scholarly journals Integrals and integral transformations related to the vector Gaussian distribution

2020 ◽  
Vol 55 (4) ◽  
pp. 457-466
Author(s):  
V. S. Mukha ◽  
N. F. Kako
Keyword(s):  
Gaussian Distribution ◽  
Regression Function ◽  
Statistical Decision Theory ◽  
Total Probability ◽  
Statistical Decision ◽  
Multivariate Gaussian Distribution ◽  
Integral Transformations ◽  
Bayes Formula ◽  
Gauss Elimination ◽  
Bayesian Estimations

This paper is dedicated to the integrals and integral transformations related to the probability density function of the vector Gaussian distribution and arising in probability applications. Herein, we present three integrals that permit to calculate the moments of the multivariate Gaussian distribution. Moreover, the total probability formula and Bayes formula for the vector Gaussian distribution are given. The obtained results are proven. The deduction of the integrals is performed on the basis of the Gauss elimination method. The total probability formula and Bayes formula are obtained on the basis of the proven integrals. These integrals and integral transformations could be used, for example, in the statistical decision theory, particularly, in the dual control theory, and as table integrals in various areas of research. On the basis of the obtained results, Bayesian estimations of the coefficients of the multiple regression function are calculated.

scholarly journals The integrals and integral transformations connected with the joint vector Gaussian distribution

2021 ◽  
Vol 57 (2) ◽  
pp. 206-216
Author(s):  
V. S. Mukha ◽  
N. F. Kako
Keyword(s):  
Probability Density Function ◽  
Decision Theory ◽  
Probability Density ◽  
Density Function ◽  
Regression Function ◽  
Statistical Decision Theory ◽  
Total Probability ◽  
Statistical Decision ◽  
Gaussian Probability Density Function ◽  
Integral Transformations

In many applications it is desirable to consider not one random vector but a number of random vectors with the joint distribution. This paper is devoted to the integral and integral transformations connected with the joint vector Gaussian probability density function. Such integral and transformations arise in the statistical decision theory, particularly, in the dual control theory based on the statistical decision theory. One of the results represented in the paper is the integral of the joint Gaussian probability density function. The other results are the total probability formula and Bayes formula formulated in terms of the joint vector Gaussian probability density function. As an example the Bayesian estimations of the coefficients of the multiple regression function are obtained. The proposed integrals can be used as table integrals in various fields of research.

scholarly journals Тotal probability formula for vector gaussian distributions

Doklady BGUIR ◽  
2021 ◽  
Vol 19 (2) ◽  
pp. 58-64
Author(s):  
V. S. Mukha ◽  
N. F. Kako
Keyword(s):  
Decision Theory ◽  
Probability Density ◽  
Direct Calculation ◽  
Statistical Decision Theory ◽  
Total Probability ◽  
Mathematical Form ◽  
Statistical Decision ◽  
Technical Problems ◽  
Gaussian Distributions ◽  
Familiar Form

The total probability formula for continuous random variables is the integral of product of two probability density functions that defines the unconditional probability density function from the conditional one. The need for calculation of such integrals arises in many applications, for instant, in statistical decision theory. The statistical decision theory attracts attention due to the ability to formulate the problems in a strict mathematical form. One of the technical problems solved by the statistical decision theory is the problem of dual control that requires calculation of integrals connected with the multivariate probability distributions. The necessary integrals are not available in the literature. One theorem on the total probability formula for vector Gaussian distributions was published by the authors earlier. In this paper we repeat this theorem and prove a new theorem that uses more familiar form of the initial data and has more familiar form of the result. The new form of the theorem allows us to obtain the unconditional mathematical expectation and the unconditional variance-covariance matrix very simply. We also confirm the new theorem by direct calculation for the case of the simple linear regression.

scholarly journals Bayesian natural selection and the evolution of perceptual systems

2002 ◽  
Vol 357 (1420) ◽  
pp. 419-448 ◽  
Author(s):  
Wilson S. Geisler ◽  
Randy L. Diehl
Keyword(s):  
Natural Selection ◽  
Decision Theory ◽  
Ideal Observer ◽  
Statistical Decision Theory ◽  
Statistical Decision ◽  
Natural Stimuli ◽  
Ideal Observers ◽  
Formal Framework ◽  
Bayesian Statistical Decision Theory ◽  
Perceptual Systems

In recent years, there has been much interest in characterizing statistical properties of natural stimuli in order to better understand the design of perceptual systems. A fruitful approach has been to compare the processing of natural stimuli in real perceptual systems with that of ideal observers derived within the framework of Bayesian statistical decision theory. While this form of optimization theory has provided a deeper understanding of the information contained in natural stimuli as well as of the computational principles employed in perceptual systems, it does not directly consider the process of natural selection, which is ultimately responsible for design. Here we propose a formal framework for analysing how the statistics of natural stimuli and the process of natural selection interact to determine the design of perceptual systems. The framework consists of two complementary components. The first is a maximum fitness ideal observer, a standard Bayesian ideal observer with a utility function appropriate for natural selection. The second component is a formal version of natural selection based upon Bayesian statistical decision theory. Maximum fitness ideal observers and Bayesian natural selection are demonstrated in several examples. We suggest that the Bayesian approach is appropriate not only for the study of perceptual systems but also for the study of many other systems in biology.

Statistical decision theory

2019 ◽  
pp. 93-109 ◽  
Author(s):  
Elías Moreno ◽  
Francisco José Vázquez-Polo ◽  
Miguel Ángel Negrín-Hernández
Keyword(s):  
Decision Theory ◽  
Statistical Decision Theory ◽  
Statistical Decision
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