On an exact probability matching property of right-invariant priors

On predictive probability matching priors

“…However, it need not be Jeffreys' prior when p > 1. Under a suitable group structure on the model, the results in [13] imply that the associated right Haar prior gives exact predictive matching, since the prediction region here is invariant. Thus in these cases the right Haar prior must also be a solution of equation (2.5).…”

“…Taking a = 1, b = 1/2 we obtain σ −2 1 (1 − ρ 2 ) −1 , which can be shown to be the right Haar prior arising from the group of transformations T −1 X on the sample space, where T is a lower triangular matrix with positive diagonal elements. This group is isomorphic to Ω and since in this case the region A is invariant it follows from [13] that this prior must be a UPMP. Similarly, all right Haar priors arising from transformations of the form T −1 M X, with M a fixed non-singular matrix, are included in (4.3).…”

“…This result therefore extends the exact Haar prior result for transformation models to the most general models for which approximate uniform matching is possible. However, it is clear from examples discussed in [4] and from the general theory for transformation models in [13] that this result will not hold in the multiparameter case. For example, the unique UPMP for the normal model with unknown mean and variance is the right Haar prior, or Jeffreys' independence prior, whereas Jeffreys' prior is the left Haar prior.…”

“…If there does exist a prior that gives rise to predictive probability matching for all α then we shall refer to it as a uniformly predictive matching prior (UPMP). Of course, in the case of a transformation model and an invariant prediction region we already know from [13] that the right Haar prior must be a solution of the PDE. It is instructive to demonstrate this directly and this is done in the Appendix for quantile matching.…”

Coherent Frequentism: A Decision Theory Based on Confidence Sets

“…In addition to precluding the error in interpretation, such matching enables the statistician to leverage the flexibility of the Bayesian approach in making self-consistent inferences, involving, for example, the probability that the parameter lies in any given region of the parameter space, on the basis of a posterior distribution firmly anchored to valid coverage rates. Priors yielding exact matching of predictive probabilities are available for many models, including location models and certain locationscale models (Datta et al, 2000;Severini et al, 2002). Although exact matching of fixed-parameter coverage rates is limited to location models (Welch and Peers, 1963;Fraser and Reid, 2002), priors yielding asymptotic matching have been identified for other models, e.g., a hierarchical normal model (Datta et al, 2000).…”