= Forward algorithm
{title2=$\alpha_t(i)$}
The <forward algorithm> computes sequence likelihoods by summing over all hidden paths through a <Hidden Markov model>. For emitting states, $\alpha_t(i)=P(o_1,\ldots,o_t,Z_t=i)$ obeys $\alpha_{t+1}(j)=b_j(o_{t+1})\sum_i\alpha_t(i)a_{ij}$. It reuses partial sums rather than enumerating exponentially many state sequences. Silent states require additional transitions without consuming an observation. Scaling or <log-sum-exp> prevents <floating-point underflow>.
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