Maximum a posteriori estimation

The maximum a posteriori estimation method (the Bayes estimation)... 

Seebauer, E.G. and Braatz, R.D. (2003a) Maximum A Posteriori Estimation of Transient Enhanced Diffusion Energetics. AIChEJ., 49, 2114-2123. 

In Trawny, Roumeliotis, and Giannakis (2005), the problem of cooperative localization under severe communication constraints is addressed. Specifically, both a minimiun mean square error and maximum a posteriori estimators are considered. The filters are able to cope with quantized process measurements, since during navigation, each robot quantizes and broadcasts its measurements and receives the quantized observations of its teammates. 

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]. 

Nonparametric EM algorithm within USC PACK suite of programs. Initially, a defined parameter space with an uninformed parameter joint density is used with individual subject data to calculate a new joint density. An iterative two-stage Bayesian algorithm (IT2B) that computes maximum a posteriori (MAP) individual parameter estimates based on population priors is also provided. 

From the function of Equation 9.19 several estimators can be defined. For instance, the Maximum a Posteriori (MAP) estimate is found by determining the value of 0, which renders p0 (0 z) maximum, while the Mean Square (MS) estimate is defined as the expected value of 0, given the data vector ... 

Sparacino, G., Tombolato, C., and CobeUi, C. 2000. Maximum-HkeHhood versus maximum a posteriori parameter estimation of physiological system models the C-peptide impulse response case study. IEEE Trans. Biom. Eng. 47 801-811. 

A convenient and popular summary of such a posterior distribution is the maximum probability interpolant (known as the MAP, maximum a posteriori probability estimate). If this is calculated using the calculus of variations, then a minimisation problem, similar to that of the Tikhonov methods is obtained. In the Bayesian formulation however, the free parameters need less ad hoc arguments for their assignment and have a clearer interpretation. 

As the Bayesian formulation was described in Section 5 it is sufficient to recall the main uses of the formulation in the maximum a posteriori (MAP) mode or in the stochastic sampling mode. The maximum likelihood estimation method is obtained by setting the prior to unity in the MAP method. The MLE method is essentially the least squares method. Without a suitable choice of prior it may be necessary to introduce further ad hoc regularisation in the case of MLE. A carefully chosen prior should regularise the problem in a satisfactory way. 

The signal has parameters 9, which are to be estimated, but the signal is also masked by additive noise, that is, the observation is x t) = As t 9) + n t). We assume p x 9) is known, but desire to estimate the most Ukely estimate of 9 based on the observations, that is 0 (x). The maximum likelihood estimator (MLE) maximizes the a posteriori probabUity density function p 9 x), which is related to the known a priori pdf p(x 9) via Bayes rule. Rather than use p(9 x), one uses log(p(x 0)) since it is a monotonic function yielding the same best estimate but simplifies the arithmetic. In the case of Gaussian noise, the MLE reduces to the problem of estimation by weighted LMS. 

In the Fisher approach, parameter estimates can be obtained by nonlinear least squares or maximum likelihood together with their precision, such as, a measure of a posteriori or numerical identifiabihty. Details and references on parameter estimation of physiologic system models can be found in Carson et al. [1983] and Landaw and DiStefano [1984]. Weighted nonlinear least squares is mostly used, in which an estimate 9 of the model parameter vector 0 is determined as... 

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