Versationalis in the Enterprise Collaboration Industry

Versational

This paper is available on arxiv under CC BY-NC-ND 4.0 DEED license.
Authors:
(1) U Jin Choi, Department of mathematical science, Korea Advanced Institute of Science and Technology & ujchoi@kaist.ac.kr;
(2) Kyung Soo Rim, Department of mathematics, Sogang University & ksrim@sogang.ac.kr. Table of Links Introduction
Organization and notation
Problem Setting and Preliminaries
Generalized Wiener algebra and Fréchet derivative
Characterization of coefficients
Coefficients from an ergodic process
Conclusion & References Conclusion In this mathematical study, we delve into the realm of statistical inference and introduce a novel approach to variational non-Bayesian inference. Most significantly, we propose a new method for uniquely determining the hidden PDF solely from a random sample while leveraging theory. Beyond merely approximating the shape of the unrevealed PDF, we yield results that provide practical assistance in inferring important moments, such as the mean and variance, from the estimated PDF. These days, methods utilizing artificial intelligence are widely employed, but they generally exhibit a drawback of depending on initial conditions, i.e., a prior distribution, and iterative computations, i.e., a backpropagation. The limitations of such approaches lie in their reliance on these conditions to achieve global optimization. We emphasize the significant contribution of the proposed method to predictive and classification models, even without any information on populations, highlighting its potential applications in various domains such as finance, economics, weather forecasting, and machine learning, all of which present unique challenges. The most well-known method for approximating hidden PDFs is the variational approach in a Bayesian context. In contrast, our study extends this problem by seeking to precisely determine unknown PDFs through a system of equations. Our approach includes proving the Fréchet differentiability of entropy to establish the uniqueness of the energy function space in the Wiener algebra. We then derive the unique determination of the energy function through the minimization of KL-divergence. Leveraging the Ergodic theorem, we elucidate that solutions to equations comprising polynomial function series are the coefficients of the energy function and numerically substantiate the convergence of partial sums of the energy function obtained from a finite number of equations. In summary, our mathematical exploration has unveiled the potential of variational Non-Bayesian inference in Wiener space. We anticipate that these mathematical ideas can offer an innovative framework for probability density estimation and predictive modeling. Ultimately, our research emphasizes its potential contribution to reshaping mathematical horizons and expanding the boundaries of knowledge in the field of statistical methodology. REFERENCES [1] K. Bharath A. Saha and S. Kurtek, A geometric variational approach to bayesian inference, Journal of the American Statistical Association 115 (2020), no. 530, 822–835. [2] P. Alquier and J. Ridgway, Concentration of tempered posteriors and of their variational approximations, The Annals of Statistics 48 (2020), no. 3, 1475–1497. [3] A. R. Barron and C. Sheu, Approximation of density functions by sequences of exponential families, The Annals of Statistics 19 (1991), no. 3, 1347–1369. [4] C. M. Bishop and M. E. Tipping, Probabilistic principal component analysis, J. R. Statist. Soc. B 61 (1999), no. 3, 611–622. [5] F. W. Olver, D. Lozier, R. F. Boisvert and C. W. Clark, NIST Handbook of Mathematical Functions, Cambridge University Press, 2010. [6] C Brouder and F Patras, One-particle irreducibility with initial correlations, (2011). [7] I. Csiszar, I-divergence geometry of probability distributions and minimization problems, Ann. Probab. 3 (1975), no. 1, 146–158. [8] G. Cybenko, Approximation by superpositions of a sigmoidal function, Mathematics of Control, Signals and Systems 2 (1989), 303–314. [9] L. Grafakos, Classical and Modern Fourier Analysis, Prentice Hall, 2003. [10] D. E. Rumelhart, G. E. Hinton and R. Williams, Learning representations by back-propagating errors, Nature 323 (1986), 533–536. [11] G. E. Hinton, A Practical Guide to Training Restricted Boltzmann Machines, University of Toronto, 2010. [12] K. Hornik, Approximation capabilities of multilayer feedforward networks, Neural Networks 4 (1991), no. 2, 251–257. [13] Y. Katznelson, An introduction to harmonic analysis (third ed.), New York: Cambridge Mathematical Library, 2004. [14] D. M. Blei, A. Kucukelbir and J. D. McAuliffe, Variational inference: A review for statisticians, Jour. of the American Statistical Association 112 (2017), no. 518, 859–877. [15] S. Kullback and R. A. Leibler, On information and sufficiency, Ann. Math. Statistics 22 (1951), no. 1, 79–86. [16] H. M. Möller and R. Tenberg, Multivariate polynomial system solving using intersections of eigenspaces, Journal of Symbolic Computation 32 (2001), no. 5, 513–531. [17] B. Mourrain, V. Y. Pan and O. Ruatta, Accelerated solution of multivariate polynomial systems of equations, SIAM Journal on Computing 32 (2003), no. 2, 435–454. [18] W. Rudin, Functional Analysis, McGraw-Hill, 1991. [19] D. W. Stroock, Probability theory, an analytic view, Cambridge University Press, 1993. [20] B. Sturmfels, Solving systems of polynomial equations (cbms regional conference series in mathematics), Amer. Math. Soc., 2002. [21] N. Wiener, Tauberian theorems, Annals of Mathematics 33 (1932), no. 1, 1–100. [22] F. Zhang and C. Gao, Convergence rates of variational posterior distributions, The Annals of Statistics 48 (2020), no. 4, 2180–2207. This paper is available on arxiv under CC BY-NC-ND 4.0 DEED license. Authors: (1) U Jin Choi, Department of mathematical science, Korea Advanced Institute of Science and Technology & ujchoi@kaist.ac.kr; (2) Kyung Soo Rim, Department of mathematics, Sogang University & ksrim@sogang.ac.kr. This paper is available on arxiv under CC