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:<math>P(X_k\ |\ o_{1:t}) = P(X_k\ |\ o_{1:k}, o_{k+1:t}) \propto P(o_{k+1:t}\ |\ X_k) P(X_k\ ,\ o_{1:k})</math>
The last step follows from an application of the [[Chain_rule_(probability)|chain rule]] and the [[conditional independence]] of <math>o_{k+1:t}</math> and <math>o_{1:k}</math> given <math>X_k</math>.
As outlined above, the algorithm involves three steps:
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