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Line 8: :<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 [[~~Bayes' rule~~Chain_rule_(probability)]] 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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