BY-NC-ND 4.0 DEED license. Authors: Authors: (1) U Jin Choi, Department of mathematical science, Korea Advanced Institute of Science and Technology & ujchoi@kaist.ac.kr; (2) Kyung Soo Rim, Department of mathematics, Sogang University & ksrim@sogang.ac.kr. Table of Links Introduction Organization and notation Problem Setting and Preliminaries Generalized Wiener algebra and Fréchet derivative Characterization of coefficients Coefficients from an ergodic process Conclusion & References Introduction Introduction Organization and notation Organization and notation Problem Setting and Preliminaries Problem Setting and Preliminaries Generalized Wiener algebra and Fréchet derivative Generalized Wiener algebra and Fréchet derivative Characterization of coefficients Characterization of coefficients Coefficients from an ergodic process Coefficients from an ergodic process Conclusion & References Conclusion & References Conclusion In this mathematical study, we delve into the realm of statistical inference and introduce a novel approach to variational non-Bayesian inference. Most significantly, we propose a new method for uniquely determining the hidden PDF solely from a random sample while leveraging theory. Beyond merely approximating the shape of the unrevealed PDF, we yield results that provide practical assistance in inferring important moments, such as the mean and variance, from the estimated PDF. These days, methods utilizing artificial intelligence are widely employed, but they generally exhibit a drawback of depending on initial conditions, i.e., a prior distribution, and iterative computations, i.e., a backpropagation. The limitations of such approaches lie in their reliance on these conditions to achieve global optimization. We emphasize the significant contribution of the proposed method to predictive and classification models, even without any information on populations, highlighting its potential applications in various domains such as finance, economics, weather forecasting, and machine learning, all of which present unique challenges. The most well-known method for approximating hidden PDFs is the variational approach in a Bayesian context. In contrast, our study extends this problem by seeking to precisely determine unknown PDFs through a system of equations. Our approach includes proving the Fréchet differentiability of entropy to establish the uniqueness of the energy function space in the Wiener algebra. We then derive the unique determination of the energy function through the minimization of KL-divergence. Leveraging the Ergodic theorem, we elucidate that solutions to equations comprising polynomial function series are the coefficients of the energy function and numerically substantiate the convergence of partial sums of the energy function obtained from a finite number of equations. In summary, our mathematical exploration has unveiled the potential of variational Non-Bayesian inference in Wiener space. We anticipate that these mathematical ideas can offer an innovative framework for probability density estimation and predictive modeling. Ultimately, our research emphasizes its potential contribution to reshaping mathematical horizons and expanding the boundaries of knowledge in the field of statistical methodology. REFERENCES [1] K. Bharath A. Saha and S. Kurtek, A geometric variational approach to bayesian inference, Journal of the American Statistical Association 115 (2020), no. 530, 822–835. [2] P. Alquier and J. Ridgway, Concentration of tempered posteriors and of their variational approximations, The Annals of Statistics 48 (2020), no. 3, 1475–1497. [3] A. R. Barron and C. Sheu, Approximation of density functions by sequences of exponential families, The Annals of Statistics 19 (1991), no. 3, 1347–1369. [4] C. M. Bishop and M. E. Tipping, Probabilistic principal component analysis, J. R. Statist. Soc. B 61 (1999), no. 3, 611–622. [5] F. W. Olver, D. Lozier, R. F. Boisvert and C. W. Clark, NIST Handbook of Mathematical Functions, Cambridge University Press, 2010. [6] C Brouder and F Patras, One-particle irreducibility with initial correlations, (2011). [7] I. Csiszar, I-divergence geometry of probability distributions and minimization problems, Ann. Probab. 3 (1975), no. 1, 146–158. [8] G. Cybenko, Approximation by superpositions of a sigmoidal function, Mathematics of Control, Signals and Systems 2 (1989), 303–314. [9] L. Grafakos, Classical and Modern Fourier Analysis, Prentice Hall, 2003. [10] D. E. Rumelhart, G. E. Hinton and R. Williams, Learning representations by back-propagating errors, Nature 323 (1986), 533–536. [11] G. E. Hinton, A Practical Guide to Training Restricted Boltzmann Machines, University of Toronto, 2010. [12] K. Hornik, Approximation capabilities of multilayer feedforward networks, Neural Networks 4 (1991), no. 2, 251–257. [13] Y. Katznelson, An introduction to harmonic analysis (third ed.), New York: Cambridge Mathematical Library, 2004. [14] D. M. Blei, A. Kucukelbir and J. D. McAuliffe, Variational inference: A review for statisticians, Jour. of the American Statistical Association 112 (2017), no. 518, 859–877. [15] S. Kullback and R. A. Leibler, On information and sufficiency, Ann. Math. Statistics 22 (1951), no. 1, 79–86. [16] H. M. Möller and R. Tenberg, Multivariate polynomial system solving using intersections of eigenspaces, Journal of Symbolic Computation 32 (2001), no. 5, 513–531. [17] B. Mourrain, V. Y. Pan and O. Ruatta, Accelerated solution of multivariate polynomial systems of equations, SIAM Journal on Computing 32 (2003), no. 2, 435–454. [18] W. Rudin, Functional Analysis, McGraw-Hill, 1991. [19] D. W. Stroock, Probability theory, an analytic view, Cambridge University Press, 1993. [20] B. Sturmfels, Solving systems of polynomial equations (cbms regional conference series in mathematics), Amer. Math. Soc., 2002. [21] N. Wiener, Tauberian theorems, Annals of Mathematics 33 (1932), no. 1, 1–100. [22] F. Zhang and C. Gao, Convergence rates of variational posterior distributions, The Annals of Statistics 48 (2020), no. 4, 2180–2207.

