An Algorithm for Nonparametric Estimation of A Multivariate Mixing Distribution with Applications to Population Pharmacokinetics

In this paper we describe a nonparametric maximum likelihood (NPML) method for estimating multivariate mixing distributions. Given $N$ independent observations, convexity theory shows that the NPML estimator is discrete with at most $N$ support points. The original infinite NPML problem then becomes the finite dimensional problem of finding the location and probability of the support points. The probability of the support points is found by a Primal-Dual Interior-Point method; the location of the support points is found by an Adaptive Grid method. Our method is able to handle high-dimensional and complex multivariate mixture models.An important application is discussed for the problem of population pharmacokinetics and a non-trivial example is treated.Our algorithm has been successfully applied in hundreds of published pharmacometric studies. In addition to population pharmacokinetics, this research also applies to empirical Bayes estimation and many other areas of applied mathematics.

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An Algorithm for Nonparametric Estimation of a Multivariate Mixing Distribution with Applications to Population Pharmacokinetics

Pharmaceutics ◽

10.3390/pharmaceutics13010042 ◽

2020 ◽

Vol 13 (1) ◽

pp. 42

Author(s):

Walter M. Yamada ◽

Michael N. Neely ◽

Jay Bartroff ◽

David S. Bayard ◽

James V. Burke ◽

...

Keyword(s):

Drug Development ◽

Population Pharmacokinetics ◽

Empirical Bayes ◽

Applied Mathematics ◽

Grid Method ◽

Bayes Estimation ◽

Population Pharmacokinetic ◽

Support Points ◽

Primal Dual ◽

Log Normal

Population pharmacokinetic (PK) modeling has become a cornerstone of drug development and optimal patient dosing. This approach offers great benefits for datasets with sparse sampling, such as in pediatric patients, and can describe between-patient variability. While most current algorithms assume normal or log-normal distributions for PK parameters, we present a mathematically consistent nonparametric maximum likelihood (NPML) method for estimating multivariate mixing distributions without any assumption about the shape of the distribution. This approach can handle distributions with any shape for all PK parameters. It is shown in convexity theory that the NPML estimator is discrete, meaning that it has finite number of points with nonzero probability. In fact, there are at most N points where N is the number of observed subjects. The original infinite NPML problem then becomes the finite dimensional problem of finding the location and probability of the support points. In the simplest case, each point essentially represents the set of PK parameters for one patient. The probability of the points is found by a primal-dual interior-point method; the location of the support points is found by an adaptive grid method. Our method is able to handle high-dimensional and complex multivariate mixture models. An important application is discussed for the problem of population pharmacokinetics and a nontrivial example is treated. Our algorithm has been successfully applied in hundreds of published pharmacometric studies. In addition to population pharmacokinetics, this research also applies to empirical Bayes estimation and many other areas of applied mathematics. Thereby, this approach presents an important addition to the pharmacometric toolbox for drug development and optimal patient dosing.

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THE CONVERGENCE RATES OF EMPIRICAL BAYES ESTIMATION FOR PARAMETERS OF TWO-SIDED TRUNCATION DISTRIBUTION FAMILIES

Acta Mathematica Scientia ◽

10.1016/s0252-9602(18)30366-7 ◽

1989 ◽

Vol 9 (4) ◽

pp. 403-413

Author(s):

Laisheng Wei

Keyword(s):

Empirical Bayes ◽

Convergence Rates ◽

Bayes Estimation ◽

Empirical Bayes Estimation

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A Note on Bayes Empirical Bayes Estimation by Means of Dirichlet Processes.

10.21236/ada170039 ◽

1985 ◽

Author(s):

Lynn Kuo

Keyword(s):

Empirical Bayes ◽

Bayes Estimation ◽

Dirichlet Processes ◽

Empirical Bayes Estimation

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Empirical Bayes Estimation With Kernel Sequence Method

10.21236/ada396449 ◽

2001 ◽

Author(s):

Shanti S. Gupta ◽

Jinjun Lu

Keyword(s):

Empirical Bayes ◽

Bayes Estimation ◽

Empirical Bayes Estimation ◽

Sequence Method

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Modeling In-Match Sports Dynamics Using the Evolving Probability Method

Applied Sciences ◽

10.3390/app11104429 ◽

2021 ◽

Vol 11 (10) ◽

pp. 4429

Author(s):

Ana Šarčević ◽

Damir Pintar ◽

Mihaela Vranić ◽

Ante Gojsalić

Keyword(s):

Monte Carlo Simulation ◽

Statistical Models ◽

Empirical Bayes ◽

Hybrid Approach ◽

Real Life ◽

Bayes Estimation ◽

Probability Method ◽

Specific Nature ◽

Sport Event ◽

Insight Into

The prediction of sport event results has always drawn attention from a vast variety of different groups of people, such as club managers, coaches, betting companies, and the general population. The specific nature of each sport has an important role in the adaption of various predictive techniques founded on different mathematical and statistical models. In this paper, a common approach of modeling sports with a strongly defined structure and a rigid scoring system that relies on an assumption of independent and identical point distributions is challenged. It is demonstrated that such models can be improved by introducing dynamics into the match models in the form of sport momentums. Formal mathematical models for implementing these momentums based on conditional probability and empirical Bayes estimation are proposed, which are ultimately combined through a unifying hybrid approach based on the Monte Carlo simulation. Finally, the method is applied to real-life volleyball data demonstrating noticeable improvements over the previous approaches when it comes to predicting match outcomes. The method can be implemented into an expert system to obtain insight into the performance of players at different stages of the match or to study field scenarios that may arise under different circumstances.

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