Processing field-of-view events and observations

Shifting to field-of-view events, we observe that an enter event only affects the shadow being entered by increasing the expected number of targets in the shadow. If there is a single shadow $s$ and an enter event happens, we merely update $P(s=a_i) = p_i$ to $P(s=a_i + 1) = p_i$ . On the other hand, an exit event does the opposite and we change $P(s=a_i) = p_i$ to $P(s = a_i - 1) = p_i$ . A complication arises here: If shadow $s_i$ splits into shadows $s_j, s_k$ and an $e_x$ event happens to shadow $s_j$ , it suggests that it is impossible for $s_j$ to have 0 target before the $e_x$ event. The affected probability mass needs to be removed and the remaining values renormalized. The null event does not change the target distribution.

Now, to propagate a probability mass through a field-of-view observation, $y$ , we essentially break the entry into three pieces according to above rules, multiplying each resulting entries with the probability $P(\mathbf{e} = e_e \mid \mathbf{y} = y), P(\mathbf{e} = e_x \mid \mathbf{y} = y)$ , and $P(\mathbf{e} = e_n \mid \mathbf{y} = y)$ , respectively. If an enter event is not possible for the observation, the two remaining entries are renormalized.