Use of Prior Information

Under certain conditions we may have some prior information about the parameter values. This information is often summarized by assuming that each parameter is distributed normally with a given mean and a small or large variance depending on how trustworthy our prior estimate is. The Bayesian objective function, SB(k), that should be minimized for algebraic equation models is 

We have assumed that the prior information can be described by the multivariate normal distribution, i.e., k is normally distributed with mean kB and co-variance matrix VB. 

The required modifications to the Gauss-Newton algorithm presented in Chapter 4 are rather minimal. At each iteration, we just need to add the following terms to matrix A and vector b, 

From a computer implementation point of view, this provides some extra flexibility to handle simultaneously parameters for which we have some prior knowledge and others for which no information is available. For the latter we simply need to input zero as the inverse of their prior variance. 

Practical experience has shown that (i) if we have a relatively large number of data points, the prior has an insignificant effect on the parameter estimates (ii) if the parameter estimation problem is ill-posed, use of prior information has a stabilizing effect. As seen from Equation 8.48, all the eigenvalues of matrix A are increased by the addition of positive terms in its diagonal. It acts almost like Mar-quadt s modification as far as convergence characteristics are concerned. 

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All three tasks are generally too complicated to be solved from first principles. They are, therefore, tackled by making use of prior information, and of information that has been condensed into knowledge. The amount of information that has to be processed is often quite large. At present, more than 41 million different compounds are known all have a series of properties, physical, chemical, or biological all can be made in many different ways, by a wide range of reactions all can be characterized by a host of spectra. This immense amount of information can be processed only by electronic means, by the power of the computer. 

Group comparison tests for proportions notoriously lack power. Trend tests, because of their use of prior information (dose levels) are much more powerful. Also, it is generally believed that the nature of true carcinogenicity (or toxicity for that matter), manifests itself as dose-response. Because of the above facts, evaluation of trend takes precedence over group comparisons. In order to achieve optimal test statistics, many people use ordinal dose levels (0,1,2..., etc.) instead of the true arithmetic dose levels to test for trend. However, such a decision should be made a priori. The following example demonstrates the weakness of homogeneity tests. 

An important lesson learned from the study is that there is a trade-off between maximizing prior information utilization and robustness concerning the accuracy of such information. Multivariate calibration methods range from explicit methods with maximum use of prior information (e.g., OLS, least robust when accurate model is not obtainable), hybrid methods with an inflexible constraint (e.g., HLA), hybrid methods with a flexible constraint (e.g., CR), and implicit methods with no prior information (e.g., PLS, most robust, but is prone to be misled by spurious correlations). We believe CR achieves the optimal balance between these ideals in practical situations. 

Gisleskog, P.O., Karlsson, M.O., and Beal, S.L. Use of prior information to stabilize a population data analysis. Journal of Pharmacokinetics and Pharmacodynamics 2003 29 473-505. 

Crossover designs have a number of problems that can invalidate their results. The chief difficulty concerns carryover, that is, the residual influence of treatments in subsequent treatment periods... When the crossover design is used it is therefore important to avoid carryover. This is best done by selective and careful use of the design on the basis of adequate knowledge of both the disease area and the new medication. The disease understudy should be chronic and stable. The relevant effects of the medication should develop fully within the treatment period. The washout periods should be sufficiently long for complete reversibility of drug effect. The fact that these conditions are likely to be met should be established in advance of the trial by means of prior information and data. ... 

In the resolution of any multicomponent system, the main goal is to transform the raw experimental measurements into useful information. By doing so, we aim to obtain a clear description of the contribution of each of the components present in the mixture or the process from the overall measured variation in our chemical data. Despite the diverse nature of multicomponent systems, the variation in then-related experimental measurements can, in many cases, be expressed as a simple composition-weighted linear additive model of pure responses, with a single term per component contribution. Although such a model is often known to be followed because of the nature of the instrumental responses measured (e.g., in the case of spectroscopic measurements), the information related to the individual contributions involved cannot be derived in a straightforward way from the raw measurements. The common purpose of all multivariate resolution methods is to fill in this gap and provide a linear model of individual component contributions using solely the raw experimental measurements. Resolution methods are powerful approaches that do not require a lot of prior information because neither the number nor the nature of the pure components in a system need to be known beforehand. Any information available about the system may be used, but it is not required. Actually, the only mandatory prerequisite is the inner linear structure of the data set. The mild requirements needed have promoted the use of resolution methods to tackle many chemical problems that could not be solved otherwise. 

Some of the motivations to pursue spectral simulation in a clinical MRS setting include providing metabolite prior information for use in parametric spectral analysis procedures, pulse sequence parameter optimization for observation of specific metabolite structures and shortening times for pulse sequence development. This section will describe, in some detail, examples of each with particular regard for the design and level of prior information inclusion of each simulation and the clinical use of the results. 

One widely used application of clinical MRS spectral simulation is the creation of prior information for spectral analysis and fitting routines. Well-defined metabolite prior information results in more consistent and complete estimations of the actual data. One example of a parametric model used to fit clinical MRS data is shown below. 

An important way to improve network performance is through the use of prior knowledge, which refers to information that one has about the desired form of the solution and which is additional to the information provided by the training data. Prior knowledge can be incorporated into the pre-processing and post-processing stages (Chapter 7), or into the network structure itself. 

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This paper discusses three major approaches in adaptive optimization that differ in the way adaptation is performed, namely (i) model-adaptation methods, where the measurements are used to refine the process model, and the updated model is used subsequently for optimization [7,17] (ii) modifier-adaptation methods, where modifier terms are added to the cost and constraints of the optimization problem, and measurements are used to update these terms [8,10,20] and (iii) direct-input-adaptation methods, where the inputs are adjusted by feedback controllers, hence not requiring optimization but a considerable amount of prior information regarding control design [9,21,25]. 

In their basic form DM exploit two types of prior information the positivity of the electron density map (this condition may be relaxed, e.g., for neutron diffraction, see Section 8.4.7), and the atomicity (the electrons are non-dispersed into the unit cell but concentrated around the nuclei). This information, apparently trivial, is very useful to succeed in all the steps of a modern DM procedure (1) scaling of the observed intensities and normalization of the structure factors (2) estimate of the structure invariants (3) application of the tangent formula (4) crystal structure completion and refinement. 

Furthermore, the distributions of the normalized structure factors are strongly affected by pseudotranslational symmetry, and for powder data also by preferred orientation effeets. The above information can also be used as prior information to perform a better powder pattern decomposition, so improving the efficieney of DM. 

Initially, the literature searcher should have a clear conception of the precise subject of his search, the time interval and sources to be covered, and the ultimate use of the information. Use of prior bibliographies often simplifies a search. Headings to be checked in subject indexes must be selected with care for each index examined, as subject headings vary in different indexes. The searcher must be discriminating in the selection of references and in establishing the point at which the search should be terminated. Presentation of results may vary depending on what form will be of most value to the reader. 

One criticism against the use of informative priors is their subjective nature, which may be perceived to introduce bias into the upcoming analysis. The choice of priors and assigning an appropriate level of informativeness is therefore of considerable importance. For population PK/PD studies, there may well be explicit, quantitative data that describes the parameter values in populations that are similar to the population in the current study. In this case it is possible to pool the available information in a meta-analytic technique to provide an appropriate level of prior information. Some care must be taken to assess for heterogeneity between studies and for applicability of studies to the current population under consideration. A brief summary of an approach is shown below. It would be impossible to include an exhaustive treatment of elicitation processes within the confines of this chapter. 

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