Maximum a posteriori

This approach is equivalent to the maximum a posteriori (MAP) approach derived by Wallner (Wallner, 1983). The position of the maximum is unchanged by a monotonic transformation and hence further simplification can be achieved by taking the logarithm of Eq. 8... 

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 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. 

As mentioned earlier, incorporating prior information does not in itself constitute a Bayesian approach. Priors have been used in non-Bayesian settings in population PK analysis and other analyses. Applications using the PRIOR subroutine in NONMEM have been described previously (3,16). In this setting the prior information can be viewed as a penalty on the likelihood function, and its implementation is similar in spirit to the maximum a posteriori (MAP) procedures used commonly... 

Gibbs sampling is a stochastic process and thus also provides several parameters to control program runtime. Gibbs uses the posterior probability of the alignment, the MAP (maximum a posteriori probability) (22), as a measure... 

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. 

Known Full Maximum likelihood, maximum a posteriori... 

Droppo, j., and Acero, a. Maximum a posteriori pitch tracking. In Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing 1998 (1998). 

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. 

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. 

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. 

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