Bayesian parameter estimation

The posterior density function is the key to Bayesian parameter estimation, both for single-response and multiresponse data. Its mode gives point estimates of the parameters, and its spread can be used to calculate intervals of given probability content. These intervals indicate how well the parameters have been estimated they should always be reported. 

The activation energies in the expressions for G and Jt were obtained by Bayesian parameter estimation, which incorporated information from density functional... 

Determining the basic PK model that best describes the data and generating post hoc empiric individual Bayesian parameter estimates. 

In metabolism and nutrition, where each experiment has been designed to be complete and population analysis is used to fill in missing values and to incorporate relative uncertainties into the estimation, a good procedure would be to first examine in detail those individual studies which are the most complete. This will familiarize the user with the behavior of the model, produce initial estimates for the system parameters, provide a chance to verify that these values are reasonable, and allow the use of tools for the identifiability of individual experiments (Jacquez and Perry, 1990). After this exercise has been complete, all experiments, including those that are incomplete, can be pooled for population analysis and testing the effects of covariates. If required, the final step would be to use the estimated distributions to obtain Bayesian parameter estimates for the individual experiments. This procedure should yield the most appropriate estimates for the incomplete experiments. 

Gull 1989) addresses these problems classical or quantified MaxEnt (in contrast to the previously described method, now dubbed historical ). It has been implemented in the MEMSYS5 package. Imaging is treated as a Bayesian parameter estimation problem for the pixel intensities. The prior probability has the form P(f) a ), the posterior probability for data D is given by P(f ) oc with the notation of Section 3. a is... 

The general principle of Bayesian parameter estimation (Beck and Katafygiotis 1998) is that uncertainties in the model parameters... 

Jalayer E, De Risi R, Elefante L, Manfredi G (2013) Robust fragility assessmtait using Bayesian parameter estimation. Paper presented at the Vienna Congress on Recent Advances in Earthquake Engineering and Structural Dynamics 2013 (VEESD 2013), Vienna, 28-30 Aug... 

However, in most cases the handling ofpg 2(0 z) and its integration in Equation 18.20 is analytically intractable and the sophisticated but computationally demanding numerical simulation techniques called Markov chain Monte Carlo must be used to determine Bayesian parameter estimates and their confidence intervals see, for example, Pillonetto et al. (2002) and Magni et al. (2001) for two recent applications. 

One drag level (Cindiv) can be used with the means and standard deviation (SD) of population parameters (Ppop) as a priori knowledge for an individual parameter estimate using the Bayesian objective function. 

If we consider the relative merits of the two forms of the optimal reconstructor, Eq. s 16 and 17, we note that both require a matrix inversion. Computationally, the size of the matrix inversion is important. Eq. 16 inverts an M x M (measurements) matrix and Eq. 17 a P x P (parameters) matrix. In a traditional least squares system there are fewer parameters estimated than there are measurements, ie M > P, indicating Eq. 16 should be used. In a Bayesian framework we are hying to reconstruct more modes than we have measurements, ie P > M, so Eq. 17 is more convenient. 

For simplicity and in order to avoid potential misrepresentation of the experimental equilibrium surface, we recommend the use of 2-D interpolation. Extrapolation of the experimental data should generally be avoided. It should be kept in mind that, if prediction of complete miscibility is demanded from the EoS at conditions where no data points are available, a strong prior is imposed on the parameter estimation from a Bayesian point of view. 

Bayesian statistics are applicable to analyzing uncertainty in all phases of a risk assessment. Bayesian or probabilistic induction provides a quantitative way to estimate the plausibility of a proposed causality model (Howson and Urbach 1989), including the causal (conceptual) models central to chemical risk assessment (Newman and Evans 2002). Bayesian inductive methods quantify the plausibility of a conceptual model based on existing data and can accommodate a process of data augmentation (or pooling) until sufficient belief (or disbelief) has been accumulated about the proposed cause-effect model. Once a plausible conceptual model is defined, Bayesian methods can quantify uncertainties in parameter estimation or model predictions (predictive inferences). Relevant methods can be found in numerous textbooks, e.g., Carlin and Louis (2000) and Gelman et al. (1997). 

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. 

