Weak Informativity and the Information in One Prior Relative to Another

Abstract

A question of some interest is how to characterize the amount of information that a prior puts into a statistical analysis. Rather than a general characterization, we provide an approach to characterizing the amount of information a prior puts into an analysis, when compared to another base prior. The base prior is considered to be the prior that best reflects the current available information. Our purpose then is to characterize priors that can be used as conservative inputs to an analysis relative to the base prior. The characterization that we provide is in terms of a priori measures of prior-data conflict.Comment: Published in at http://dx.doi.org/10.1214/11-STS357 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

Measuring Statistical Evidence Using Relative Belief

“…Here "lying in the tails" is interpreted as indicating that a prior-data conflict exists since the data is not coming from a distribution where the prior assigns most of the belief. In Evans and Jang [17] it is shown how this approach to assessing prior-data conflict can be used to characterize weakly informative priors and also how to modify a prior, when such a conflict is obtained, in a way that is not data dependent, to avoid such a conflict. Further details and discussion on all of this can be found in Evans [13].…”

Statistical Reasoning: Choosing and Checking the Ingredients, Inferences Based on a Measure of Statistical Evidence with Some Applications

“…There are a variety of refinements of (9) that allow for looking at particular components of a prior to isolate where a problem with the prior may be. In [22], a method is developed for replacing a prior when a prior-data conflict has been detected. This does not mean simply replacing a prior by one that is more diffuse, however, as is demonstrated in Section 4.1.…”

Prior Elicitation, Assessment and Inference with a Dirichlet Prior

“…While opinions vary on this, the point-of-view taken here is that properly collected data are primary in determining inferences, and so models and priors that are contradicted by the data need to be modified when this occurs. The issue is somewhat less relevant for priors, as with enough data the effect of the prior is minimal, but on the other hand it often turns out to be relatively easy to modify the prior so that the conflict is avoided, see [12].…”

“…Intuitively, the closer τ is to 0, then the less information the prior is putting into the analysis. This idea can be made precise in terms of the weak informativity of one prior with respect to another as developed in [12]. As such, if prior-data conflict is obtained with the prior specified by a value of (ξ 1 , .…”