current summand p u03bb x 1 x r s 1 s r is the joint probability of all r training utterances and the corresponding set of word sequences for the current summand and where u03b4 s r s r is the kronechker delta function that equals one if s r s r and zero otherwise under growth transformation optimization instead of optimizing the objective function of eq 5 directly an auxiliary function is optimized where the auxiliary function is defined as f u03bb u03bb u2032 g u03bb u2212 o u03bb u2032 h u03bb d eq 7 where f u03bb u03bb u2032 is the auxiliary function g u03bb is the numerator of the objective function i e o u03bb u2032 is the objective function using the previous parameter set u03bb u2032 and d is a quantity that is independent of the parameter set the auxiliary function of eq 7 can be used to form a further auxiliary function that guarantees an increase in the auxiliary function when the further auxiliary function increases this further auxiliary function is then maximized by taking the derivative of the further auxiliary function with respect to the mean and variance of each state of the hidden markov model resulting in update equations of where u03bc i is the mean for state i t r is the number of time points or frames of utterance r x t is the feature vector for utterance r at time point t u03bc i u2032 is the mean for state i for the previous version of the acoustic model u03c3 i is the variance for state i u03c3 i u2032 is the variance for the previous model for state i and where where p u03bb u2032 s r x r is the probability of word sequence candidate s r.