On the Application of Information Geometry to the Manifold Induced by the Parameters of the Mean Square Error of Probability Functions

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The manuscript under review proposes a differential-geometric framework for the study of a parametrized family of Gaussian probability distributions, with particular emphasis on the geometric interpretation of the mean square error (MSE). More precisely, the authors consider a family X of Gaussian distributions parametrized by the mean and the standard deviation, and they investigate how the MSE associated with this family can be described through geometric methods. According to the presentation in the paper, Lemma 1 establishes that the mean square error can be represented, or approximated, by a quadratic polynomial, while Lemma 2 identifies the coefficients of this polynomial in terms of the moments E(X) and E(X^2). Based on these preliminary results, the authors then attempt to derive geometric consequences for the parameter space associated with the family of Gaussian distributions under consideration.

The underlying idea of introducing geometric methods into the study of parametrized probabilistic models is mathematically meaningful and potentially interesting. The use of differential-geometric tools to study spaces of parametrized objects has a long and well-established tradition in several areas of mathematics, including geometric invariant theory, moduli theory, and information geometry. In this context, the attempt to interpret statistical quantities such as the mean square error in geometric terms could lead to interesting perspectives, especially for researchers working at the intersection of probability theory, statistics, and geometric methods in information theory. In particular, understanding whether the MSE induces, approximates, or relates to a geometric structure on the parameter space could potentially provide conceptual insight into the local and global behavior of statistical models.

Nevertheless, despite the conceptual interest of the proposed approach, I believe that the manuscript in its current form still suffers from several structural and mathematical issues that should be addressed before the paper can be considered for publication. My comments are detailed below.

First, the formulation and interpretation of Lemma 1 require substantial clarification. The statement that the mean square error “is represented by” a quadratic polynomial is currently too vague from a mathematical point of view. It is not clear whether the authors are claiming:

• an exact algebraic identity,
• a local Taylor approximation,
• an asymptotic expansion,
• a least-squares approximation,
• or another type of approximation procedure.

This distinction is fundamental because each interpretation leads to very different mathematical consequences. If the statement is intended as an approximation result, then the authors must specify: the domain on which the approximation is valid, the variables with respect to which the expansion is performed, the regularity assumptions required, the order of the approximation, and, most importantly, an explicit control of the associated error term.

At present, no remainder estimate or convergence analysis is provided. This omission is particularly problematic because the entire geometric interpretation developed later in the manuscript depends on the validity of this quadratic approximation. For example, if the approximation is only local, then the induced geometric conclusions may only be valid in a neighborhood of a reference point in parameter space. Conversely, if the approximation is global, then significantly stronger arguments are required. Without a rigorous discussion of these issues, it is difficult to evaluate the mathematical reliability of the geometric claims presented in the subsequent sections.

Second, Lemma 2 also requires a more detailed discussion. Although the coefficients of the quadratic polynomial are expressed in terms of E(X) and E(X^2), the mathematical significance of these coefficients is not sufficiently analyzed. The manuscript would be strengthened if the authors explained why these particular moments naturally arise, whether the coefficients possess an invariant geometric interpretation, and how they behave under transformations of the parameter space.

In addition, it would be useful to discuss whether these coefficients encode curvature-type information or whether they can be interpreted statistically in terms of variability, concentration, or stability of the family of distributions. At present, the role of these coefficients appears largely computational rather than conceptual.

Third, the overall structure of Section 3 should be reconsidered. The section contains the main mathematical developments of the paper, yet these developments are presented primarily through intermediate lemmas and informal discussion, without culminating in a precise theorem that clearly states the principal contribution of the article. This significantly weakens the mathematical organization of the manuscript.

In my opinion, the authors should reformulate Section 3 around at least one central theorem explicitly describing the main geometric result derived from the previous lemmas. Currently, the reader is left to infer the main conclusion indirectly from the surrounding discussion, which makes it difficult to identify the exact novelty of the paper. A more theorem-oriented structure would greatly improve both the readability and the mathematical rigor of the manuscript.

