Robust Bayes analysis - AbsoluteAstronomy.com

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

Bayesian inference

In statistics, Bayesian inference is a method of statistical inference. It is often used in science and engineering to determine model parameters, make predictions about unknown variables, and to perform model selection...

to uncertainty about the precise details of the analysis. 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 probability

In Bayesian statistical inference, a prior probability distribution, often called simply the prior, of an uncertain quantity p is the probability distribution that would express one's uncertainty about p before the "data"...

. Likewise the appropriate likelihood function

Likelihood function

In statistics, a likelihood function is a function of the parameters of a statistical model, defined as follows: the likelihood of a set of parameter values given some observed outcomes is equal to the probability of those observed outcomes given those parameter values...

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

Bayes' theorem

In probability theory and applications, Bayes' theorem relates the conditional probabilities P and P. It is commonly used in science and engineering. The theorem is named for Thomas Bayes ....

. 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. 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 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). 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

Robust statistics provides an alternative approach to classical statistical methods. The motivation is to produce estimators that are not unduly affected by small departures from model assumptions.- Introduction :...

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

Omniscience omniscient point-of-view in writing) is the capacity to know everything infinitely, or at least everything that can be known about a character including thoughts, feelings, life and the universe, etc. In Latin, omnis means "all" and sciens means "knowing"...

” 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

Conjugate prior

In Bayesian probability theory, if the posterior distributions p are in the same family as the prior probability distribution p, the prior and posterior are then called conjugate distributions, and the prior is called a conjugate prior for the likelihood...

families of distributions, (ii) parametric but non-conjugate families, (iii) density-ratio (bounded density distributions), (iv) ε-contamination, mixture

Mixture (probability)

In probability theory and statistics, a mixture is a combination of two or more probability distributions. The concept arises in two contexts:* A mixture defining a new probability distribution from some existing ones, as in a mixture density...

, quantile

Quantile function

In probability and statistics, the quantile function of the probability distribution of a random variable specifies, for a given probability, the value which the random variable will be at, or below, with that probability...

classes, etc., and (v) bounds on cumulative distributions

Probability box

A probability box is a characterization of an uncertain number consisting of both aleatoric and epistemic uncertainties that is often used in risk analysis or quantitative uncertainty modeling where numerical calculations must be performed...

. Although calculating the solutions to robust Bayes problems can, in some cases, be computationally intensive, there are several special cases in which the requisite calculations are, or can be made, straightforward.

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See also