What is maximum likelihood PCA? There are many papers on this topic, such as this one (pdf). However, I could not find out what exactly maximum likelihood PCA is, how it is applied and for which purpose.
Can anyone explain it?
 A: Possibly this article (which is online readable) is also helpful. In the following the abstract:

Abstract
The maximum likelihood PCA (MLPCA) method has been devised in
chemometrics as a generalization of the well-known PCA method in order
to derive consistent estimators in the presence of errors with known
error distribution. For similar reasons, the total least squares (TLS)
method has been generalized in the ﬁeld of computational mathematics
and engineering to maintain consistency of the parameter estimates in
linear models with measurement errors of known distribution. The basic
motivation for TLS is the following. Let a set of multi-dimensional
data points (vectors) be given. How can one obtain a linear model that
explains these data? The idea is to modify all data points in such a
way that some norm of the modiﬁcation is minimized subject to the
constraint that the modiﬁed vectors satisfy a linear relation.
Although the name “total least squares” appeared in the literature
only 25 years ago, this method of ﬁtting is certainly not new and has
a long history in the statistical literature, where the method is
known as “orthogonal regression”, “errors-in-variables regression” or
“measurement error modeling”.
The purpose of this paper is to explore
the tight equivalences between MLPCA and element-wise weighted TLS
(EW-TLS). Despite their seemingly different problem formulation, it is
shown that both methods can be reduced to the same mathematical kernel
problem, i.e. ﬁnding the closest (in a certain sense) weighted low
rank matrix approximation where the weight is derived from the
distribution of the errors in the data. Different solution approaches,
as used in MLPCA and EW-TLS, are discussed. In particular, we will
discuss the weighted low rank approximation (WLRA), the MLPCA, the
EW-TLS and the generalized TLS (GTLS) problems. These four approaches
tackle an equivalent weighted low rank approximation problem, but
different algorithms are used to come up with the best approximation
matrix. We will compare their computation times on chemical data and
discuss their convergence behavior.
© 2004 Elsevier B.V. All rights reserved.
Keywords: TLS; MLPCA; Rank reduction; Measurement errors

