In statistics, robust Bayesian analysis, also called Bayesian sensitivity analysis, is a type of

sensitivity analysis Sensitivity analysis is the study of how the uncertainty in the output of a mathematical model or system (numerical or otherwise) can be divided and allocated to different sources of uncertainty in its inputs. A related practice is uncertainty ana ...

applied to the outcome from Bayesian inference or Bayesian optimal decisions.

Sensitivity analysis

Robust Bayesian analysis, also called Bayesian sensitivity analysis, investigates the robustness of answers from a

Bayesian analysis Bayesian inference is a method of statistical inference in which Bayes' theorem is used to update the probability for a hypothesis as more evidence or information becomes available. Bayesian inference is an important technique in statistics, and ...

to uncertainty about the precise details of the analysis.Berger, J.O. (1994)
"An overview of robust Bayesian analysis"
(with discussion). ''Test'' 3: 5-124.Pericchi, L.R. (2000)

An answer is ''robust'' if it does not depend sensitively on the assumptions and calculation inputs on which it is based. Robust Bayes methods acknowledge that it is sometimes very difficult to come up with precise distributions to be used as

priors Prior (or prioress) is an ecclesiastical title for a superior in some religious orders. The word is derived from the Latin for "earlier" or "first". Its earlier generic usage referred to any monastic superior. In abbeys, a prior would be l ...

. Likewise the appropriate

likelihood function The likelihood function (often simply called the likelihood) represents the probability of random variable realizations conditional on particular values of the statistical parameters. Thus, when evaluated on a given sample, the likelihood funct ...

that should be used for a particular problem may also be in doubt. In a robust Bayes approach, a standard Bayesian analysis is applied to all possible combinations of prior distributions and likelihood functions selected from ''classes'' of priors and likelihoods considered empirically plausible by the analyst. In this approach, a class of priors and a class of likelihoods together imply a class of posteriors by pairwise combination through

Bayes' rule In probability theory and statistics, Bayes' theorem (alternatively Bayes' law or Bayes' rule), named after Thomas Bayes, describes the probability of an event, based on prior knowledge of conditions that might be related to the event. For exampl ...

. Robust Bayes also uses a similar strategy to combine a class of probability models with a class of utility functions to infer a class of decisions, any of which might be the answer given the uncertainty about best probability model and

utility function As a topic of economics, utility is used to model worth or value. Its usage has evolved significantly over time. The term was introduced initially as a measure of pleasure or happiness as part of the theory of utilitarianism by moral philosoph ...

. In both cases, the result is said to be robust if it is approximately the same for each such pair. If the answers differ substantially, then their range is taken as an expression of how much (or how little) can be confidently inferred from the analysis. Although robust Bayes methods are clearly inconsistent with the Bayesian idea that uncertainty should be measured by a single additive probability measure and that personal attitudes and values should always be measured by a precise utility function, they are often accepted as a matter of convenience (e.g., because the cost or schedule do not allow the more painstaking effort needed to get a precise measure and function).Walley, P. (1991). ''Statistical Reasoning with Imprecise Probabilities''. Chapman and Hall, London. Some analysts also suggest that robust methods extend the traditional Bayesian approach by recognizing incertitude as of a different kind of uncertainty. Analysts in the latter category suggest that the set of distributions in the prior class is not a class of reasonable priors, but that it is rather a reasonable class of priors. The idea is that no single distribution is reasonable as a model of ignorance, but considered as a whole, the class is a reasonable model for ignorance. Robust Bayes methods are related to important and seminal ideas in other areas of statistics such as

robust statistics Robust statistics are statistics with good performance for data drawn from a wide range of probability distributions, especially for distributions that are not normal. Robust statistical methods have been developed for many common problems, suc ...

and resistance estimators. The arguments in favor of a robust approach are often applicable to Bayesian analyses. For example, some criticize methods that must assume the analyst is "

omniscient Omniscience () is the capacity to know everything. In Hinduism, Sikhism and the Abrahamic religions, this is an attribute of God. In Jainism, omniscience is an attribute that any individual can eventually attain. In Buddhism, there are diff ...

" about certain facts such as model structure, distribution shapes and parameters. Because such facts are themselves potentially in doubt, an approach that does not rely too sensitively on the analysts getting the details exactly right would be preferred. There are several ways to design and conduct a robust Bayes analysis, including the use of (i) parametric conjugate families of distributions, (ii) parametric but non-conjugate families, (iii) density-ratio (bounded density distributions), (iv) ε-contamination, mixture,

quantile In statistics and probability, quantiles are cut points dividing the range of a probability distribution into continuous intervals with equal probabilities, or dividing the observations in a sample in the same way. There is one fewer quantile ...

classes, etc., and (v) bounds on cumulative distributions.Basu, S., and A. DasGupta (1995).
Robust Bayesian analysis with distribution bands
. ''Statistics and Decisions'' 13: 333–349. Although calculating the solutions to robust Bayesian problems can, in some cases, be computationally intensive, there are several special cases in which the requisite calculations are, or can be made, straightforward.

Sensitivity analysis

See also

References

Other reading