Variational Non-Bayesian Inference of the Probability Density Function: Conclusion & References

This paper is available on arxiv under CC BY-NC-ND 4.0 DEED license.
Authors:
(1) U Jin Choi, Department of mathematical science, Korea Advanced Institute of Science and Technology & ujchoi@kaist.ac.kr;
(2) Kyung Soo Rim, Department of mathematics, Sogang University & ksrim@sogang.ac.kr. Table of Links Introduction
Organization and notation
Problem Setting and Preliminaries
Generalized Wiener algebra and Fréchet derivative
Characterization of coefficients
Coefficients from an ergodic process
Conclusion & References PROOF. By Taylor’s series expansion of the exponential function, the triangle inequality, and the uniform convergence, we have This paper is available on arxiv under CC BY-NC-ND 4.0 DEED license. Authors: (1) U Jin Choi, Department of mathematical science, Korea Advanced Institute of Science and Technology & ujchoi@kaist.ac.kr; (2) Kyung Soo Rim, Department of mathematics, Sogang University & ksrim@sogang.ac.kr. This paper is available on arxiv under CC BY-NC-ND 4.0 DEED license. Authors: Authors: (1) U Jin Choi, Department of mathematical science, Korea Advanced Institute of Science and Technology & ujchoi@kaist.ac.kr; (2) Kyung Soo Rim, Department of mathematics, Sogang University & ksrim@sogang.ac.kr. Table of Links Introduction Organization and notation Problem Setting and Preliminaries Generalized Wiener algebra and Fréchet derivative Characterization of coefficients Coefficients from an ergodic process Conclusion & References Introduction Introduction Organization and notation Organization and notation Problem Setting and Preliminaries Problem Setting and Preliminaries Generalized Wiener algebra and Fréchet derivative Generalized Wiener algebra and Fréchet derivative Characterization of coefficients Characterization of coefficients Coefficients from an ergodic process Coefficients from an ergodic process Conclusion & References Conclusion & References PROOF. By Taylor’s series expansion of the exponential function, the triangle inequality, and the uniform convergence, we have

