Quantum probability for statisticians; some new ideas

arXiv:2503.02658 · quant-ph · Submitted 2025-03-04 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: Today's paper: "Quantum probability for statisticians; some new ideas".

Mira: Quantum probability for statisticians;

Kai: First, who's behind it and why it matters.

Title and authors: Kai: Now, moving into the actual substance of "Quantum probability for statisticians; some new ideas," they lay out a set of postulates that appear to imply essential parts of the ordinary formalism, especially the formalism behind quantum probabilities. Mira, can you summarize what these specific postulates are doing in plain English?

Mira: They are setting up a system where theoretical variables are categorized as accessible or inaccessible to an actor C, and they posit that there exists this inaccessible variable phi such that every accessible variable can be seen as a function of it.

Lev: That structure immediately raises questions about how we define the probability space itself; if everything boils down to functions of one thing, does that simplify the complexity of modeling outcomes or just shift the difficulty around?

Kai: They go on to define what a maximal accessible variable is—one where you can't write it as a function of another accessible variable using a non-invertible function—and then postulate that these maximal variables are "very close to what Bohr called complementary variables."

Mira: That connection to complementarity is important because it suggests that the structure derived from these postulates naturally produces relationships reminiscent of quantum probability rules, specifically linking them to Born’s formula.

Lev: If they derive a variant of Born's formula using these postulates, it means we are trying to bridge the gap between abstract quantum formalism and concrete probability calculations in a mathematically rigorous way.

Kai: They establish this link through Postulate five which introduces the "likelihood principle," assuming an epistemic setting where an experimentalist C’s decisions are influenced by a superior being D, and then connecting that to the Dutch Book Principle via Postulate six.

Mira: That's a heavy assumption because it grounds quantum probability in a specific kind of rational epistemic setting involving this entity D, which essentially acts as the source of the 'real' probability q.

Lev: For running this on hardware, we’d need to figure out how to model this "superior being D" or how to mathematically operationalize that likelihood principle when dealing with noisy experimental data from a physical system.

Kai: They then generalize this into the generalized likelihood effect F a(u;z, τ) using the spectral theorem for continuous variables, which is what allows them to get Theorem two showing a simple version of Born’s formula for two different discrete maximal accessible variables <ref:2503.02658#pg0>.

The paper's summary: Kai: So we've seen the structure and the derivation of Born's formula; now let’s look at how these ideas are being applied, because this paper suggests several concrete improvements for statistical inference, especially in areas like model reduction.

Mira: They suggest applying this framework to model reduction by reducing a large parameter space phi to an orbit under a group G acting on it, choosing an orbit where the subparameter of interest is "permissible," which motivates models like partial least squares regression because their assumptions satisfy Theorem one.

Lev: From my perspective as someone dealing with high-dimensional data, this idea of selecting a permissible orbit based on group action seems like a way to systematically handle parameter redundancy without relying on purely heuristic criteria for model selection.

Kai: They also apply this to decision theory by modeling two different maximal decision variables, θ and η, using associated self-adjoint operators A θ and A η in a Hilbert space H, defining the "maximal case" as one where each eigenvalue is single.

Mira: This correspondence between eigenvectors and specific decisions gives us a way to link the abstract algebraic structure of quantum theory directly to concrete choices an actor makes in a statistical experiment.

Lev: If we use this for decision theory, it means we could potentially design AI systems that evaluate different analytical paths not just by expected loss, but by how they relate to these structural constraints imposed by the maximal variables.

Kai: Furthermore, they show specific applications in medical experiments where they find the maximal parameter that can be estimated is defined as theta b = sign(mu b - one/three(mu a + mu c + mu d)).

The paper's improvements: Kai: We’ve covered a lot about how this paper proposes a new structure linking quantum probability to statistics, from the foundational postulates down to specific applications in model reduction and decision theory. Mira, what do you see as the biggest conceptual takeaway from "Quantum probability for statisticians; some new ideas"?

Mira: I think the most significant contribution is demonstrating that by introducing structure into a parameter space via group actions and maximal variables, we can derive quantum probabilities that are relevant to statistical inference, moving beyond the idea that they are just exotic tools.