Bayesian probability theory157 can also be applied to the problem of NMR parameter estimation this approach incorporates prior knowledge of the NMR parameters and is particularly useful at short aquisition times158 and when the FID contains few data points.159 Bayesian analysis gives more precise estimates of the NMR parameters than do methods based on the discrete Fourier transform (DFT).160 The amplitudes can be estimated independently of the phase, frequency and decay constants of the resonances.161 For the usual method of quadrature detection, it is appropriate to apply this technique to the two quadrature signals in the time domain.162-164... 

In this chapter, Bayesian and likelihood-based approaches have been described for parameter estimation from multiresponse data with unknown covariance matrix S. The Bayesian approaches permit objective estimates of 6 and E by use of the noninformative prior of Jeffreys (1961). Explicit estimation of unknown covariance elements is optional for problems of Types 1 and 2 but mandatory for Types 3 and 4. 

Bayesian two-stage Rich data, sparse data, or a mixture of both Properly accounts for data imbalance. Good parameter estimates are usually obtained Computationally intense... 

Sensitivity analysis is about asking how sensitive your model is to perturbations of assumptions in the underlying variables and structure. Models developed under any platform should be subject to some form of sensitivity analysis. Those constructed under a Bayesian framework may be subject to further sensitivity analysis associated with assumptions that may be made in the specihcation of the prior information. In general, therefore, a sensitivity analysis will involve some form of perturbation of the priors. There are generally scenarios where this may be important. First, the choice of a noninformative prior could lead to an improper posterior distribution that may be more informative than desired (see Gelman (18) for some discussion on this). Second, the use of informative priors for PK/PD analysis raises the issue of introduction of bias to the posterior parameter estimates for a specihed subject group that is, the prior information may not have been exchangeable with the current data. 

Maitre et al. (15) proposed an improvement on the traditional approach. The approach consists of using individual Bayesian posthoc PK or PK/PD parameters from a population modeling software such as NONMEM and plotting these parameter estimates against covariates to look for any possible model parameter covariate relationship. The individual model parameter estimates are obtained using a base model—a model without covariates. The covariates are in turn tested to determine individual significant covariate predictors, which are in turn used to form a full model. The final irreducible model is obtained by backward elimination. The drawback for this approach is the same as that for the traditional approach. 

Step 2. Plot the individual Bayesian parameters versus covariates (those estimated from the base model in step 1). 

Step 2. Determination of a basic PK (or PK/PD) model using NONMEM, for example, and obtaining Bayesian individual parameter estimates. 

With GAM the data (covariate and individual Bayesian PM parameter estimates) would be subjected to a stepwise (single-term addition/deletion) modeling procedure. Each covariate is allowed to enter the model in any of several functional representations. The Akaike information criterion (AIC) is used as the model selection criterion (22). At each step, the model is changed by addition or deletion of a covariate that results in the largest decrease in the AIC. The search is stopped when the AIC reached a minimum value. 

NONMEM was used to estimate the parameters for each bootstrap data set. Individual Bayesian parameters were generated. These estimates along with covariates formed a new data set. 

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... 

In many cases, if the prior knowledge of 0 is vague, such a prior is said to be diffuse or uninformative, and the Bayesian estimates for the model parameters simplify to the frequentist estimate of the model parameters. In other words, as the standard deviation of the prior distribution increases, the posterior parameter estimates will tend to converge to the parameter estimates in the absence of such prior knowledge. For example, a 1-compartment model with absorption was fit to the data in Table 3.5 using SAAM II where the model was now reparameterized in terms of clearance and not kio. Equal weights were assumed and starting values of 5000 mL/h, 110 L, and 1.0 per hour for Cl, Vd, and ka, respectively, were used. 

Figure 3.13 Model parameter estimates as a function of the prior standard deviation for clearance. A 1-compartment model with absorption was fit to the data in Table 3.5 using a proportional error model and the SAAM II software system. Starting values were 5000 mL/h, 110 L, and 1.0 per hour for clearance (CL), volume of distribution (Vd), and absorption rate constant (ka), respectively. The Bayesian prior mean for clearance was fixed at 4500 mL/h while the standard deviation was systematically varied. The error bars represent the standard error of the parameter estimate. The open symbols are the parameter estimates when prior information is not included in the model.

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