Variational Non-Bayesian Inference of the Probability Density: Characterization of Coefficients

This paper is available on arxiv under CC BY-NC-ND 4.0 DEED license.
Authors:
(1) U Jin Choi, Department of mathematical science, Korea Advanced Institute of Science and Technology & ujchoi@kaist.ac.kr;
(2) Kyung Soo Rim, Department of mathematics, Sogang University & ksrim@sogang.ac.kr. Table of Links Introduction
Organization and notation
Problem Setting and Preliminaries
Generalized Wiener algebra and Fréchet derivative
Characterization of coefficients
Coefficients from an ergodic process
Conclusion & References To state the main theorem of this section we need the following lemma. This completes the proof. This paper is available on arxiv under CC BY-NC-ND 4.0 DEED license. Authors: (1) U Jin Choi, Department of mathematical science, Korea Advanced Institute of Science and Technology & ujchoi@kaist.ac.kr; (2) Kyung Soo Rim, Department of mathematics, Sogang University & ksrim@sogang.ac.kr. This paper is available on arxiv under CC BY-NC-ND 4.0 DEED license. Authors: Authors: (1) U Jin Choi, Department of mathematical science, Korea Advanced Institute of Science and Technology & ujchoi@kaist.ac.kr; (2) Kyung Soo Rim, Department of mathematics, Sogang University & ksrim@sogang.ac.kr. Table of Links Introduction Organization and notation Problem Setting and Preliminaries Generalized Wiener algebra and Fréchet derivative Characterization of coefficients Coefficients from an ergodic process Conclusion & References Introduction Introduction Organization and notation Organization and notation Problem Setting and Preliminaries Problem Setting and Preliminaries Generalized Wiener algebra and Fréchet derivative Generalized Wiener algebra and Fréchet derivative Characterization of coefficients Characterization of coefficients Coefficients from an ergodic process Coefficients from an ergodic process Conclusion & References Conclusion & References To state the main theorem of this section we need the following lemma. This completes the proof.

Variational Non-Bayesian Inference: Generalized Wiener Algebra and Fréchet Derivative

This paper is available on arxiv under CC BY-NC-ND 4.0 DEED license.
Authors:
(1) U Jin Choi, Department of mathematical science, Korea Advanced Institute of Science and Technology & ujchoi@kaist.ac.kr;
(2) Kyung Soo Rim, Department of mathematics, Sogang University & ksrim@sogang.ac.kr. Table of Links Introduction
Organization and notation
Problem Setting and Preliminaries
Generalized Wiener algebra and Fréchet derivative
Characterization of coefficients
Coefficients from an ergodic process
Conclusion & References The left-hand sides of the equations (6.4) and (6.5) depend only on X, while the right-hand sides are comprised of sums of homogeneous polynomials in the components of y. Therefore, the equations reveal themselves as algebraic systems of polynomial series in the components of y. We summarize it as follows, in which the result does not assume a statistical model for the unrevealed PDF. In the following subsection, we will use a random sample drawn from the bivariate normal distribution as a hidden PDF to verify the numerical approximations. Two systems of (6.6) and (6.7) are composed of linear combinations of homogeneous polynomials. There have been many research results to approximate their solutions, and there are various algorithms available ([16, 17, 20]). In this experiment, we use the solve () function in MATLAB to find the solution. Finally, we demonstrate the feasibility of the methods presented with the following numerical results. This paper is available on arxiv under CC BY-NC-ND 4.0 DEED license. Authors: (1) U Jin Choi, Department of mathematical science, Korea Advanced Institute of Science and Technology & ujchoi@kaist.ac.kr; (2) Kyung Soo Rim, Department of mathematics, Sogang University & ksrim@sogang.ac.kr. This paper is available on arxiv under CC BY-NC-ND 4.0 DEED license. Authors: Authors: (1) U Jin Choi, Department of mathematical science, Korea Advanced Institute of Science and Technology & ujchoi@kaist.ac.kr; (2) Kyung Soo Rim, Department of mathematics, Sogang University & ksrim@sogang.ac.kr. Table of Links Introduction Organization and notation Problem Setting and Preliminaries Generalized Wiener algebra and Fréchet derivative Characterization of coefficients Coefficients from an ergodic process Conclusion & References Introduction Introduction Organization and notation Organization and notation Problem Setting and Preliminaries Problem Setting and Preliminaries Generalized Wiener algebra and Fréchet derivative Generalized Wiener algebra and Fréchet derivative Characterization of coefficients Characterization of coefficients Coefficients from an ergodic process Coefficients from an ergodic process Conclusion & References Conclusion & References The left-hand sides of the equations (6.4) and (6.5) depend only on X , while the right-hand sides are comprised of sums of homogeneous polynomials in the components of y . Therefore, the equations reveal themselves as algebraic systems of polynomial series in the components of y . We summarize it as follows, in which the result does not assume a statistical model for the unrevealed PDF. X y y In the following subsection, we will use a random sample drawn from the bivariate normal distribution as a hidden PDF to verify the numerical approximations. Two systems of (6.6) and (6.7) are composed of linear combinations of homogeneous polynomials. There have been many research results to approximate their solutions, and there are various algorithms available ([16, 17, 20]). In this experiment, we use the solve () function in MATLAB to find the solution. Finally, we demonstrate the feasibility of the methods presented with the following numerical results. solve ()

