Robust Bayes

In a regular application of Bayes s rule, a prior estimate of probability and a likelihood function are combined to produce a posterior estimate of probability, which may then be used as an input in a risk analysis. Bayes s rule is 

When there are many possible values of 0 and the prior p(0) is a probability distribution and the likelihood function p(E Q) is defined on the same axis, 

Application of Uncertainty Analysis to Ecological Risk of Pesticides 

FIGURE 6.1 Bayes combination of a prior distribution and a likelihood function to obtain a posterior distribution for 0. The vertical axis (not shown) is probability density. 

The classes specified in a robust Bayesian analysis can be defined in a variety of ways, depending on the nature of the analyst s uncertainty. For instance, one could specify parametric classes of distributions in one of the conjugate families (e.g., all the beta distributions having parameters in certain ranges). Alternatively, one could specify parametric classes of distributions but not take advantage of the conjugacies. 

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Robust Bayes redresses some of the most commonly heard criticisms of the Bayesian approach. For instance, robust Bayes relaxes the requirement for an analyst to specify a particular prior distribution and reflects the analyst s confidence about the choice of the prior. Bayesian methods generally preserve zero probabilities. That is, any values of the real line for which the prior distribution is surely zero will remain with zero probability in the posterior, no matter what the likelihood is and no matter what new data may arrive. This preservation of zero probabilities means that an erroneous prior conception about what is possible is immutable in the face of... 

Another potentially serious limitation of robust Bayes methods is that their computational costs can be large. The complexity of the requisite calculations depends on how the class of priors and the class of likelihoods are specified. In some cases, the use of computers may lessen the burden on human analysts, although there does not yet exist convenient software for this purpose. 

Finally, although both probability bounds analysis and robust Bayes methods are fully legitimate applications of probability theory and, indeed, both find their foundations in classical results, they may be controversial in some quarters. Some argue that a single probability measure should be able to capture all of an individual s uncertainty. Walley (1991) has called this idea the dogma of ideal precision. The attitude has never been common in risk analysis, where practitioners are governed by practical considerations. However, the bounding approaches may precipitate some contention because they contradict certain attitudes about the universal applicability of pure probability. 

Robust Bayes A school of thought among Bayesian analysts in which epistemic uncertainty about prior distributions or likelihood functions is quantified and projected through Bayes rule to obtain a class of posterior distributions. 

The group means and covariances can also be estimated robustly, for example, by the minimum covariance determinant (MCD) estimator (see Section 2.3.2). The resulting discriminant rule will be less influenced by outlying objects and thus be more robust (Croux and Dehon 2001 He and Fung 2000 Hubert and Van Driessen 2004). Note that Bayes discriminant analysis as described is not adequate if the data set has more variables than objects or if the variables are highly correlating, because we need to compute the inverse of the pooled covariance matrix in Equation 5.2. Subsequent sections will present methods that are able to deal with this situation. 

The importance of this result is that it leads to an overall objective criterion for sample size determination that averages criteria based on specific model assumptions. Thus it provides a solution that is robust to model uncertainty. Closed-form calculations of (8) are intractable, so we have developed numerical approximations to the conditional entropies Ent(6k n, yk, MLk) and Ent(9k n, yk, MGk). The computations of the expected Bayes risk are performed via stochastic simulations and the exact objective function is estimated by curve fitting as suggested by Miiller and Parmigiani (1995). These details are available on request from the authors. 

The linear mixed effect model assumes that the random effects are normally distributed and that the residuals are normally distributed. Butler and Louis (1992) showed that estimation of the fixed effects and covariance parameters, as well as residual variance terms, were very robust to deviations from normality. However, the standard errors of the estimates can be affected by deviations from normality, as much as five times too large or three times too small (Verbeke and Lesaffre, 1997). In contrast to the estimation of the mean model, the estimation of the random effects (and hence, variance components) are very sensitive to the normality assumption. Verbeke and Lesaffre (1996) studied the effects of deviation from normality on the empirical Bayes estimates of the random effects. Using computer simulation they simulated 1000 subjects with five measurements per subject, where each subject had a random intercept coming from a 50 50 mixture of normal distributions, which may arise if two subpopulations were examined each having equal variability and size. By assuming a unimodal normal distribution of the random effects, a histogram of the empirical Bayes estimates revealed a unimodal distribution, not a bimodal distribution as would be expected. They showed that the correct distributional shape of the random effects may not be observed if the error variability is large compared to the between-subject variability. 

A naive Bayes classifier is a simple probabilistic classifier based on the so-called Bayes theorem with strong independence assumptions and is particularly suited when the dimensionality of the inputs is high. The naive Bayes model assumes that, given a class r = j, the features X, are independent. Despite its simplicity, the naive Bayes classifier is known to be a robust method even if the independence assumption does not hold (Michalski and Kaufman, 2001). 

The adaptive MCMC simulation method is applied to updating the robust reliability for a two-story structural frame, depicted in Figure 2.21. The bay width and story height are 5.0 m and 2.5 m, respectively. The Young s modulus and mass density are taken to be 200 GPa and 7800 kg/m, respectively. The beams have a cross-sectional area of 0.01 vn5 and a moment of inertia of 6.0 X 10 m but they are 0.02 m and 1.5 x 10 m" for the columns. As a result, the structure has modal frequencies of 5.20 and 15.4 Hz. The structure is assumed to have 1.0% of critical damping for all modes. A simple model with two degrees of freedom is used in the system identification in order to estimate the inter-story stiffnesses and to assess the reliability of the structure. Specifically, the stiffness matrix is given by ... 

Box, G.E.P. 1980. Sampling and Bayes inference in scientific modelling and robustness (with discussion). J. Roy. Stat. Society A 143 383-430. 

San Francisco Bay Guardian, 02-28-1996, Epicenter The Plastics Inevitable, by Leighton Klein ... so producing plastic requires the constant extraction of oil, coal, and natural gas. Repairing anything made of plastic is nearly impossible, and the notion of recychng it is a robust fallacy at best. ... 

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