Based on recalling two characteristic features of Bayesian statistical inference in commutative probability theory, a stability property of the inference is pointed out, and it is argued that that stability of the Bayesian statistical inference is an essential property which must be preserved under generalization of Bayesian inference to the non‐commutative case. Mathematical no‐go theorems are recalled then which show that, in general, the stability can not be preserved in non‐commutative context. Two possible interpretations of the impossibility of generalization of Bayesian statistical inference to the non‐commutative case are offered, none of which seems to be completely satisfying.
Based on recalling two characteristic features of Bayesian statistical inference in commutative probability theory, a stability property of the inference is pointed out, and it is argued that that stability of the Bayesian statistical inference is an essential property which must be preserved under generalization of Bayesian inference to the non‐commutative case. Mathematical no‐go theorems are recalled then which show that, in general, the stability can not be preserved in non‐commutative context. Two possible interpretations of the impossibility of generalization of Bayesian statistical inference to the non‐commutative case are offered, none of which seems to be completely satisfying.
This paper attempts to develop a theory of sufficiency in the setting of non-commutative algebras parallel to the ideas in classical mathematical statistics. Sufficiency of a coarse-graining means that all information is extracted about the mutual relation of a given family of states. In the paper su cient coarse-grainings are characterized in several equivalent ways and the non-commutative analogue of the factorization theorem is obtained. As an application we discuss exponential families. Our factorization theorem also implies two further important results, previously known only infinite Hilbert space dimension, but proved here in generality: the Koashi-Imoto theorem on maps leaving a family of states invariant, and the characterization of the general form of states in the equality case of strong subadditivity.
We offer a realist interpretation of non-relativistic quantum mechanics in which dynamical properties are properly possessed by the system in question, and are supposed to have definite values at any time. Like the QBists, we employ Bayesian probability, but we adopt something closer to the Bayesian statistics of E. T. Jaynes than to the subjective Bayesian statistics of B. de Fenetti employed by the QBists. Accordingly, we view calculated Bayesian probabilities as rational degrees of expectation of dynamical property values rather than as personal degrees of expectation of future (measurement) experiences. Probabilities are, for us, based on knowledge of the value of some dynamical property of the system, not on knowledge of previous experiences unassociated with system dynamical properties. As some Bayesians might, we take a probability equal to 1 not generally to indicate certainty but only (full) expectation; and we disallow probabilities of conjunctions of propositions claiming incompatible properties. Then, by reinterpreting and adding a little to the quantum formalism, we argue that we can maintain the advantages of the QBist interpretation. So, for us (as for the QBists), there is no unexplained collapse of the wave function, no need for ‘spooky action at a distance’, and no problem raised by the double slit experiment, the Kochen-Specker paradox or Bell type theorems. By holding on to a realist perspective, modelling (of ideal measurements, of system preparation processes etc.) is possible, and we can claim certain dynamical laws of quantum mechanics without leading to contradiction.
Quantum theory can be viewed as a generalization of classical probability theory, but the analogy as it has been developed so far is not complete. Whereas the manner in which inferences are made in classical probability theory is independent of the causal relation that holds between the conditioned variable and the conditioning variable, in the conventional quantum formalism, there is a significant difference between how one treats experiments involving two systems at a single time and those involving a single system at two times. In this article, we develop the formalism of quantum conditional states, which provides a unified description of these two sorts of experiment. In addition, concepts that are distinct in the conventional formalism become unified: Channels, sets of states, and positive operator valued measures are all seen to be instances of conditional states; the action of a channel on a state, ensemble averaging, the Born rule, the composition of channels, and nonselective state-update rules are all seen to be instances of belief propagation. Using a quantum generalization of Bayes' theorem and the associated notion of Bayesian conditioning, we also show that the remote steering of quantum states can be described within our formalism as a mere updating of beliefs about one system given new information about another, and retrodictive inferences can be expressed using the same belief propagation rule as is used for predictive inferences. Finally, we show that previous arguments for interpreting the projection postulate as a quantum generalization of Bayesian conditioning are based on a misleading analogy and that it is best understood as a combination of belief propagation (corresponding to the nonselective state-update map) and conditioning on the measurement outcome.
An understanding of quantum theory in terms of new, underlying descriptions capable of explaining the existence of non-classical correlations, non-commutativity of measurements and other unique and counter-intuitive phenomena remains still a challenge at the foundations of our description of physical phenomena . Among some proposals, the idea that quantum states are essentially states of knowledge in a Bayesian framework is an intriguing possibility due to its explanatory power. In this work, the formalism of quantum theory is derived from the application of Bayesian probability theory to “fragile” systems, that is, systems that are perturbed by the measurement. Complex Hilbert spaces , non-commuting operators and the trace rule for expectations all arise naturally from the use of linear algebra to solve integral equations involving classical probabilities over hidden variables. The non-fragile limit of the theory, where all measurements are commutative and the theory becomes analogous to classical statistical theory is discussed as well.
Quantum mechanics is a fundamentally probabilistic theory (at least so far as the empirical predictions are concerned). It follows that, if one wants to properly understand quantum mechanics, it is essential to clearly understand the meaning of probability statements. The interpretation of probability has excited nearly as much philosophical controversy as the interpretation of quantum mechanics. 20th century physicists have mostly adopted a frequentist conception. In this paper it is argued that we ought, instead, to adopt a logical or Bayesian conception. The paper includes a comparison of the orthodox and Bayesian theories of statistical inference. It concludes with a few remarks concerning the implications for the concept of physical reality.
We introduce quantum Markov categories as a structure that refines and extends a synthetic approach to probability theory and information theory so that it includes quantum probability and quantum information theory. In this broader context, we analyze three successively more general notions of reversibility and statistical inference: ordinary inverses, disintegrations, and Bayesian inverses. We prove that each one is a strictly special instance of the latter for certain subcategories, providing a categorical foundation for Bayesian inversion as a generalization of reversing a process. We unify the categorical and C∗-algebraic notions of almost everywhere (a.e.) equivalence. As a consequence, we prove many results including a universal no-broadcasting theorem for S-positive categories, a generalized Fisher--Neyman factorization theorem for a.e. modular categories, a relationship between error correcting codes and disintegrations, and the relationship between Bayesian inversion and Umegaki's non-commutative sufficiency.