Lev: From a research implementation standpoint, the main limitation I see is how we operationalize this concept of an inaccessible variable phi; building a measurement scheme around something truly inaccessible sounds like it might lead to intractable experimental requirements for real hardware.

Kai: That’s a fair point about the practical realization, Lev, but they do acknowledge that they are sketching ideas for applications in machine learning, which suggests there's hope for computational exploration even if the full quantum foundation is hard to build.

Mira: They also state a limitation regarding continuous parameters: they have to approximate them using finite sets of values while trying to maintain the structural integrity through spectral theorems and density operators.

Lev: That approximation aspect is critical; we can work with discrete states, but if the underlying reality is continuous, that approximation might introduce systematic errors that are hard to quantify without a full quantum formalism.

Kai: So, to wrap up "Quantum probability for statisticians; some new ideas," the paper provides a framework suggesting model selection and parameter estimation can be guided by group-theoretic symmetries and the concept of maximal accessible variables.

Mira: It offers a way to interpret statistical knowledge as epistemic knowledge structured by quantum principles, which is a significant conceptual refinement for how we view knowledge acquisition in science.

Lev: I just reiterate that while the theoretical structure is rich, moving this onto real physical systems will require solving the problem of defining and measuring those inaccessible variables phi first.

Conclusion: Kai: So, to summarize our discussion on "Quantum probability for statisticians; some new ideas," we've seen how the paper builds a foundation from postulates about accessible and inaccessible variables to derive a link between quantum theory and Born’s rule through concepts like maximal variables and the likelihood principle.

Mira: Essentially, they argue that by introducing this structure into the parameter space, statistical theory gains a richer framework that incorporates quantum probability rules, which is a major step toward viewing statistical science as an epistemic theory.

Lev: I still have to emphasize that realizing this on real hardware hinges on overcoming the hurdle of defining those inaccessible variables phi in a measurable way, because without that definition, it remains purely mathematical speculation for now.

Kai: That’s right; the paper suggests we can use model reduction guided by group theory and decision theory based on maximal variables to handle complex statistical problems in fields like machine learning more structurally.

Mira: It opens up new ways to think about priors and parameter estimation by using symmetry arguments derived from quantum probability, which could lead to more informed Bayesian results even when data is limited.

Lev: If we manage that operationalization, the potential for better structural understanding of complex systems in areas like error correction or materials science becomes quite interesting.

Kai: We’ve discussed the theory behind "Quantum probability for statisticians; some new ideas," which proposes a powerful way to structure statistical thinking using concepts from quantum foundations.

Mira: It certainly provides a deep theoretical underpinning for why certain symmetries and structures are important when we look at how we model knowledge acquisition.

Lev: And for hardware, the next step is definitely figuring out the physical constraints imposed by that theory on what's actually possible to measure in a lab setting.

quant-ph

Submitted: 2025-03-04

Updated: 2026-10-07

Comments: 30 pages

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 67/100

The gist: Quantum probability for statisticians; some new ideas argues that quantum probabilities are relevant to statistical settings by proposing new foundations for quantum theory and exploring potential

Key concepts

Accessible vs. Inaccessible Variables
Variables are classified based on whether an actor can obtain accurate estimates for them. Accessible variables can be measured or estimated reliably, while inaccessible variables cannot be obtained with accuracy. The theory posits that all accessible variables depend on a single, underlying inaccessible variable.
Maximal Accessible Variable
This is a specific type of accessible variable defined as one where no other accessible variable can be expressed as a function of it using a non-invertible function. Postulate 4 asserts the existence of these maximal variables, which are related to Bohr's concept of complementary variables.
Likelihood Principle
This principle is assumed in a rational epistemic setting where an experimentalist's choices are influenced by a 'superior being.' It connects quantum theory to probability by assuming that the superior being's probability for an outcome is the real probability, linking it to the Dutch Book Principle.
Model Reduction in ML
This technique reduces a large parameter space by choosing an orbit of a group acting on it where a specific subparameter of interest is 'permissible.' This allows researchers to motivate models like partial least squares regression by showing their assumptions satisfy the paper's foundational theorem.