Fourth, the geometric framework itself requires additional development. Since the paper claims to adopt a differential-geometric perspective, the manuscript should provide a more systematic introduction to the geometric structures being considered. For instance, the authors should clarify: what the underlying manifold is, what coordinates are being used, whether the parameter space carries a smooth structure, what geometric objects are defined on this space, and which differential-geometric tools are being employed.

At present, the terminology of geometry is used somewhat informally, but many of the relevant geometric notions are not rigorously introduced. This makes it difficult to determine whether the proposed framework constitutes a fully developed geometric theory or only a geometric analogy. The paper would benefit substantially from a more explicit mathematical formalization of the geometric setting.

Fifth, the numerical analysis presented in Section 4 should be more tightly connected to the theoretical results obtained earlier in the paper. At present, the numerical section appears somewhat disconnected from the conceptual developments. 

Finally, the discussion and conclusion sections should be significantly expanded. The manuscript currently provides only a limited comparison with the existing literature, which makes it difficult to evaluate the originality of the contribution. In particular, the authors should situate their work more explicitly within the context of information geometry and geometric statistics. Several important questions remain insufficiently addressed:

• How does the proposed framework differ from classical Fisher-information-based geometry?
• Is the quadratic approximation related to existing second-order statistical approximations?
• What advantages does the proposed approach offer compared to established methods?
• Which aspects of the construction are genuinely new?

The authors should also discuss the limitations of their framework. For example, it would be important to clarify whether the proposed methods extend beyond Gaussian families and whether the approach remains valid for more general statistical models. Addressing these questions would considerably strengthen the conceptual impact of the paper.

In addition, the conclusion should include a more detailed discussion of open problems and future research directions. Since the paper introduces a potentially interesting geometric viewpoint, it would be valuable for the authors to indicate possible extensions of the theory, such as: applications to non-Gaussian statistical families, connections with information-theoretic divergences, higher-dimensional parameter spaces, or relationships with established geometric structures in statistics and machine learning.

Author Response

 

Reviewer 1.

 

The manuscript under review proposes a differential-geometric framework for the study of a parametrized family of Gaussian probability distributions, with particular emphasis on the geometric interpretation of the mean square error (MSE). More precisely, the authors consider a family X of Gaussian distributions parametrized by the mean and the standard deviation, and they investigate how the MSE associated with this family can be described through geometric methods. According to the presentation in the paper, Lemma 1 establishes that the mean square error can be represented, or approximated, by a quadratic polynomial, while Lemma 2 identifies the coefficients of this polynomial in terms of the moments E(X) and E(X^2). Based on these preliminary results, the authors then attempt to derive geometric consequences for the parameter space associated with the family of Gaussian distributions under consideration.

 

The underlying idea of introducing geometric methods into the study of parametrized probabilistic models is mathematically meaningful and potentially interesting. The use of differential-geometric tools to study spaces of parametrized objects has a long and well-established tradition in several areas of mathematics, including geometric invariant theory, moduli theory, and information geometry. In this context, the attempt to interpret statistical quantities such as the mean square error in geometric terms could lead to interesting perspectives, especially for researchers working at the intersection of probability theory, statistics, and geometric methods in information theory. In particular, understanding whether the MSE induces, approximates, or relates to a geometric structure on the parameter space could potentially provide conceptual insight into the local and global behavior of statistical models.

 

Response: We thank the reviewer for taking the time to fully read and understand the main arguments of our contribution.

 

Nevertheless, despite the conceptual interest of the proposed approach, I believe that the manuscript in its current form still suffers from several structural and mathematical issues that should be addressed before the paper can be considered for publication. My comments are detailed below.

 

Response: We explain in the rest of this response letter the changes we have included into all the reviewer’s issues. We hope that we have clarified most, if not all, of the reviewer’s more than justified concerns. We again thank the reviewer for pointing out the weakest points in our manuscript.

 

First, the formulation and interpretation of Lemma 1 require substantial clarification. The statement that the mean square error “is represented by” a quadratic polynomial is currently too vague from a mathematical point of view. It is not clear whether the authors are claiming:

 

an exact algebraic identity,

a local Taylor approximation,

an asymptotic expansion,

a least-squares approximation,

or another type of approximation procedure.

