When the HMM algorithms are run on long sequences, underflow errors often occur because we need to multiply many small positive numbers. To address this issue, we often represent all forward and backward probabilities in the logarithm domain. In this case, we can do arithmetic operations as follows:

Use these routines to rewrite the forward–backward Algorithm 12.15 in the logarithm domain. In other words, use ˜t ¹iº = log ????
t ¹iº
and ˜t ¹iº = log ????
t ¹iº
in place of t ¹iº and t ¹iº.

log(ab) = log(a)+ log(b) log(a/b) = log(a) - log(b) log(a + b) = log(a) +log [1 exp (log(b) - log(a))].