Information-Geometric Models in Data Analysis and Physics

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The article is written on the topic of information geometry. The article seems interesting. However, I don't think that the abstract and introduction reflect the content very well. They focus on philosophical aspects, while the text contains an overview of some results related to some mathematical models. If the proofs of some mathematical statements are new results of the authors, then, in my opinion, this should be reflected in the introduction. For example, a significant part of the text of the article is devoted to proving formula (65). In my opinion, this result is not sufficiently presented in the introduction.
Some of the statements, such as the sentence from the abstract “Ultimately, this could lead to groundbreaking discoveries and significant advancements that reshape our understanding of reality itself,” seem to me to be slighty speculative. The text of the article would be strengthened if, in addition to the description of the models, more examples with references to the relevant literature were mentioned. For example, it would be interesting if it were mentioned what manifolds arise when applying the considered models in quantum mechanics. I recommend that the authors rework the text of the article.

Author Response

First, we would like to thank the first reviewer for their comments.

This article is intended to be the first of several, and we agree with the reviewer’s opinion that the specific contribution of this first article, its scope, and purpose should be better clarified. To this end, we have rewritten parts of the abstract and introduction.

We have also attempted to clarify the most novel and specific contributions, particularly the introduction of parametric models through a complex square root of a probability density function, along with the corresponding Riemannian metric, which is an extension of the usual information metric —a complex version of information.

In the appendix, we have summarized the basic results of tensor analysis and differential geometry to help readers coming from other areas of knowledge. More specific and detailed examples may be made explicit in future papers. We have expanded on some points, providing more explicit indications of developments that will be made in the near future in other articles.

We have tried to clarify the basic purpose of this work, which is to persuade the reader of the basic relationship between data analysis/statistics and any subsequent physical-theoretical development, since this must take into account, more or less explicitly, the observer, who, in many aspects, will behave as a statistical machine. In other words, data analysis and statistics are disciplines that provide supposedly efficient ways of analyzing information conveyed by data obtained from experiments or surveys.

Moreover, living beings, particularly humans, as a result of natural selection associated
with the evolutionary process, have become efficient machines for analyzing information from their environment. The brain-senses complex analyzes and represents vast amounts of information: in a certain sense, it performs data analysis naturally and spontaneously. This is why data analysis/statistical methods are at the base of any observation process, and
it seems natural to think that this “built-in” statistics will influence any theoretical approach to developing the laws of nature.

In this sense, data analysis/statistics is a kind of pre-physics that must be a constitutive influence in any formulation of the laws of nature (the laws of reality) that we intend to develop. That is why we aim to establish a conceptual framework that will serve both to process the usual information obtained from experiments or surveys and to formulate fundamental physical laws.

We have also attempted to minimize overtly speculative statements, as mentioned by the reviewer, while highlighting those lines that are perceived as suggestive in subsequent work.

Reviewer 2 Report

Comments and Suggestions for Authors

Referee Report on

“Information-Driven Geometric Models: Transforming Data Analysis and Physical Theories”
by D. Bernal-Casas and J. M. Oller

 

Summary

This manuscript proposes a geometric framework for modeling inference and measurement, grounded in information geometry and variational principles. The authors construct a Riemannian manifold ( \mathcal{O} ) representing the observer’s parameter space, and define a plausibility distribution ( \Psi(\theta) ) that evolves via gradient alignment with a source information field ( I_w(\theta) ). The framework is positioned as a meta-theory of inference, with potential implications for data analysis, quantum mechanics, and foundational physics.

 

Major Comments

1. Ontological Status and Physical Relevance

The manuscript presents a compelling epistemic model of inference, but its ontological claims remain underdeveloped. Specifically:

• It is unclear whether the framework can reproduce the empirical predictions of established physical theories such as quantum mechanics, general relativity, or thermodynamics.
• The absence of unitary evolution, operator algebra, and entanglement formalism suggests that the framework is not translatable into the Hilbert space machinery that underpins all experimentally verified quantum formulations.
• If the framework cannot reproduce known physical results, its claim to describe “real reality” must be qualified. It may instead serve as a conceptual scaffold for modelling observer–system interaction.

Recommendation: The authors should clarify whether their framework is intended as a supplement to existing physical theories, a reinterpretation of their foundations, or a candidate for replacement. Explicit discussion of its empirical scope and limitations is essential.

 

2. Measurement and Observer Dynamics

The manuscript offers a novel geometric interpretation of measurement as gradient alignment rather than state collapse. This is philosophically rich but technically ambiguous:

• The variational principle governing ( \Psi(\theta) ) lacks discussion of solution behaviour, stability, and boundary conditions.
• The role of the environment is modelled as deformation of the information field, but without a formal treatment of system–environment coupling or decoherence.
• The observer is treated as an active agent shaping the geometry of inference, yet the framework does not specify how this geometry interacts with ontic physical systems.

