Forward algorithm

The forward algorithm, in the context of a hidden Markov model, is used to calculate a 'belief state': the probability of a state at a certain time, given the history of evidence. The process is also known as filtering. The forward algorithm is closely related to, but distinct from, the Viterbi algorithm.

For an HMM such as this one:

this probability is written as ${\displaystyle P(x_{t}|y_{1:t})}$. Here ${\displaystyle x(t)}$ is the hidden state which is abbreviated as ${\displaystyle x_{t}}$ and ${\displaystyle y_{1:t}}$ are the observations ${\displaystyle 1}$ to ${\displaystyle t}$. A belief state can be calculated at each time step, but doing this does not, in a strict sense, produce the most likely state sequence, but rather the most likely state at each time step, given the previous history.

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