This Title All WIREs
How to cite this WIREs title:
WIREs Comp Stat

Bayesian inference: an approach to statistical inference

Full article on Wiley Online Library:   HTML PDF

Can't access this content? Tell your librarian.

Abstract The original Bayes used an analogy involving an invariant prior and a statistical model and argued that the resulting combination of prior with likelihood provided a probability description of an unknown parameter value in an application; the combination in particular contexts with invariance can currently be called a confidence distribution and is subject to some restrictions when used to construct confidence intervals and regions. The procedure of using a prior with likelihood has now, however, been widely generalized with invariance being extended to less restrictive criteria such as non‐informative, reference, and more. Other generalizations are to allow the prior to represent various forms of background information that is available or elicited from those familiar with the statistical context; these can reasonably be called subjective priors. Still further generalizations address an anomaly where marginalization with a vector parameter gives results that contradict the term probability; these are Dawid, Stone, Zidek marginalization paradoxes; various priors for this are called targeted priors. A special case where the prior describes a random source for the parameter value is however just probability analysis but is frequently treated as a Bayes procedure. We survey the argument in support of probability characteristics and outline various generalizations of the original Bayes proposal. Copyright © 2010 John Wiley & Sons, Inc. This article is categorized under: Statistical and Graphical Methods of Data Analysis > Bayesian Methods and Theory

The extreme value EV(θ,1) model: (a) the distribution of y given θ; (b) the posterior distribution of θ given y0; the p‐value p(θ) from panel (a) is equal to the survivor value s(θ) in panel (b).

[ Normal View | Magnified View ]

Actual proportion for β = 90 and 10%.

[ Normal View | Magnified View ]

Actual(ρ) proportion with quantile level β = 50%.

[ Normal View | Magnified View ]

(a) The model is N(θ;I); region for p(θ) calculation indicated. (b) The posterior distribution for θ is N(y0;I); region for s(θ) calculation indicated.

[ Normal View | Magnified View ]

Normal with bounded mean: the actual proportions for the β = 90% and for β = 10% are strictly less than the alleged.

[ Normal View | Magnified View ]

Normal with bounded mean: the actual proportion for the β = 50% quantile is strictly less than 50%.

[ Normal View | Magnified View ]

The normal (θ, 1) with θ≥ θ0 = 0: (a) the likelihood function L(θ); (b) p‐value function p(θ) = Φ(y0 − θ); (c) s‐value function s(θ) = Φ(y0 − θ)/Φ(y0).

[ Normal View | Magnified View ]

The p‐value p(θ) from panel (a) and the survivor value s(θ) from panel (b) of Figure 1 are equal and are plotted against θ.

[ Normal View | Magnified View ]

Browse by Topic

Statistical and Graphical Methods of Data Analysis > Bayesian Methods and Theory

Access to this WIREs title is by subscription only.

Recommend to Your
Librarian Now!

The latest WIREs articles in your inbox

Sign Up for Article Alerts