Editing Baum–Welch algorithm (section)

====Forward procedure====
Let <math>\alpha_i(t)=P(Y_1=y_1,\ldots,Y_t=y_t,X_t=i\mid\theta)</math>, the probability of seeing the observations <math>y_1,y_2,\ldots,y_t</math> and being in state <math>i</math> at time <math>t</math>. This is found recursively:
#<math>\alpha_i(1)=\pi_i b_i(y_1),</math>
#<math>\alpha_i(t+1)=b_i(y_{t+1}) \sum_{j=1}^N \alpha_j(t) a_{ji}.</math>

Since this series converges exponentially to zero, the algorithm will numerically underflow for longer sequences.<ref>{{cite web|url=https://www.ece.ucsb.edu/Faculty/Rabiner/ece259/Reprints/tutorial%20on%20hmm%20and%20applications.pdf|title=A Tutorial on Hidden Markov Models and Selected Applications in Speech recognition|last=Rabiner|first=Lawrence|date=February 1989|publisher=Proceedings of the IEEE|access-date=29 November 2019}}</ref> However, this can be avoided in a slightly modified algorithm by scaling <math>\alpha</math> in the forward and <math>\beta</math> in the backward procedure below.