= Belief state
{title2=$B\subseteq S$}
Without a prior <probability distribution>, a belief state is the <set> of physical states still compatible with an observed history. If the physical <search transition model> is deterministic, $T(s,a)$, and the sensor map is $O$, action $a$ followed by observation $o$ gives
$$
B'_{a,o}=\{T(s,a):s\in B,\ O(T(s,a))=o\}.
$$
For a <set>-valued outcome relation, replace the image by the <set union> of all possible successors before filtering. A guaranteed-success <goal test> requires every member of the surviving <belief state> to satisfy the physical goal. A probabilistic belief is instead a posterior <probability distribution>, requiring a specified prior and observation law.
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