Variational Non-Bayesian Inference: Coefficients From an Ergodic Process

This paper is available on arxiv under CC BY-NC-ND 4.0 DEED license.
Authors:
(1) U Jin Choi, Department of mathematical science, Korea Advanced Institute of Science and Technology & ujchoi@kaist.ac.kr;
(2) Kyung Soo Rim, Department of mathematics, Sogang University & ksrim@sogang.ac.kr. Table of Links Introduction
Organization and notation
Problem Setting and Preliminaries
Generalized Wiener algebra and Fréchet derivative
Characterization of coefficients
Coefficients from an ergodic process
Conclusion & References 3. Problem Setting and Preliminaries. This section will set the mathematical model that is being addressed and provide basic background information relevant to understanding it. This paper is available on arxiv under CC BY-NC-ND 4.0 DEED license. Authors: (1) U Jin Choi, Department of mathematical science, Korea Advanced Institute of Science and Technology & ujchoi@kaist.ac.kr; (2) Kyung Soo Rim, Department of mathematics, Sogang University & ksrim@sogang.ac.kr. This paper is available on arxiv under CC BY-NC-ND 4.0 DEED license. Authors: Authors: (1) U Jin Choi, Department of mathematical science, Korea Advanced Institute of Science and Technology & ujchoi@kaist.ac.kr; (2) Kyung Soo Rim, Department of mathematics, Sogang University & ksrim@sogang.ac.kr. Table of Links Introduction Organization and notation Problem Setting and Preliminaries Generalized Wiener algebra and Fréchet derivative Characterization of coefficients Coefficients from an ergodic process Conclusion & References Introduction Introduction Organization and notation Organization and notation Problem Setting and Preliminaries Problem Setting and Preliminaries Generalized Wiener algebra and Fréchet derivative Generalized Wiener algebra and Fréchet derivative Characterization of coefficients Characterization of coefficients Coefficients from an ergodic process Coefficients from an ergodic process Conclusion & References Conclusion & References 3. Problem Setting and Preliminaries. This section will set the mathematical model that is being addressed and provide basic background information relevant to understanding it.

