Questing for Objective Inference: A Missing (Data) Perspective for a Missing Grail

Bayes Rule gives us a coherent way of using evidence to update our prior beliefs. Such beliefs may contain initial biases, but in the long run, evidence drowns out bias. As Keynes warned though, “In the long run we are all dead.” Scientific inquiry demands more—that we begin with a tabula rasa free of bias. Thus the Big Question: Is objective inference possible? The father of statistics, R.A. Fisher, answered with Fiducial inference, a most controversial idea because it seems to violate the laws of probability. Some see it as Fisher’s greatest mistake. Others see within it a way to unify Bayesian and Frequentist thinking. Did Fisher, a mathematical genius, really forget his probability laws? We show that Fiducial calculations can be easily understood if we adopt the powerful missing-data perspective. This idea has reappeared several times in the literature to intuitively motivate Fiducial. But each time, many of the rich tools and concepts associated with missing data problems remained uninvented. The timing is now optimal to fully deploy these tools to flesh out a principled Fiducial theory. Within the past decade, Fiducial arguments have been employed to tackle varied statistical challenges from meta-analysis to multiple testing. This empirical success validates its merit. But now we need to consolidate our understanding: When does it work? Why does it work? We will showcase via a series of publications and a workshop how the analytic toolkit developed for missing data problems provides a natural way to understand and study Fiducial inference. Initial explorations already illuminate several situations where intuitive arguments have overreached—such is the power of a systematic approach to foundation building. We hope that this new tact will usher in a fruitful era for Fiducial research as a fundamental step towards a full understanding of the feasibility of objective probabilistic inference.