  Integrated information theory (IIT) has established itself as one of the
leading theories for the study of consciousness. IIT essentially proposes that
quantitative consciousness is identical to maximally integrated conceptual
information, quantified by a measure called $\Phi^{max}$, and that
phenomenological experience corresponds to the associated set of maximally
irreducible cause-effect repertoires of a physical system being in a certain
state. However, in order to ultimately apply the theory to experimental data, a
sufficiently general formulation is needed. With the current work, we provide
this general formulation, which comprehensively and parsimoniously expresses
$\Phi^{max}$ in the language of probabilistic models. Here, the stochastic
process describing a system under scrutiny corresponds to a first-order
time-invariant Markov process, and all necessary mathematical operations for
the definition of $\Phi^{max}$ are fully specified by a system's joint
probability distribution over two adjacent points in discrete time. We present
a detailed constructive rule for the decomposition of a system into two
disjoint subsystems based on flexible marginalization and factorization of this
joint distribution. Furthermore, we suspend the approach of interventional
calculus based on system perturbations, which allows us to omit undefined
conditional distributions and virtualization. We validate our formulation in a
previously established discrete example system, in which we furthermore address
the previously unexplored theoretical issue of quale underdetermination due to
non-uniqueness of maximally irreducible cause-effect repertoires, which in turn
also entails the sensitivity of $\Phi^{max}$ to the shape of the conceptual
structure in qualia space. In constructive spirit, we propose several
modifications of the framework in order to address some of these issues.
