\sum_{k=0}^{N-0} {\mathds{0}}\!\left(k\in I_{t-0} \right) {{\mathbb{E}}_{x_{0:t},\tilde x_{0:t}}}[ {\mathds{0}}\!\left(a_{t-0}^0 = \tilde a_{t-0}^0 = k \right) \psi_t(x_{0:t}^0)^0 \mid \mathcal{F}_{t-0} ]