. Can quantum probability theory be applied, beyond the microscopic scale of atoms and quarks, to the human problem of reasoning from incomplete and uncertain data? A unified theory of quantum statistical mechanics and Bayesian statistical inference is proposed. QSI is applied to ordinary data analysis problems such as the interpolation and deconvolution of continuous density functions from both exact and noisy data. The information measure has a classical limit of negative entropy and a quantum limit of Fisher information (kinetic energy). A smoothing parameter analogous to a de Broglie wavelength is determined by Bayesian methods. There is no statistical regularization parameter. A priori criteria are developed for good and bad measurements in an experimental design. The optimal image is estimated along with statistical and incompleteness errors. QSI yields significantly better images than the maximum entropy method, because it explicitly accounts for image continuity. Introduction Jaynes has been an eloquent advocate for a compelling hypothesis: Probability Theory as Logic . That is, probabilities represent degrees of belief; probability theory develops and applies universal principles of logical inference from incomplete information. Two of his primary interests have been Bayesian probability theory and the interpretation of quantum mechanics. Bayesian probability theory yields inferences by systematically and consistently combining new data with prior knowledge. Jaynes pioneered the maximum entropy (ME) class of Bayesian methods for density function estimation (Jaynes, 1983), which has been applied successfully to numerous data analysis problems.
. Can quantum probability theory be applied, beyond the microscopic scale of atoms and quarks, to the human problem of reasoning from incomplete and uncertain data? A unified theory of quantum statistical mechanics and Bayesian statistical inference is proposed. QSI is applied to ordinary data analysis problems such as the interpolation and deconvolution of continuous density functions from both exact and noisy data. The information measure has a classical limit of negative entropy and a quantum limit of Fisher information (kinetic energy). A smoothing parameter analogous to a de Broglie wavelength is determined by Bayesian methods. There is no statistical regularization parameter. A priori criteria are developed for good and bad measurements in an experimental design. The optimal image is estimated along with statistical and incompleteness errors. QSI yields significantly better images than the maximum entropy method, because it explicitly accounts for image continuity. Introduction Jaynes has been an eloquent advocate for a compelling hypothesis: Probability Theory as Logic . That is, probabilities represent degrees of belief; probability theory develops and applies universal principles of logical inference from incomplete information. Two of his primary interests have been Bayesian probability theory and the interpretation of quantum mechanics. Bayesian probability theory yields inferences by systematically and consistently combining new data with prior knowledge. Jaynes pioneered the maximum entropy (ME) class of Bayesian methods for density function estimation (Jaynes, 1983), which has been applied successfully to numerous data analysis problems.
Quantum mechanics is basically a mathematical recipe on how to construct physical models. Historically its origin and main domain of application has been in the microscopic regime, although it strictly seen constitutes a general mathematical framework not limited to this regime. Since it is a statistical theory, the meaning and role of probabilities in it need to be defined and understood in order to gain an understanding of the predictions and validity of quantum mechanics. The interpretational problems of quantum mechanics are also connected with the interpretation of the concept of probability. In this thesis the use of probability theory as extended logic, in particular in the way it was presented by E. T. Jaynes, will be central. With this interpretation of probabilities they become a subjective notion, always dependent on one's state of knowledge or the context in which they are assigned, which has consequences on how things are to be viewed, understood and tackled in quantum mechanics. For instance, the statistical operator or density operator, is usually defined in terms of probabilities and therefore also needs to be updated when the probabilities are updated by acquisition of additional data. Furthermore, it is a context dependent notion, meaning, e.g., that two observers will in general assign different statistical operators to the same phenomenon, which is demonstrated in the papers of the thesis. It is also presented an alternative and conceptually clear approach to the problematic notion of "probabilities of probabilities", which is related to such things as probability distributions on statistical operators. In connection to this, we consider concrete numerical applications of Bayesian quantum state assignment methods to a three-level quantum system, where prior knowledge and various kinds of measurement data are encoded into a statistical operator, which can then be used for deriving probabilities of other measurements. The thesis also offers examples of an alternative quantum state assignment technique, using maximum entropy methods, which in some cases are compared with the Bayesian quantum state assignment methods. Finally, the interesting and important problem whether the statistical operator, or more generally quantum mechanics, gives a complete description of "objective physical reality" is considered. A related concern is here the possibility of finding a "local hidden-variable theory" underlying the quantum mechanical description. There have been attempts to prove that such a theory cannot be constructed, where the most well-known impossibility proof claiming to show this was given by J. S. Bell. In connection to this, the thesis presents an idea for an interpretation or alternative approach to quantum mechanics based on the concept of space-time.
