  Recently foundational issues of applicability of the formalism of quantum
mechanics (QM) to cognitive psychology, decision making, and psychophysics
attracted a lot of interest. In particular, in \cite{DKBB} the possibility to
use of the projection postulate and representation of "mental observables" by
Hermitian operators was discussed in very detail. The main conclusion of the
recent discussions on the foundations of "quantum(-like) cognitive psychology"
is that one has to be careful in determination of conditions of applicability
of the projection postulate as a mathematical tool for description of
measurements of observables represented by Hermitian operators. To represent
some statistical experimental data (both physical and mental) in the
quantum(-like) way, one has to use generalized quantum observables given by
positive operator-valued measures (POVMs). This paper contains a brief review
on POVMs which can be useful for newcomers to the field of quantum(-like)
studies. Especially interesting for cognitive psychology is a variant of the
formula of total probability (FTP) with the interference term derived for
incompatible observables given by POVMs. We present an interpretation of the
interference term from the psychological viewpoint. As was shown before, the
appearance of such a term (perturbing classical FTP) plays the important role
in cognitive psychology, e.g., recognition of ambiguous figures and the
disjunction effect. The interference term for observables given by POVMs has
much more complicated structure than the corresponding term for observables
given by Hermitian operators. We elaborate cognitive interpretations of
different components of the POVMs-interference term and apply our analysis to a
quantum(-like) model of decision making.
