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

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by Bayesian InferenceApril 19th, 2024

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 & [email protected];

(2) Kyung Soo Rim, Department of mathematics, Sogang University & [email protected].

- Introduction
- Organization and notation
- Problem Setting and Preliminaries
- Generalized Wiener algebra and Fréchet derivative
- Characterization of coefficients
- Coefficients from an ergodic process
- Conclusion & References

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.

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