 

This distinction is fundamental because each interpretation leads to very different mathematical consequences. If the statement is intended as an approximation result, then the authors must specify: the domain on which the approximation is valid, the variables with respect to which the expansion is performed, the regularity assumptions required, the order of the approximation, and, most importantly, an explicit control of the associated error term.

 

Response: We sincerely thank the reviewer for pointing out this unclarified point. We have now added description of the rational behind the method, in Sect. 2 (Pags. 5 – 6). There, we discuss that the definition of MSE is based on a quadratic expression, and the fitting polynomial for MSE is always possible. Of course, the representation of a distribution in terms of MSE is limited, since it is based in first and second statistical momenta (we discuss this in Sect. 5) .

 

At present, no remainder estimate or convergence analysis is provided. This omission is particularly problematic because the entire geometric interpretation developed later in the manuscript depends on the validity of this quadratic approximation. For example, if the approximation is only local, then the induced geometric conclusions may only be valid in a neighborhood of a reference point in parameter space. Conversely, if the approximation is global, then significantly stronger arguments are required. Without a rigorous discussion of these issues, it is difficult to evaluate the mathematical reliability of the geometric claims presented in the subsequent sections.

 

Response: We thank the reviewer for pointing out the weak justification behind lemma 1 in the first version. We have now extended both the justification and its interpretation. We now give more details in the individual steps of the proof.

 

Second, Lemma 2 also requires a more detailed discussion. Although the coefficients of the quadratic polynomial are expressed in terms of E(X) and E(X^2), the mathematical significance of these coefficients is not sufficiently analyzed. The manuscript would be strengthened if the authors explained why these particular moments naturally arise, whether the coefficients possess an invariant geometric interpretation, and how they behave under transformations of the parameter space.

 

Response: We have now extended the interpretation and offered mode details for Lemma 2.

 

In addition, it would be useful to discuss whether these coefficients encode curvature-type information or whether they can be interpreted statistically in terms of variability, concentration, or stability of the family of distributions. At present, the role of these coefficients appears largely computational rather than conceptual.

 

Response: We thank the reviewer for this insightful observation. We have included a discussion about the curvature of the psi (k1,k2) space in Section 3, where we discuss the curvature of psi, defined by the lienar and constant parameters k1,k2.

 

Third, the overall structure of Section 3 should be reconsidered. The section contains the main mathematical developments of the paper, yet these developments are presented primarily through intermediate lemmas and informal discussion, without culminating in a precise theorem that clearly states the principal contribution of the article. This significantly weakens the mathematical organization of the manuscript.

 

In my opinion, the authors should reformulate Section 3 around at least one central theorem explicitly describing the main geometric result derived from the previous lemmas. Currently, the reader is left to infer the main conclusion indirectly from the surrounding discussion, which makes it difficult to identify the exact novelty of the paper. A more theorem-oriented structure would greatly improve both the readability and the mathematical rigor of the manuscript.

 

Response: We have modified Sect. 3. We added Lemma 3 to show the explicit relation between k2 and k1, the constant and linear coefficients of the MSE polynomial. We have also included additional differential geometry aspects such as curvature of the psi manifold. A new figure was included to extend the impact of curvature in the psi manifold.

 

Fourth, the geometric framework itself requires additional development. Since the paper claims to adopt a differential-geometric perspective, the manuscript should provide a more systematic introduction to the geometric structures being considered. For instance, the authors should clarify: what the underlying manifold is, what coordinates are being used, whether the parameter space carries a smooth structure, what geometric objects are defined on this space, and which differential-geometric tools are being employed.

 

Response: We have modified the text so more rigour is present in the defnitions and the interpretations.

 

At present, the terminology of geometry is used somewhat informally, but many of the relevant geometric notions are not rigorously introduced. This makes it difficult to determine whether the proposed framework constitutes a fully developed geometric theory or only a geometric analogy. The paper would benefit substantially from a more explicit mathematical formalization of the geometric setting.

