Since the observed spectral density function , where denotes
log-frequency, is considered to be
generated from the model of multiple harmonic structures, the log-likelihood difference
in accordance with an update of the model
Although Dempster formulated EM algorithm  in order to maximize the mean
log-likelihood considering as a probabilistic density function,
it can also be formulated in a same way even if is replaced with
spectral density function.
By taking expectation of both sides with respect to
which represents the probability of the -labeled Gaussian
distribution from which is generated, -function will be derived
in the right-hand side.
Given -function as
thus it yields
which maximizes the
function, the log-likelihood of the model of multiple harmonic structures with
respect to every will be monotonously increased. A posteriori
in equation (3) is given as
is a Gaussian distribution.
Log-likelihood of the
, is given as
By the iterative procedure of the two steps as follows,
the model parameter
locally converges to ML estimates.
$B!Z(BIterative procedure of EM algorithm$B![(B
Initialize the model parameter
with equation (3).
to obtain the next estimate
and repeat from the Expectation-step.