Working in the log domain.

Since probabilities can become extremely small, it is necessary to remain in the log-domain. Staying in the log-domain is a problem when summations are required. Let $l_i=\log {\cal N}({\bf z}_k,$$\mu$$_{i},$$\Sigma$$_i)$. The summation

$$\log S=\log\left[{\displaystyle \sum_{i=1}^{L}} \;\alpha_{i} \exp(l_i) \right]$$

which appears in the first step of the E-M algorithm should be implemented as

$$\log S = M + \log\left\{\displaystyle \sum_{i=1}^{L} \;\alpha_{i} \exp(l_i-M) \right\},$$

where $M=\max_i l_i$.