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The '''forward–backward algorithm''' is an [[Statistical_inference | inference]] [[algorithm]] for [[hidden Markov models]] which computes the [[posterior probability|posterior]] [[marginal probability|marginals]] of all hidden state variables given a sequence of observations/emissions <math>o_{1:t}:= o_1,\dots,o_t</math>, i.e. it computes, for all hidden state variables <math>X_k \in \{X_1, \dots, X_t\}</math>, the distribution <math>P(X_k\ |\ o_{1:t})</math>. This inference task is usually called ''smoothing''. The algorithm makes use of the principle of [[dynamic programming]] to compute efficiently the values that are required to obtain the posterior marginal distributions in two passes. The first pass goes forward in time while the second goes backward in time; hence the name ''forward–backward algorithm''.
The term ''forward–backward algorithm'' is also used to refer to any algorithm belonging to the general class of algorithms that operate on sequence models in a forward–backward manner. In this sense, the descriptions in the remainder of this article refer but to one specific instance of this class.
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