Special Considerations in Bayesian Inference
In statistics, an inference is described as robust if it is not affected substantially by changing the assumptions used in drawing it. Frequentists as well as Bayesians must worry about robustness of their inferences. One concern for both camps is that as
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Special Considerations in Bayesian Inference
5.1 Robustness to Prior Specifications In statistics, an inference is described as robust if it is not affected substantially by changing the assumptions used in drawing it. Frequentists as well as Bayesians must worry about robustness of their inferences. One concern for both camps is that assumptions regarding the form of the likelihood can affect inference. For example, suppose that our data are scores obtained by 100 undergraduates on a calculus exam, and that we wish to use these data to estimate the mean of the scores that would have been obtained if all undergraduates who took calculus I that year had taken this exam. We might get very different estimates under each of the following assumptions: 1. The 100 scores are independent draws from a normal distribution. 2. The population of scores is likely to include extreme outliers, so the 100 scores are independent draws from a t distribution with 5 degrees of freedom. 3. Students who took the same calculus class are likely to have more similar scores than students in different classes; thus, rather than treating the 100 scores as independent, the likelihood must account for correlation within classes. Furthermore, frequentists make assumptions about characteristics of the population of interest when they perform power and sample-size calculations during the design of a study. In addition to all the robustness issues encountered by frequentist statisticians, the Bayesian must consider whether inference is robust to different prior specifications. Would different choices of parametric family for the prior lead to different inference? How sensitive is inference to different values of prior parameters within a particular parametric family of priors? A valuable tool in addressing these kinds of questions is a sensitivity analysis— an explicit comparison of important characteristics of the posterior distribution obtained under all plausible prior distributions under consideration.
M.K. Cowles, Applied Bayesian Statistics: With R and OpenBUGS Examples, Springer Texts in Statistics 98, DOI 10.1007/978-1-4614-5696-4 5, © Springer Science+Business Media New York 2013
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5 Special Considerations in Bayesian Inference Table 5.1 Posterior summaries for binomial likelihood, data = 17 successes and 43 failures, under different prior specifications Prior density U(0, 1) Beta(1.25, 5) Beta(2.5, 10) Beta(5, 20) Beta(10, 40)
Pr(π > 0.10|y) 0.867 0.842 0.884 0.937 0.981
E(π |y) 0.154 0.147 0.152 0.160 0.170
95% equal tail credible set (0.070, 0.263) (0.068 0.249) (0.075, 0.250) (0.087, 0.250) (0.103, 0.249)
Returning to the example of the quitting-school survey from Chap. 6, Table 5.1 presents a sensitivity analysis of the effects of the five different beta prior specifications that we entertained on posterior inference about π . Note that whether inference is “affected substantially” by changes in assumptions is a subjective determination that depends upon the primary purpose of the analysis. Usually a statistical study wi
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