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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 overcome some of these issues.
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