Reproducible Bayes, Higher order likelihood and Inference methodology

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Statistics has a wealth of procedures that range from recording an average of collected data, to examining the experimental results on a drug, to the verifying that is needed in the recent search for the Higgs Boson; the amount of data can range from some minimal few to the truly massive. There is always some purpose for  a procedure. And all serious participants would want to know how well the procedures accomplished or achieved the purpose. For an exploratory use this means calibration of the procedure, in the every day sense of calibration. And for verifying it means, in part at least, reproducibility. For some calibration this might amount to the familiar 19 out of 20: and for some verification it could involve 1 in 3 million for some opposite. All of this involves some level of modelling to assess what can happen with a procedure. The Higgs search  intrinsically involved Poisson counts and our inference group was instrumental in promoting higher accuracy for that analysis, and switching to the needed one-sided confidence and testing procedures from the initial two-sided. This research program is  focused on the calibration of exploratory procedures and the verification of reproducibility, all in the context of full or partial modelling of the  object under investigation. We have previously developed highly accurate assessment procedures and investigated the statistical arguments being used. In particular the arbitrariness involved in obtaining testing procedures has now been removed for a broad class of regular statistical models, and the Bayesian procedures for such models have been clarified and a long standing and problematic proposal by Jeffreys for prior information has  been reinstated by a simple procedural adjustment that had not been foreseen in the 65 years of its availability. The research program  also broadens this to the large data contexts where information is assembled locally in space and time and convenience, and  the resulting pieces of partially dependent  information are to be combined accurately and efficiently. This issue arose as composite likelihood and is now  being extended to the more important composite significance with huge needs in the context of large to massive data. For a wide range of models it is known that frequency and Bayes give closely equivalent results for linear parameters but then change in opposite directions in the presence of parameter curvature: as the frequency analysis is reproducible this says bluntly that the Bayes is usually not reproducible. The research program is seeking a broad context implementation of corrections for this, to allow  the  convenience of the Bayes approach; the methodology is now  established and our research seeks wide implementation.  The key tool of exponential models can be examined to second rather than third order. The simplified model then gives  second order procedures easily, close to  third. This technique will be examined widely.