alpha }^{\left(2\right)}(j,c)=\{\begin{array}{cc}1\hfill & \text{if}j=n+1\hfill \\ \underset{j\le k\le n}{\max }\underset{c\in \{1,\dots ,C\}}{\max }{\alpha }^{\left(1\right)}(j,k){\psi }_{jk}^{\left(2\right)}\left(0\right)e^{{\omega }_c}{\alpha }^{\left(2\right)}(k+1,c)\hfill & \text{if}c=0\hfill \\ \underset{j\le k\le n}{\max }{\alpha }^{\left(1\right)}(j,k){\psi }_{jk}^{\left(2\right)}\left(1\right){\alpha }^{\left(2\right)}(k+1,c-1)\hfill & \text{if}c>0\hfill \end{array}\phantom{\}}$