Hierarchical Markov models

Girard et al. (34) proposed a hierarchical Markov model for patient compliance with oral medications that was conditioned on a set of individual-specific nominal daily dose times. The individual random effects for the model were assumed to be multivariate normally distributed. Assuming first-order Markov hypothesis (see... 

Probabilistic models have been developed for characterizing compliance. The most commonly cited probabilistic approach is the hierarchical Markov model. Other more recently developed approaches range from a random sampling probabilistic model approach, to likelihood approaches, Bayesian approaches, and a missing dosing history approach. It is up to the pharmacometrician to choose the method that would best characterize his/her nonadherence data. The application example reinforces the importance of compliance to prescribed drug therapy, and how steady-state pharmacokinetics can be disrupted in the presence of noncompliance. 

Failure to account for nonadherence to study drug administration schedules will lead to biased and imprecise trial simulation outcome measures (19). Models to assimilate compliance often involve a hierarchical Markov model, where the probability for an individual to take a scheduled dose is conditional on whether this individual had taken the previous dose (20,21). The model may also contain covariates as predictors of compliance. For example, compliance has been shown to be affected by dosing frequency, where an increased frequency (e.g., three times daily vs. once daily) has been associated with worse compliance (22, 23). Alternatively, the consequence of missing a once-a-day dose may have more significant impact on efficacy. PK/PD-based simulations play an important role in understanding the balance of these situations. 

Carlin, B.P. Hierarchical longitudinal modelling. In Markov Chain Monte Carlo in practice. (Gilks, W.R., Richardson,... 

Usual microscopic models of chemical reaction assumes the existence of ordered energy barriers. Chemical reaction is used to be considered as diffusion in the phase space, Kramers (1940). It is not clear, whether how to switch such kinds of microscopic models to traditional CDS models. One of the main open problems in chemical kinetic is to derive CDS model from microscopic picture. It seems to be credible, that ordered energy barriers can lead to traditional Markov models. Random fluctuations in the energy barrier can have ultrametric structure (e.g. Zumofen et al 1986) and may lead to much more complex, i.e. hierarchical dynamic models (Vilgis 1987). Chemical reaction can be interpreted as anomalous, i.e. ultradiffusion (Huberman and Kerszberg 1985).While normal diffusion is characterized by the relation... 

In the next subsection, I describe how the basic elements of Bayesian analysis are formulated mathematically. I also describe the methods for deriving posterior distributions from the model, either in terms of conjugate prior likelihood forms or in terms of simulation using Markov chain Monte Carlo (MCMC) methods. The utility of Bayesian methods has expanded greatly in recent years because of the development of MCMC methods and fast computers. I also describe the basics of hierarchical and mixture models. 

Bayesian Hierarchical Modeling with Markov Chain Monte Carlo Methods... 

BAYESIAN HIERARCHICAL MODELING WITH MARKOV CHAIN MONTE CARLO METHODS... 

Hierarchical protein strucmre superposition STRUCTAL-based program Markov transition model of evolution... 

We should always use proper priors in the hierarchical model, particularly for scale parameters. When improper priors are used in the hierarchical model and the Gibbs sampler is used, each node looks like it has a proper posterior. However, overall, the posterior is improper. This means the Gibbs sampler represents a Markov chain that has all null recurrent states so it does not have a steady state distribution to converge to. 

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