Recommendation: The authors should elaborate on how their model handles imperfect isolation, entanglement-like coherence, and the emergence of classicality. A comparison with open quantum systems and reduced density matrices would be instructive.

 

3. Statistical Uncertainty and Entropy

The framework defines uncertainty via gradient misalignment and curvature, diverging from the von Neumann entropy used in quantum mechanics. While this is conceptually innovative:

• It is unclear whether the proposed measure of uncertainty can be operationalized or compared with standard entropy-based metrics.
• The absence of spectral theory and probabilistic collapse limits the framework’s ability to model statistical mixtures or entangled subsystems.

Recommendation: The authors should clarify whether their geometric uncertainty measure can be extended to multipartite systems, and whether it admits a thermodynamic or information-theoretic interpretation.

 

4. Purpose and Scope of the Meta-Theory

The manuscript repeatedly emphasizes the extraction of structure from data but does not articulate how this structure interfaces with ontological models of reality. This raises a foundational concern:

Extracting structure is not a self-sufficient goal—it must enable the application of physical theories to empirical phenomena.

Recommendation: The authors should explicitly state the purpose of their meta-theory: Is it a tool for data analysis, a philosophical model of inference, or a candidate for physical law? Clarifying this would sharpen the manuscript’s contribution and situate it within the broader landscape of foundational science.

 

Minor Comments

• The notation shifts between ( \mathcal{O} ), ( \mathcal{M} ), and ( \mathcal{H} ) without clear hierarchy.
• The derivation of the Riemannian metric from divergence functions omits intermediate steps.
• The kernel construction is promising but lacks justification for kernel choice and empirical validation.
• < !--StartFragment -->

  The references are partially appropriate and moderately relevant, but the manuscript would benefit from:

  • Broader engagement with adjacent foundational literature.
  • More recent citations in quantum gravity and epistemic modeling.
  • Clearer integration of references into the conceptual narrative.
  < !--EndFragment -->

 

Recommendation

Major Revision.
The manuscript presents a conceptually rich and mathematically intriguing framework, but its physical relevance, empirical scope, and foundational claims require substantial clarification. Addressing the issues above would significantly strengthen its contribution to both data science and theoretical physics.

Author Response

First, we would like to thank the second reviewer very much for their valuable comments.

We have rewritten the introduction, provided a more detailed explanation of the section on physical applications, and attempted to clarify some of the issues raised by the referee. We are trying to frame this article within an ongoing research project that we hope will be published in various outlets, addressing some of the aspects discussed.

In particular, we are already working on a second part of the article, which will be presented in this same special issue. This will present and discuss a solution to the variational problem, exploring connections with fundamental physics equations and expanding on a paper first presented by the authors in Mathematics in 2024.

The primary purpose of this article is to develop a sufficiently broad and rich conceptual framework on which to base two apparently different topics: the first, a data analysis topic, either medical, sociological, biological, etc., primarily based on geometric techniques that help visualize the relationships between the objects studied, and the second, to serve as a
foundation upon which fundamental physical theories can subsequently be built. We will briefly develop these ideas.

The starting point assumes an implicit epistemological reflection: we accept that in any realistic theory of knowledge, we must distinguish between the object to be known, the knowing subject, and the environment, without forgetting any of these. The knowing subject is a fundamental and obvious reality that must be taken into account. In a way, it possesses its properties due to thousands of years of natural selection during the evolutionary process in which it emerged. This natural selection enables it to efficiently analyze vast amounts of information that reach it through the senses and that it represents internally thanks to its brain. The knowing subject naturally and spontaneously performs an accurate data/statistical analysis. It is, we might say, a built-in statistical machine. Thousands of years of natural selection will have shaped its properties according to some principle of optimality, following some natural constraints based on the subject’s own physical limitations. That is why any physical description that can be made to describe nature must take this reality into account.

Detecting the optimality principle, with predictable constraints, that selection has led to in shaping the subject’s properties, would be a good objective to base data/statistics analysis on. Previous work by the authors partially suggests that a convenient variational principle allows us to recover some fundamental physical equations. This is, in broad outline, some of the ideas that inspire the present work.

We have attempted to clarify the paper's main contributions and indicate the steps to be taken in our future research, where we believe the reasons for the methods presented here will be more explicitly appreciated. As a reminder, in the paper, we introduce parametric families in terms of square roots, not necessarily real, of typical probabilistic density functions.

This approach allows, on the one hand, to construct a Riemannian geometry in the parameter space, formula (24), which is nothing more than an extension of the informational geometry induced by the ordinary Fisher information matrix, and on the other hand, the extension of the classical notion of information in terms of the logarithm of these square roots, formula (69). The work continues by presenting the construction of a Reproducing Kernel Hilbert Space (RKHS), starting from a given kernel, as shown in formula (25). This RKHS can be seen as both a richer and simpler model than L^2(\Theta, μ), but one that allows for the same Riemannian metric in parameter space. As an aside, in later physical applications, we will think of this RKHS as the most convenient abstract model for identifying objects in the physical world, viewing them as vectors in this infinite-dimensional Hilbert space.