Variational Non-Bayesian Inference: Problem Setting and Preliminaries

This paper is available on arxiv under CC BY-NC-ND 4.0 DEED license.
Authors:
(1) U Jin Choi, Department of mathematical science, Korea Advanced Institute of Science and Technology & ujchoi@kaist.ac.kr;
(2) Kyung Soo Rim, Department of mathematics, Sogang University & ksrim@sogang.ac.kr. Table of Links Introduction
Organization and notation
Problem Setting and Preliminaries
Generalized Wiener algebra and Fréchet derivative
Characterization of coefficients
Coefficients from an ergodic process
Conclusion & References This paper presents a research study focused on uncovering the hidden population distribution from the viewpoint of a variational non-Bayesian approach. It asserts that if the hidden probability density function (PDF) has continuous partial derivatives of at least half the dimension’s order, it can be perfectly reconstructed from a stationary ergodic process: First, we establish that if the PDF belongs to the Wiener algebra, its canonical ensemble form is uniquely determined through the Fréchet differentiation of the KullbackLeibler divergence, aiming to minimize their cross-entropy. Second, we utilize the result that the differentiability of the PDF implies its membership in the Wiener algebra. Third, as the energy function of the canonical ensemble is defined as a series, the problem transforms into finding solutions to the equations of analytic series for the coefficients in the energy function. Naturally, through the use of truncated polynomial series and by demonstrating the convergence of partial sums of the energy function, we ensure the efficiency of approximation with a finite number of data points. Finally, through numerical experiments, we approximate the PDF from a random sample obtained from a bivariate normal distribution and also provide approximations for the mean and covariance from the PDF. This study substantiates the excellence of its results and their practical applicability. 1. Introduction. Statistical inference enables us to draw meaningful conclusions about a population’s distribution by studying a representative sample. Numerous facets of scientific research implicitly or explicitly require the estimation of population’s distribution from a sample. By analyzing a subset of individuals or observations, we can gather information that applies to the larger group, even when studying the entire population is impractical or impossible. Drawing from the works of two authors ([8, 12]), machine learning methods have gained popularity for predicting hidden information from data. Today, most emerging artificial neural networks incorporate statistical methods. Nevertheless, these computations do not achieve global optimization of learning results because of the inherent challenges related to setting initial conditions and handling the small magnitudes of high-dimensional differential values. To address this issue, many scholars have applied statistical inferences. Among these methods, variational Bayesian method is a commonly used technique for estimating posterior information. It’s worth noting that there are numerous excellent studies on the method. One can refer to the citations [1, 2, 14, 22] as examples of the most recent works. Our study aims to identify the hidden probability density function (PDF) without the need for a prior distribution in a non-Bayesian context. This is achieved solely from a stationary ergodic process using entropy minimization. Additionally, we aim to estimate moments of the PDF, such as the mean and variance. We provide a proof of the norm convergence of sequences of approximated PDFs from a finite sample size, which is useful in practical applications. In most probability models arising from natural phenomena, assuming the boundedness of a random variable within a specific window becomes reasonable when considering a sufficiently wide range of variability. Moreover, we assume ergodicity in the context of a stationary process, as is common in many scientific communities. The basic framework of the idea to find the hidden PDF is as follows: First, we express the PDF in the form of a canonical ensemble, resembling a mechanical heat system. Second, by embedding the problem in an infinite-dimensional function space, we establish the existence of the Fréchet derivative of an entropy induced from the energy function of the PDF (for more information on the Fréchet derivative, refer to [18]). Third, we uniquely determine the energy function by minimizing the Kullback-Leibler divergence (KL-divergence) ([7, 15]). The concept of KL-divergence, valuable in information theory, finds widespread use, even in artificial intelligence (for example, see [10], [4], [11]). Finally, we derive a system of polynomial series that corresponds to the sample means of complex exponentials and present their numerical solutions from random samples in a bivariate normal distribution. Using the approximated PDFs that we obtained, we compare approximations of the mean and variance from the bivariate normal distribution. This paper is available on arxiv under CC BY-NC-ND 4.0 DEED license. Authors: (1) U Jin Choi, Department of mathematical science, Korea Advanced Institute of Science and Technology & ujchoi@kaist.ac.kr; (2) Kyung Soo Rim, Department of mathematics, Sogang University & ksrim@sogang.ac.kr. This paper is available on arxiv under CC