1 Answers
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.
The forward and backward algorithms should be placed within the context of probability as they appear to simply be names given to a set of standard mathematical procedures within a few fields. For example, neither "forward algorithm" nor "Viterbi" appear in the Cambridge encyclopedia of mathematics. The main observation to take away from these algorithms is how to organize Bayesian updates and inference to be efficient in the context of directed graphs of variables.
For an HMM such as this one:
this probability is written as p {\displaystyle p}. Here x {\displaystyle x} is the hidden state which is abbreviated as x t {\displaystyle x_{t}} and y 1 : t {\displaystyle y_{1:t}} are the observations 1 {\displaystyle 1} to t {\displaystyle t}.