Bayes risk

The solutions are found by averaging out and folding back (Raiffa and Schlaifer, 1961), so that we compute the expected loss at the chance nodes (open circles), given everything to the left of the node. We determine the best actions by minimizing the expected loss at the decision nodes. The first decision is the choice of the inference method a and the optimal decision a (or Bayes action) is found by minimizing the expected loss E L(n, 6, y, a, c), where the expectation is with respect to the conditional distribution of 0 given n and y. The expected loss evaluated in the Bayes action a is called the Bayes risk and we denote it by... 

This quantity is also a function of the data y, and the optimal sample size is chosen by minimizing the expected Bayes risk E R(n, y, a, c), where this expectation is with respect to the marginal distribution of the data. 

As stated above, in order to choose the optimal sample size n = i + 2> we need to minimize with respect to n and 2 the expected Bayes risk (where tt and 2 are the numbers of tissue samples for conditions 1 and 2) that is,... 

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. 

Figure 6 shows an example of the stochastic estimation of the Bayes risk as a function of the sample sizes ti and 2, where the data were obtained by resampling from the data set of 102 prostatectomy samples described in Section 6.1. From the results on the reproducibility, we estimated that a sample of size n induces a reproducibility of about (22.5 log( ) — 4)%, so we used as loss function... 

Figure 6. (a) The surface for the estimated Bayes risk (vertical axis) as a function of the number of samples n and n2 for the two conditions (b) the contour plot of the same surface. 

The Bayes risk of the test strategy is the strategy s mean risk for all states ... 

A Bayes test, say Sjr, which minimizes the Bayes risks among all tests can be obtained as follows For each Sj. Thus, we need to have estimators for H (e, z). 

The minimum Bayes risk of this testing problem is then given by... 

It is not difficult to derive the following expressions for the risk function and the Bayes risk ... 

Abramowicz-Gerigk T., Burciu Z. 2012. Analysis of trends in navigational safety assessment an empirical and Bayes risk-based approach. 9th International Conference on Hydrodynamics in Ship Design 4th International Symposium on Ship Manoeuvring 7-12, September, Ilawa, Poland. 

HORN-ROSS P L, HOGGATT K J, LEE M M (2002) Phytoestrogeus and thyroid cancer risk the San Francisco bay area thyroid cancer study. Cancer Epidemiol Biomarkers Prev. 11 43-9. 

An inverse correlation between thyroid cancer risk and phytoestrogens was recently proposed as a result of a multi-ethnic population-based case control study conducted in the San Francisco Bay Area (Hom-Ross et al., 2002). In this study, dietary habits and phytoestrogen consumption were assessed by a food-frequency questionnaire and by a nutrient database. The outcome of the study was that soy-based foods and alfalfa sprouts were associated with a reduction of thyroid cancer risk, whereas a Western diet did not influence cancer risk. No difference was observed between American and Asian women or between pre- and postmenopausal women. Furthermore, among the few compounds examined, the isoflavones genistein and daidzein and the lignan secoisolariciresinol were the phytoestrogens most frequently associated with risk reduction (Horn-Ross et al., 2002). 

Friedman [12] introduced a Bayesian approach the Bayes equation is given in Chapter 16. In the present context, a Bayesian approach can be described as finding a classification rule that minimizes the risk of misclassification, given the prior probabilities of belonging to a given class. These prior probabilities are estimated from the fraction of each class in the pooled sample ... 

Professor Waitz, who was working in Monowitz as an internee doctor, advised me against seeking admission to the sick bay so as not to run the risk of being selected, i.e., to be sent to Birkenau. I made friends with an I.G. man named Malzer and discussed our prison existence with him. He knew.. . . The Amsterdam chemist Beinima worked in the chemistry squad. He was very ill (jaundice and tuberculosis) and was chosen to be picked out. 

Hall, L.W., Jr., M.C. Scott, and W.D. Killen. 1998. Ecological risk assessment of copper and cadmium in surface waters of Chesapeake Bay watershed. Environ. Toxicol. Chem. 17 1172-1189. 

Heaton, S.N., S.J. Bursian, J.P. Giesy, D.E. Tillitt, J.A. Revder, P.D. Jones, D.A. Verbrugge, T.J. Kubiak, and R.J. Aulerich. 1995. Dietary exposure of mink to carp from Saginaw Bay, Michigan. 1. Effects on reproduction and survival and the potential risks to wild mink populations. Arch. Environ. Contam. Toxicol. 28 334-343. 

Empirical Bayes methodology and other kinds of shrinkage estimation may be considered in situations where there is some, perhaps limited information for a situation of specific interest, but also a desire to give some weight to data from situations less representative. The term shrinkage expresses the idea that an estimate from the situation of specific interest is shrunk toward some prior estimate such as an estimate from less strictly representative situations. As yet the methods have seen little or no use for pesticide ecological risk assessment in regulatory contexts. 

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... 

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. 

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