BY-NC-ND 4.0 DEED license. Authors: Authors: (1) U Jin Choi, Department of mathematical science, Korea Advanced Institute of Science and Technology & ujchoi@kaist.ac.kr; (2) Kyung Soo Rim, Department of mathematics, Sogang University & ksrim@sogang.ac.kr. Table of Links Introduction Organization and notation Problem Setting and Preliminaries Generalized Wiener algebra and Fréchet derivative Characterization of coefficients Coefficients from an ergodic process Conclusion & References Introduction Introduction Organization and notation Organization and notation Problem Setting and Preliminaries Problem Setting and Preliminaries Generalized Wiener algebra and Fréchet derivative Generalized Wiener algebra and Fréchet derivative Characterization of coefficients Characterization of coefficients Coefficients from an ergodic process Coefficients from an ergodic process Conclusion & References Conclusion & References This paper presents a research study focused on uncovering the hidden population distribution from the viewpoint of a variational non-Bayesian approach. It asserts that if the hidden probability density function (PDF) has continuous partial derivatives of at least half the dimension’s order, it can be perfectly reconstructed from a stationary ergodic process: First, we establish that if the PDF belongs to the Wiener algebra, its canonical ensemble form is uniquely determined through the Fréchet differentiation of the KullbackLeibler divergence, aiming to minimize their cross-entropy. Second, we utilize the result that the differentiability of the PDF implies its membership in the Wiener algebra. Third, as the energy function of the canonical ensemble is defined as a series, the problem transforms into finding solutions to the equations of analytic series for the coefficients in the energy function. Naturally, through the use of truncated polynomial series and by demonstrating the convergence of partial sums of the energy function, we ensure the efficiency of approximation with a finite number of data points. Finally, through numerical experiments, we approximate the PDF from a random sample obtained from a bivariate normal distribution and also provide approximations for the mean and covariance from the PDF. This study substantiates the excellence of its results and their practical applicability. 1. Introduction. Statistical inference enables us to draw meaningful conclusions about a population’s distribution by studying a representative sample. Numerous facets of scientific research implicitly or explicitly require the estimation of population’s distribution from a sample. By analyzing a subset of individuals or observations, we can gather information that applies to the larger group, even when studying the entire population is impractical or impossible. Drawing from the works of two authors ([8, 12]), machine learning methods have gained popularity for predicting hidden information from data. Today, most emerging artificial neural networks incorporate statistical methods. Nevertheless, these computations do not achieve global optimization of learning results because of the inherent challenges related to setting initial conditions and handling the small magnitudes of high-dimensional differential values. To address this issue, many scholars have applied statistical inferences. Among these methods, variational Bayesian method is a commonly used technique for estimating posterior information. It’s worth noting that there are numerous excellent studies on the method. One can refer to the citations [1, 2, 14, 22] as examples of the most recent works. Our study aims to identify the hidden probability density function (PDF) without the need for a prior distribution in a non-Bayesian context. This is achieved solely from a stationary ergodic process using entropy minimization. Additionally, we aim to estimate moments of the PDF, such as the mean and variance. We provide a proof of the norm convergence of sequences of approximated PDFs from a finite sample size, which is useful in practical applications. In most probability models arising from natural phenomena, assuming the boundedness of a random variable within a specific window becomes reasonable when considering a sufficiently wide range of variability. Moreover, we assume ergodicity in the context of a stationary process, as is common in many scientific communities. The basic framework of the idea to find the hidden PDF is as follows: First, we express the PDF in the form of a canonical ensemble, resembling a mechanical heat system. Second, by embedding the problem in an infinite-dimensional function space, we establish the existence of the Fréchet derivative of an entropy induced from the energy function of the PDF (for more information on the Fréchet derivative, refer to [18]). Third, we uniquely determine the energy function by minimizing the Kullback-Leibler divergence (KL-divergence) ([7, 15]). The concept of KL-divergence, valuable in information theory, finds widespread use, even in artificial intelligence (for example, see [10], [4], [11]). Finally, we derive a system of polynomial series that corresponds to the sample means of complex exponentials and present their numerical solutions from random samples in a bivariate normal distribution. Using the approximated PDFs that we obtained, we compare approximations of the mean and variance from the bivariate normal distribution.