Terminology

Summary

Quantum probability for statisticians; some new ideas argues that quantum probabilities are relevant to statistical settings by proposing new foundations for quantum theory and exploring potential applications in areas like machine learning and model reduction.

How it works

The paper proposes a completely new approach towards quantum foundations derived from a set of postulates, emphasizing that these postulates imply essential parts of the ordinary formalism, in particular the formalism behind quantum probabilities. This foundation is based on characterizing theoretical variables as accessible or inaccessible to an actor C. Accessible variables are those that can be obtained with accurate estimates, while inaccessible ones cannot. The theory posits that there exists an inaccessible variable φ such that all the accessible variables can be seen as functions of φ.

The core postulates proposed are:

  1. If η is a theoretical variable and γ = f(η) for some function f, then γ is also a theoretical variable.

  2. If θ is accessible to C and λ = f(θ) for some function f, then λ is also accessible to C.

  3. In the given context there exists an inaccessible variable φ such that all the accessible ones can be seen as functions of φ, and there is a group K acting upon φ.

Definition 1 defines a maximal accessible variable θ as one where there is no other accessible variable λ such that θ = f(λ) for some non-invertible function f. Postulate 4 then asserts: There exist maximal accessible variables relative to this partial ordering. This leads to the result that two different maximal accessible variables are very close to what Bohr called complementary variables.

How it works

The paper derives a variant of Born’s formula using these postulates, establishing a link between quantum theory and probability. Postulate 5 introduces the likelihood principle, which is assumed to hold in a rational epistemic setting where an experimentalist C's decisions are influenced by a superior being D. Postulate 6 connects this principle to the Dutch Book Principle, assuming D's probability for an outcome E is q, and q is taken as the real probability for E.

The generalized likelihood effect F a(u;z, τ) is defined based on these postulates. For a continuous variable θ a, the likelihood effect can be defined using the spectral theorem: F a(u;z, τ) = Z σA p(zτ,θ a = u)dEA(u). Using Gleason’s Theorem and these postulates, Theorem 2 provides a simple version of Born’s formula for two different discrete maximal accessible variables θ a and θ b: P(θ b = v j θ a = ui) = ⟨a;ib; j⟩2.

How it works

The paper demonstrates how quantum probabilities can be applied to statistical inference through model reduction and decision theory. In model reduction, the idea is to reduce a large parameter space φ to an orbit of a group G acting on the parameter space, choosing an orbit where a subparameter ζ of interest is permissible. This principle allows for the motivation of models like partial least squares regression by showing that their assumptions satisfy Theorem 1.

In decision theory, two different maximal decision variables θ and η are modeled using quantum theory. The situation is described by associated self-adjoint operators A θ and A η in a Hilbert space H. The maximal case for a decision process is defined as one where each eigenvalue of the corresponding operator is single. This correspondence links eigenvectors to the question Which decision process? and Which action did this decision process lead to?.

How it works

The paper discusses specific statistical applications, including medical experiments and machine learning. In a medical experiment involving continuous inaccessible parameters µa,µb,µc, and µd, it is shown that the maximal parameter which can be estimated is θ b = sign(µ b − 1/3(µa + µc + µd)). The paper compares this result with a full Bayesian approach and a quantum probability solution based on symmetry arguments.

For machine learning, model reduction is applied to neural networks. By expanding the weights w in terms of eigenvectors of the covariance matrix Σ, model reduction is introduced by hypothesizing There are exactly m nonzero terms in (24). The theory shows that this constraint can be motivated by a joint quantum model for the reduced and arbitrary models, satisfying Theorem 1.

How it works

The paper concludes by discussing continuous parameters and complementarity. For continuous parameters, the Hilbert space is taken as L2(R,dµ), and the operator A θ corresponds to multiplication by f ∈ L2(R,dµ) by θ. Complementarity in a statistical context is defined as two parameters being "really different and both are maximal accessible variables.

Improvements for AI systems

Based on the provided scientific paper, here are specific improvements for Artificial Intelligence (AI) systems that can be derived from its theoretical framework:

The core improvement involves shifting from traditional statistical inference to a Quantum Probability framework, which treats model selection and parameter estimation as problems within a Hilbert space structure defined by maximal accessible variables and associated operators.

