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We have demonstrated a conditional maximum likelihood algorithm for a
conditioned mixture of Gaussians. Essentially, this optimized the
likelihood of the data given a model
.
If a prior on the model
is given, we can perform MAP
estimation simply by computing a bound on the prior as well. Recall
that we had bounded the conditional loglikelihood (as in
Equation 7.6) using a Q function. We depict the
inclusion of a prior in Equation 7.32. Note how this does not
change the derivation of the maximum conditional likelihood solution
and merely adds one more bound for the prior to the N bounds
computed for each data point for the ML^{c} solution. In fact, the
optimization of the prior is identical as it would be for a joint
density maximum a posteriori problem (MAP) and the fact that we are
optimizing MAP^{c} does not affect it. Thus, a prior on a conditioned
mixture model being update with CEM will can be bounded exactly as a
prior on a normal mixture model being updated with EM. In theory, it
is thus possible to add to the above CEM derivation a variety of
priors including noninformative priors, conjugate priors, entropic
priors and so on.



(7.32) 
Next: Implementation and Interpretation
Up: CEM for Gaussian Mixture
Previous: Bounding Matrices: The Gate
Tony Jebara
19990915