Variational Non-Bayesian Inference of the Probability Density Function: Introduction

This paper is available on arxiv under CC BY-NC-ND 4.0 DEED license.
Authors:
(1) U Jin Choi, Department of mathematical science, Korea Advanced Institute of Science and Technology & ujchoi@kaist.ac.kr;
(2) Kyung Soo Rim, Department of mathematics, Sogang University & ksrim@sogang.ac.kr. Table of Links Introduction
Organization and notation
Problem Setting and Preliminaries
Generalized Wiener algebra and Fréchet derivative
Characterization of coefficients
Coefficients from an ergodic process
Conclusion & References 2. Organization and notation. This article is organized as follows: In the next section, we formulate the problem by defining the energy and partition function for the hidden PDF in a generalized Wiener algebra. We observe that the nonlinearity of energy functions poses challenges in maximizing likelihood functions. To overcome this issue, we adopt the KLdivergence between two distribution functions. In Section 4, we embed energy functions into a generalized Wiener algebra, thereby transforming the nonlinearity of the energy function into a system of equations in the function space and simultaneously converting the min-max problem into a problem of solving equations. We showthe Fréchet differentiability of the KL-divergence with respect to coefficients in the generalized Wiener algebra. From this property, we then uniquely obtain the existence such that its energy function yields a KL-divergence of 0. We establish a characterization of the coefficients of the energy function by expressing them in relation to the Fourier coefficients found within the (classical) Wiener algebra, as demonstrated in Section 5. This serves as an example of the generalized Wiener algebra. In the concluding Section 6, when dealing with a random sample that consists of realizations of a stationary ergodic process, we formulate a system of polynomial series with the coefficients of the energy function as variables, which consists of infinitely many equations in infinitely many variables. From the convergence property of the partial sums of an energy function in Section 4, the truncated system, consisting of a finite number of equations with partial sums, offers an avenue for coefficient approximations. The estimated coefficients effectively serve as estimators for the coefficients. More precisely, the hidden PDF can be approximated using these estimators. Furthermore, we provide a numerical example that emphasizes the approximation of the PDF, as well as the calculation of the mean and variance from the PDF using a random sample. This sample is generated from a bivariate normal distribution. In this paper, we consistently employ the following general notations (Table 1). Some notations will be elaborated upon in more detail when they are introduced. This paper is available on arxiv under CC BY-NC-ND 4.0 DEED license. Authors: (1) U Jin Choi, Department of mathematical science, Korea Advanced Institute of Science and Technology & ujchoi@kaist.ac.kr; (2) Kyung Soo Rim, Department of mathematics, Sogang University & ksrim@sogang.ac.kr. This paper is available on arxiv under CC BY-NC-ND 4.0 DEED license. Authors: Authors: (1) U Jin Choi, Department of mathematical science, Korea Advanced Institute of Science and Technology & ujchoi@kaist.ac.kr; (2) Kyung Soo Rim, Department of mathematics, Sogang University & ksrim@sogang.ac.kr. Table of Links Introduction Organization and notation Problem Setting and Preliminaries Generalized Wiener algebra and Fréchet derivative Characterization of coefficients Coefficients from an ergodic process Conclusion & References Introduction Introduction Organization and notation Organization and notation Problem Setting and Preliminaries Problem Setting and Preliminaries Generalized Wiener algebra and Fréchet derivative Generalized Wiener algebra and Fréchet derivative Characterization of coefficients Characterization of coefficients Coefficients from an ergodic process Coefficients from an ergodic process Conclusion & References Conclusion & References 2. Organization and notation. This article is organized as follows: In the next section, we formulate the problem by defining the energy and partition function for the hidden PDF in a generalized Wiener algebra. We observe that the nonlinearity of energy functions poses challenges in maximizing likelihood functions. To overcome this issue, we adopt the KLdivergence between two distribution functions. In Section 4, we embed energy functions into a generalized Wiener algebra, thereby transforming the nonlinearity of the energy function into a system of equations in the function space and simultaneously converting the min-max problem into a problem of solving equations. We showthe Fréchet differentiability of the KL-divergence with respect to coefficients in the generalized Wiener algebra. From this property, we then uniquely obtain the existence such that its energy function yields a KL-divergence of 0. We establish a characterization of the coefficients of the energy function by expressing them in relation to the Fourier coefficients found within the (classical) Wiener algebra, as demonstrated in Section 5. This serves as an example of the generalized Wiener algebra. In the concluding Section 6, when dealing with a random sample that consists of realizations of a stationary ergodic process, we formulate a system of polynomial series with the coefficients of the energy function as variables, which consists of infinitely many equations in infinitely many variables. From the convergence property of the partial sums of an energy function in Section 4, the truncated system, consisting of a finite number of equations with partial sums, offers an avenue for coefficient approximations. The estimated coefficients effectively serve as estimators for the coefficients. More precisely, the hidden PDF can be approximated using these estimators. Furthermore, we provide a numerical example that emphasizes the approximation of the PDF, as well as the calculation of the mean and variance from the PDF using a random sample. This sample is generated from a bivariate normal distribution. In this paper, we consistently employ the following general notations (Table 1). Some notations will be elaborated upon in more detail when they are introduced.

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