Bayes factors for a test about the drift of the Brownian motion under noninformative priors

Brownian motions are useful in modeling many stochastic phenomena. We address the problem of default testing for the sign of the drift, if any, in the mean of the process using the Bayesian approach. Conventional Bayes factors for hypotheses testing, however, cannot typically accommodate the use of standard noninformative priors, as such priors are defined only up to arbitrary constants which affect the values of the Bayes factors. To address this problem for some common noninformative priors, we shall use Intrinsic Bayes factors due to Berger and Pericchi (1996, J. Amer. Statist. Assoc. 91, 109-122) and fractional Bayes factors due to O'Hagan (1995, J. Roy. Statist. Soc. Ser. B 57(1), 99-138), assuming discrete observations are available from the process on a coarse time scale.Fractional Bayes factor Intrinsic Bayes factor Probability matching prior Reference Prior Wiener process

Prior-free probabilistic prediction of future observations

<div><p>Prediction of future observations is a fundamental problem in statistics. Here we present a general approach based on the recently developed inferential model (IM) framework. We employ an IM-based technique to marginalize out the unknown parameters, yielding prior-free probabilistic prediction of future observables. Verifiable sufficient conditions are given for validity of our IM for prediction, and a variety of examples demonstrate the proposed method's performance. Thanks to its generality and ease of implementation, we expect that our IM-based method for prediction will be a useful tool for practitioners.</p></div