 

Fifth, the numerical analysis presented in Section 4 should be more tightly connected to the theoretical results obtained earlier in the paper. At present, the numerical section appears somewhat disconnected from the conceptual developments.

 

Finally, the discussion and conclusion sections should be significantly expanded. The manuscript currently provides only a limited comparison with the existing literature, which makes it difficult to evaluate the originality of the contribution. In particular, the authors should situate their work more explicitly within the context of information geometry and geometric statistics. Several important questions remain insufficiently addressed:

 

How does the proposed framework differ from classical Fisher-information-based geometry?

Is the quadratic approximation related to existing second-order statistical approximations?

What advantages does the proposed approach offer compared to established methods?

Which aspects of the construction are genuinely new?

 

Response: We have extended the disucsion section, including the issues kindly raised by the reviewer.

 

The authors should also discuss the limitations of their framework. For example, it would be important to clarify whether the proposed methods extend beyond Gaussian families and whether the approach remains valid for more general statistical models. Addressing these questions would considerably strengthen the conceptual impact of the paper.

 

Response: Again, we thank the reviewer for this suggestion. We have addressed these issues and discussed about the limitations. For example, we have made explicit the limitations when third or higher statistical momenta are needed to describe a distribution.

 

In addition, the conclusion should include a more detailed discussion of open problems and future research directions. Since the paper introduces a potentially interesting geometric viewpoint, it would be valuable for the authors to indicate possible extensions of the theory, such as: applications to non-Gaussian statistical families, connections with information-theoretic divergences, higher-dimensional parameter spaces, or relationships with established geometric structures in statistics and machine learning.

 

Response: We thank the reviewr for these suggestions. We have extended this section.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

his paper proposes a novel manifold ψ induced by the polynomial coefficients of the mean squared error (MSE) of probability distributions, offering a new geometric viewpoint for distribution comparison within information geometry. The core idea is original and experimentally supported. However, several revisions in notation, logic, writing, and references are required to enhance mathematical rigor and academic quality.

1. Give the detailed explanation of the evaluation of MSE(a) in its support can be described by a second-degree polynomial on last line in page 4.
2. Add recent references focused on information geometry, statistical manifolds, and MSE-based geometric comparison; avoid over‑reliance on textbooks and older works.
3. In the introduction, the choice of example points lacks clear motivation. Please explain why certain points are selected instead of others to strengthen the illustrative logic.

4. where is the Figs.  4-A before page 7

5. Figure 4 is not clear, please provide a color image
6. Standardize and unify the writing of references, such as consistent abbreviations in magazines, etc

Author Response

Reviewer 2.

This paper proposes a novel manifold ψ induced by the polynomial coefficients of the mean squared error (MSE) of probability distributions, offering a new geometric viewpoint for distribution comparison within information geometry. The core idea is original and experimentally supported. However, several revisions in notation, logic, writing, and references are required to enhance mathematical rigor and academic quality.

 

Response: We thank the reviewer for the comments and suggestions made. We thank the reviewer for considering our proposal original.

 

Give the detailed explanation of the evaluation of MSE(a) in its support can be described by a second-degree polynomial on last line in page 4.

Add recent references focused on information geometry, statistical manifolds, and MSE-based geometric comparison; avoid over‑reliance on textbooks and older works.

In the introduction, the choice of example points lacks clear motivation. Please explain why certain points are selected instead of others to strengthen the illustrative logic.

 

Response: We thank the reviewer for raising these issues. We have extensively modified the paper to take into account the issues. For instance, we have extended the motivation behind our contribution.

 

where is the Figs. 4-A before page 7

Figure 4 is not clear, please provide a color image

 

Response: We have provided color images. Thanks.

 

Standardize and unify the writing of references, such as consistent abbreviations in magazines, etc

 

Response: We have updated some of the references. Thanks.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The authors have carefully addressed my comments. In my view, the nature of the approximations and convergence is now much clearer. Furthermore, the conclusions have been strengthened.

Reviewer 2 Report

Comments and Suggestions for Authors

Accept in present form