Estimation, Bayesian

Bayes theorem is fundamental in parameter estimation. Published over two centuries ago (Bayes 1763) from the last work of an English clergyman, this theorem is a powerful tool for data-based analysis of mathematical models. 

FYom observations and any available prior information, Bayes theorem infers a probability distribution for the parameters of a postulated model. This posterior distribution tells all that can be inferred about the parameters on the basis of the given information. From this function the most probable parameter values can be calculated, as well as various measures of the precision of the parameter estimation. The same can be done for any quantity predicted by the model. 

Bayes demonstrated his theorem by inferring a posterior distribution for the parameter p of Eq. (4.3-2) from the observed number k of successes in n Bernoulli trials. His distribution formula expresses the probability, given k and n, that p lies somewhere between any two degrees of probability that can be named. The subtlety of the treatment delayed its impact until the middle of the twentieth century, though Gauss (1809) and Laplace (1810) used related methods. Stigler (1982, 1986) gives lucid discussions of Bayes classic paper and its various interpretations by famous statisticians. 

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Colombo, A. G. and O. Saracco. Bayesian Estimation of the Time-independent Failure Rate of an Item Taking into Account Its Quality and Operational Constraints. Proceedings of the 4th Euredata Conference, 1983. 

Mosleh, Kazarians, and Gekler obtained a Bayesian estimate of the failure rate, Z, of a coolant recycle pump in llie hazard/risk study of a chemical plant. The estimate was based on evidence of no failures in 10 years of operation. Nuclear industry experience with pumps of similar types was used to establish tire prior distribution of Z. Tliis experience indicated tliat tire 5 and 95 percentiles of lire failure rate distribution developed for tliis category were 2.0 x 10" per hour (about one failure per 57 years of operation) and 98.3 x 10 per hour (about one failure per year). Extensive experience in other industries suggested the use of a log-nonnal distribution witli tlie 5 and 95 percentile values as llie prior distribution of Z, tlie failure rate of the coolant recycle pump. 

The mean of tlie posterior distribution of Z is Bayesian estimate of the failure rate per year. If E(Z B) is tlie mean of the posterior distribution, then... 

Tlie number of defective items in a sample of size n produced by a certain machine has a binomial distribution with parameters n and p, where p is the probability tliat an item produced is defective. For die case of 2 observed defectives in a sample of size 20, obtain die Bayesian estimate of p if die prior distribution of p is specified by the pdf... 

Isepamicin Population PK and Bayesian estimates of individual PK parameters were used to calculate various surrogate markers of isepamicin exposure to be tentatively correlated with clinical outcome and nephrotoxicity. No correlation was found between peak, AUC, or their ratio with MIC and clinical efficacy suggesting the drug should not be monitored using PK parameters... 

Zobrist J, Reichert P (2006) Bayesian estimation of export coefficients from diffuse and point sources in Swiss watersheds. J Hydrol 329 207-223... 

L, volume of peripheral compartment is 3.4 L. The estimated terminal elimination half-life for the reference patient was 20 days (480 hours), which is similar to the terminal elimination half-life for human IgG (18 to 23 days). Bayesian estimates of terminal elimination half-life ranged from 11 to 38 days for the 123 patients included in the population analysis. 

Kloek T, Van Dijk HK. 1978. Bayesian estimates of equation system parameters an application of integration by Monte Carlo. Econometrics 46 1-20. 

Conjugate pair In Bayesian estimation, when the observation of new data changes only the parameters of the prior distribution and not its statistical shape (i.e., whether it is normal, beta, etc.), the prior distribution on the estimated parameter and the distribution of the quantity (from which observations are drawn) are said to form a conjugate pair. In case the likelihood and prior form a conjugate pair, the computational burden of Bayes rule is greatly reduced. 

Merle, Y. and Mentre, F. (1999) Optimal sampling times for Bayesian estimation of the pharmacokinetic parameters of nortriptyline during therapeutic drug monitoring. / Pharmacokinet Biopharm 27 85-101. 

This is the net Bose-Einstein factor. It will be used in the following to form Bayesian estimators (see, e.g., Frieden, 1983) of the object nm. ... 

G.E.P. Box and N.R. Draper, The Bayesian estimation of common parameters from several responses. Biometrika, 52 (1965) 355-365. 

Asymptotics take on a different meaning in the Bayesian estimation context, since parameter estimators do not converge to a population quantity. Nonetheless, in a Bayesian estimation setting, as the sample size increases, the likelihood function will dominate the posterior density. What does this imply about the Bayesian estimator when this occurs. 

The Bayesian estimator must converge to the maximum likelihood estimator as the sample size grows. The posterior mean will generally be a mixture of the prior and the maximizer of the likelihood function. We do note, however, that the likelihood will only dominate an informative prior asymptotically - the Bayesian estimator in this case will ultimately be a mixture of a prior with a finite precision and a likelihood based estimator whose variance converges to zero (thus, whose precision grows infinitely). Thus, the domination will not be complete in a finite sample. 

In most models developed for pharmacokinetic and pharmacodynamic data it is not possible to obtain a closed form solution of E(yi) and var(y ). The simplest algorithm available in NONMEM, the first-order estimation method (FO), overcomes this by providing an approximate solution through a first-order Taylor series expansion with respect to the random variables r i,Kiq, and Sij, where it is assumed that these random effect parameters are independently multivariately normally distributed with mean zero. During an iterative process the best estimates for the fixed and random effects are estimated. The individual parameters (conditional estimates) are calculated a posteriori based on the fixed effects, the random effects, and the individual observations using the maximum a posteriori Bayesian estimation method implemented as the post hoc option in NONMEM [10]. 

The following chapters and the package GREGPLUS apply these principles to practical models and various data structures. Least squares, multiresponse estimation, model discrimination, and process function estimation are presented there as special forms of Bayesian estimation. 

An objective function S(6) is presented here for use in Bayesian estimation of the parameter vector 0 in a mathematical model... 

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