Forward–backward algorithm: Difference between revisions

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To understand this, we note that <math>\mathbf{f_{0:t}}(i) \cdot \mathbf{b_{t:T}}(i)</math> provides the probability for observing the given events in a way that passes through state <math>x_i</math> at time t. This probability includes the forward probabilities covering all events up to time t as well as the backward probabilities which include all future events. This is the numerator we are looking for in our equation, and we divide by the total probability of the observation sequence to normalize this value and extract only the probability that <math>X_t=x_i</math>. These values are sometimes called the "smoothed values" as they combine the forward and backward probabilities to compute a final probability.

The values <math>\mathbf{\gamma_t}(i)</math> thus provide the probability of being in each state at time t. As such, they are useful for determining the most probable state at any time. It should be noted, however, that theThe term "most probable state" is somewhat ambiguous. While the most probable state is the most likely to be correct at a given point, the sequence of individually probable states is not likely to be the most probable sequence. This is because the probabilities for each point are calculated independently of each other. They do not take into account the transition probabilities between states, and it is thus possible to get states at two moments (t and t+1) that are both most probable at those time points but which have very little probability of occurring together, i.e. <math> \mathbf{P}(X_t=x_i,X_{t+1}=x_j) \neq \mathbf{P}(X_t=x_i) \mathbf{P}(X_{t+1}=x_j) </math>. The most probable sequence of states that produced an observation sequence can be found using the [[Viterbi algorithm]].

==Example ==