Let
be a collection of data.
The Qfunction is defined as the expected ``complete" logPDF where
the expectation is carried out over the conditional distribution
of the ``missing data", given , using the current best estimate of the
PDF parameters , and the logPDF is written in terms of the
new values of the parameters to be estimated,
:
Expanding,
where
are
are the assignments not associated with sample .
The inner summation is a marginalization
Thus,
where the conditional model probabilities
are defined as
The maximization of
can be carried out on the quantity
where we have added data weights, ,
which define a probabilistic weights for each data sample.
This could be interpreted as adjusting the influence of a training sample
as though sample was replicated times, or can be thought
of as the probabilistic certainty that sample is indeed valid.
By collecting and
together
into a quantity , we have

(13.2) 
where
The algorithm in Table 13.1, maximizes (13.2) over
at each iteration.
While correct, it is representative only. Actual
computation requires careful attention to
numerical issues which are discussed below.
Table 13.1:
Update Equations for Gaussian Mixtures. This is
representative only. Actual implementation requires attention to
numerical issues discussed in the text.

Baggenstoss
20170519