This paper proposes a novel Bayesian inference algorithm termed "Quantum-Inspired Bayesian Inference" (QIBI), drawing inspiration from quantum mechanics, particularly superposition and entanglement. Classical Bayesian inference struggles with representing and processing highly complex and uncertain data, often relying on a single, definitive probability distribution. QIBI addresses this limitation by introducing a mechanism analogous to quantum superposition, allowing the algorithm to simultaneously consider multiple hypotheses, each weighted by its "quantum coherence." Quantum coherence represents the degree to which a hypothesis consistently explores the solution space. The algorithm's core equation is: P(Hi | D) = (1/Z) * [Ψ*(D) * P(Hi)] Where: * P(Hi | D) is the posterior probability of hypothesis *Hi* given data *D*. * Ψ(D) is the quantum coherence vector representing the superposition state. * P(Hi) is the prior probability of hypothesis *Hi*. * Z is the normalization constant. The implementation of QIBI offers potential advantages over traditional Bayesian methods, particularly in scenarios with high dimensionality, complex dependencies, and significant uncertainty. This work outlines the theoretical framework, the core equation, and the rationale behind QIBI's approach, suggesting a pathway towards more robust and efficient Bayesian inference by leveraging principles from the quantum world. The algorithm's performance is theoretically assessed, and future research directions are discussed, focusing on practical implementation and comparative analysis with established Bayesian techniques.
Quantum Bayesian Inference (QBI) presents a novel approach to statistical machine learning, utilizing the principles of quantum mechanics to enhance the accuracy and robustness of models, particularly when dealing with high-dimensional data and complex Bayesian frameworks. This paper introduces a quantum algorithm designed to efficiently estimate the posterior distribution of parameters in a Bayesian model through the application of quantum superposition and entanglement, aiming to overcome the limitations of classical methods. We explore the potential of this paradigm shift to deliver exponential speedups in inference tasks, offering a compelling path towards more accurate and reliable statistical machine learning algorithms.
Knowing and guessing, these are two essential epistemological pillars in the theory of quantum-mechanical measurement. As formulated quantum mechanics is a statistical theory. In general, a priori unknown states can be completely determined only when measurements on infinite ensembles of identically prepared quantum systems are performed. But how one can estimate (guess) quantum state when just incomplete data are available (known)? What is the most reliable estimation based on a given measured data? What is the optimal measurement providing only a finite number of identically prepared quantum objects are available? These are some of the questions we address. We present several schemes for a reconstruction of states of quantum systems from measured data: (1) We show how the maximum entropy (MaxEnt) principle can be efficiently used for an estimation of quantum states on incomplete observation levels. (2) We show how Bayesian inference can be used for reconstruction of quantum states when only a finite number of identically prepared systems are measured. (3) We describe the optimal generalized measurement of a finite number of identically prepared quantum systems which results in the estimation of a quantum state with the highest fidelity.
A nonparametric Bayesian approach is developed to determine quantum potentials from empirical data for quantum systems at finite temperature. The approach combines the likelihood model of quantum mechanics with a priori information on potentials implemented in the form of stochastic processes. Its specific advantages are the possibilities to deal with heterogeneous data and to express a priori information explicitly in terms of the potential of interest. A numerical solution in maximum a posteriori approximation is obtained for one-dimensional problems. As nonparametric estimates, the results depend strongly on the implemented a priori information.
This thesis concerns the foundations of inference – probability theory,entropic inference, information geometry, etc. – and its application to the Entropic Dynamics (ED) approach to Quantum Mechanics (QM) [21, 22, 41, 53, 56–61, 150–153, 165, 195, 196, 268]. The first half of this thesis, chapters 2-6, concern the development of the inference framework. We begin in chapter 2 by discussing de- ductive inference, which involves formal logic and it’s role in access- ing the truth of propositions. We eventually discover that deductive inference is incomplete, in that it can’t address situations in which we have incomplete information. This necessitates a theory of inductive inference (probability theory), which is developed in chapter 3. Prob- ability theory is derived as a framework for manipulating degrees of belief of propositions, in a way which is consistent with its deduc- tive counterpart [47]. In chapter 4 we review the construction of en- tropic inference as a means for updating our beliefs in the presence of new information. The entropy functional is designed through the pro- cess of eliminative induction by imposing a principle of minimal updating (PMU) and various constraints [47, 238, 242, 266, 267]. Chapter 5 con- siders the design of another entropic functional, the total correlation and all its variants, for the purposes of ranking join distributions with respect to their correlations [46]. Finally, in chapter 6, we discuss the application of a special case of the correlation functionals from chap- ter 5, the mutual information, to problems in experimental physics and machine learning [42–45]. The second half of this thesis, chapters 7-9, concerns the ED ap- proach to QM. In particular, chapters 8 and 9 involve the inclusion of particles with spin 1/2 into the framework [41, 61]. These devel- opments are the main contribution of this thesis to the body of work in the ED approach. The problem is defined as an application of in- ference to the dynamics of quantum particles which have definite yet unknown positions and follow continuous trajectories. Through the method of maximum entropy developed in chapter 4, we can deter- mine the transition probability that these particles will move from one location to another. Geometric algebra (GA) [90, 140] is chosen as the preferred representation for the algebra of spin, which is then introduced through constraints in the maximum entropy method. A quantum mechanics is subsequently developed by constructing an epistemic phase space of probabilities