Another aspect we would like to highlight is the intrinsic Taylor expansions (formulas (31) and (33)), which naturally provide covariant tensors associated with variables defined in the parameter space. In particular, we can highlight the first-order expansions that, via the gradient, allow us to interpret the representation spaces obtained in classical data analysis,
but especially the second-order covariant tensors obtained in second-order expansions. These second-order covariant tensors, through the natural operation of raising indices, have associated second-order mixed tensors, identifiable with fields of linear operators acting in each tangent space of the parameter space. It is in this context that we hope to relate the present development to classical spectral analysis of linear operators in a Hilbert space, at least as an approximation. In other words, the mathematical edifice outlined presents rich potential that we will explore in the future, both in fundamental physics, where we hope to reobtain or reinterpret classical results, opening up new perspectives, and in classical data analysis, which will help generate meaningful graphical representations for applied scientists.

Some of these observations have been introduced more or less explicitly throughout the article to clarify the reviewer's interesting questions and observations.

We have to specially remark that we are already working on a second paper where we will develop a solution to the variational problem posed, which will result in a system of partial differential equations on the manifold, a paper that would be viewed as an extension of a paper presented by the authors in 2024 in this journal. Some additional references are also introduced.

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

I am satisfied with the corrections made by the authors. I have no comments on the revised version of the article.

Author Response

We again appreciate the comments of the referee in his first report and are satisfied that we have addressed the observations and comments reasonably.

We have included a sentence in the acknowledgments section to appreciate the comments.

Reviewer 2 Report

Comments and Suggestions for Authors

Referee Follow-Up Note

Manuscript Title: Information-Geometric Models in Data Analysis and Physics
Authors: D. Bernal-Casas and J. M. Oller

 

Summary of Revision Assessment

The revised manuscript demonstrates a thoughtful and constructive response to the initial referee report. The authors have improved the clarity of exposition, expanded the mathematical formalism, and adopted a more philosophically modest tone regarding the framework’s physical implications. These changes enhance the manuscript’s readability and conceptual coherence.

However, several foundational concerns remain only partially addressed:

 

1. Empirical Scope and Physical Relevance

While the revised version acknowledges that the framework does not yet reproduce the predictive machinery of quantum mechanics, general relativity, or thermodynamics, it still lacks a clear articulation of how it might eventually interface with these theories. The ontological status of the framework remains speculative, and its empirical applicability is deferred to future work.

2. Measurement and System–Environment Interaction

The manuscript retains its triadic model of measurement and expands the discussion of observer-generated geometry. However, it does not offer a formal treatment of decoherence, entanglement, or reduced density matrices. The environment is modeled geometrically rather than dynamically, limiting the framework’s compatibility with open quantum systems.

3. Uncertainty and Entropy

The revised version elaborates on geometric measures of uncertainty, including curvature and gradient misalignment. Yet it still does not establish operational comparability with standard entropy metrics such as von Neumann entropy, nor does it address multipartite systems or thermodynamic analogues.

4. Purpose and Positioning

The manuscript now more clearly positions itself as a mathematical scaffold for inference and data representation, rather than a replacement for physical law. This clarification is welcome and aligns the manuscript more appropriately with its current capabilities.

 

Recommendation

The revision reflects meaningful progress and intellectual engagement with the referee’s concerns. However, further development is needed to establish the framework’s empirical relevance and physical compatibility. I recommend that the manuscript be considered for publication contingent on its revised scope—as a conceptual and mathematical contribution to information geometry and inference modelling, rather than a foundational physical theory.

 

Author Response

First, we would like to thank the reviewer very much again for their valuable comments in the first and second reports.

We are encouraged to note that some of the objections expressed by the referee in his first report have been partially resolved.

We recognize that because the article addresses fundamental issues, it is sometimes difficult to strike the right
tone, especially when it is intended to be only a modest first contribution in a direction that we hope will prove fruitful over time.

We would certainly like to have a clear path already outlined for developing the variational principle and connecting it with the major existing physical theories. For the time being, in our immediate work, we will limit ourselves to attempting to derive some existing basic equations: the Schrödinger, Klein-Gordon, and Dirac equations, and to the extent possible, we will explore their compatibility with general relativity and thermodynamics. In addition, it is certainly quite a challenge to establish and deepen the framework proposed by the referee, which encompasses decoherence, entanglement, reduced density matrices, and open quantum systems. Moreover, we agree and appreciate the referee’s suggestion that it would be beneficial to delve deeper into other standard entropy metrics, such as von Neumann entropy.

We have included reflections in this regard at the end of the paper.

We have also included a sentence in the acknowledgments section to express our appreciation for the comments.