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In the first pass, the forward–backward algorithm computes a set of forward probabilities which provide, for all <math>k \in \{1, \dots, t\}</math>, the probability of ending up in any particular state given the first <math>k</math> observations in the sequence, i.e. <math>P(X_k\ |\ o_{1:k})</math>. In the second pass, the algorithm computes a set of backward probabilities which provide the probability of observing the remaining observations given any starting point <math>k</math>, i.e. <math>P(o_{k+1:t}\ |\ X_k)</math>. These two sets of probability distributions can then be combined to obtain the distribution over states at any specific point in time given the entire observation sequence:
:<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}
The last step follows from an application of the [[Bayes' rule]] and the [[conditional independence]] of <math>o_{k+1:t}</math> and <math>o_{1:k}</math> given <math>X_k</math>.
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