MCMC techniques

Often, maximum-likelihood and maximum-posterior estimates are quite consistent and provide good predictive models for many practical applications (Needham et al., 2008). The confidence intervals for each parameter can also be obtained based either on maximum-likelihood or maximum-posterior estimators. Note again that the parameters for Bayesian networks may be learned even when the data information is incomplete using statistical techniques such as the EM algorithm and the Markov chain Monte Carlo (MCMC) technique (Murphy and Mian, 1999). 

This chapter has merely introduced the subject of Bayesian statistics, a field that is far broader than the subject of estimating parameters from data with normally-distributed errors discussed here. For further reading, comprehensive graduate-level overviews of Bayesian statistics are provided by Robert (2001) and Leonard Hsu (2001). A text suitable for undergraduates is Bolstad (2004). Among specialized texts. Box Tiao (1973) treats in further depth the problem of parameter estimation however, it does not discuss advanced MCMC techniques. For more on Bayesian Monte Carlo methods, consult Chen era/. (2000). For a more philosophical, conceptual treatment of Bayesian statistics see Bernardo Smith (2000). 

For a particular data set, subsets with high posterior probability must be identified. This can be a computational challenge with h possible effects, there are 2h different subsets. In order to identify promising subsets, MCMC methods for simulating from the posterior distribution on subsets may be used as a stochastic search. Section 3 outlines efficient techniques for exploring the subset space using MCMC methods. 

Once a PBPK model is developed and implemented, it should be tested for mass balance consistency, as weU as through simulated test cases that can highlight potential errors. These test cases often include software boundary conditions, such as zero dose and high initial tissue concentrations. Some parameters in the PBPK model may have to be estimated through available in vivo data via standard techniques such as nonlinear regression or maximum likelihood estimation (30). Furthermore, in vivo data can be used to update existing (or prior) PBPK model parameter estimates in a Bayesian framework, and thus help in the rehnement of the PBPK model. The Markov chain Monte Carlo (MCMC) (31-34) is one of the... 

Generating Samples in Robust Reliability Based on Adaptive MCMC Simulation Technique... 

---