Here are the specific improvements and capabilities:

  1. textbfQuantum Model Reduction for High-Dimensional Regression (PLS Regression):

The paper explicitly links Partial Least Squares (PLS) regression to the structure imposed by Theorem 1 of Subsection 2.2, which is motivated by quantum probability principles.

  • The improved system will use the concept of permissible parameters and maximal accessible variables to perform model reduction in high-dimensional datasets (where predictors > units).

  • By leveraging the group theory structure (e.g., orthogonal transformations) acting on the parameter space, the AI can automatically identify and select a reduced set of parameters that are most statistically relevant, replacing arbitrary model selection criteria with information derived from maximal variables.

  • The AI will be able to find the best possible model reduction under a least squares criterion more robustly than traditional methods when dealing with collinearity.

  1. textbfQuantum-Inspired Priors for Bayesian Inference:

The paper demonstrates that quantum probabilities (via Theorem 3) provide a prior distribution for one set of parameters based on knowledge of another set of maximal accessible parameters (e.g., using the contrast variables in the rash-medicine example).

  • The improved system can integrate these quantum priors into Bayesian frameworks. Instead of relying solely on abstract, inaccessible parameters, it can use quantum priors derived from symmetry and group actions (like those found in the translation groups acting on parameter spaces).

  • This will lead to more informed and potentially more accurate posterior distributions for experimental units when data is limited or noisy.

  1. textbfQuantum Decision Theory for Complex Modeling:

The paper discusses modeling complex decisions (like choosing a model, method, or reporting a p-value) using quantum decision theory, where decisions are mapped to maximal accessible variables in the mind of an actor/group.

  • The AI can be designed with quantum decision modules that evaluate different analytical paths (e.g., comparing different regression models) not just based on expected loss functions, but on a framework that respects the constraints imposed by maximal decision variables and complementary pairs.

  • This allows for the modeling of complex, context-dependent choices in statistical analysis (e.g., model selection in machine learning) using a more structurally rich mathematical theory than standard frequentist or classical Bayesian approaches allow.

  1. textbfGeneral Epistemic Interpretation of Neural Networks:

The paper suggests that neural networks can be seen as hidden variable models for quantum systems.

  • The improved system will use the machinery of Theorem 1 (Hilbert spaces, symmetric operators, and spectral decomposition) to interpret the weights and hidden layers not just as arbitrary parameters, but as components associated with specific maximal accessible variables in a high-dimensional space.

  • This provides a rigorous mathematical foundation for understanding why certain parameter combinations or features are accessible or inaccessible based on the structure of the underlying statistical model, potentially leading to more interpretable and theoretically sound deep learning architectures.

  1. textbfHandling Continuous Parameters via Finite Approximations:

For continuous parameters, the theory suggests approximating them with finite sets of values (discrete variables) while maintaining structural integrity through approximations involving spectral theorems and density operators (Theorem 3).

  • The AI can be designed to operate on a discrete representation of continuous data or parameters. This allows the system to utilize the powerful algebraic tools derived from Hilbert spaces and operators (like those in the spectral theorem) even when dealing with real-valued, continuous statistical variables.

In summary, the improved AI system will move beyond purely correlational learning to a framework where:

  • Model selection is guided by group-theoretic symmetries (Quantum Model Reduction).

  • Prior knowledge is structured according to quantum probability principles (Quantum Priors).

  • Complex analytical decisions are modeled using decision theory based on maximal accessible variables.

Abstract

It is argued from several points of view that quantum probabilities might play a role in statistical settings. New approaches toward quantum foundations have postulates that appear to be equally valid in macroscopic settings. One such approach is described here in detail, while one other is briefly sketched. In particular, arguments behind the Born rule, which gives the basis for quantum probabilities, are given. A list of ideas for possible statistical applications of quantum probabilities is provided and discussed. A particular area is machine learning, where there exists substantial literature on links to quantum probability. Here, an idea about model reduction is sketched and is motivated from a quantum probability model. Quantum models can play a role in model reduction, where the partial least squares regression model is a special case. It is shown that for certain experiments, a Bayesian prior given by a quantum probability can be motivated. Quantum decision theory is an emerging discipline that can be motivated by this author's theory of quantum foundations.

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