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At a recent conference on Bayes, fiducial and frequentist inference, David Cox presented eight illustrative examples, chosen to highlight potential difficulties for the theory of inference. We discuss these examples in light of the efforts of the conference, and related meetings, to study the similarities and differences between the approaches to inference. Emphasis is placed on the goal of finding a distribution for an unknown parameter. In memory of Steve Fienberg DF: Steve was an undergraduate student long ago in classes for which I was fortunate to be the instructor; he had that extraordinary enthusiasm for the discipline and life, which makes the classroom scene a joy and a reason in itself. We remained close friends ever since. He was persistent in the pursuit of directions and needs in all areas of our discipline and tireless in bringing together people who could participate and contribute. He will be deeply remembered in his multiple roles. NR: When I met Steve in recent years, at conferences or committee meetings, he always seemed to be busily tapping away on his iPad, and when he looked up he would say “Annals of Applied Statistics”. (Although as it turned out he was editing several journals at the same time.) Steve was an inspiring mentor to me, about editorial work and much else. He showed by example how much fun a professional life can be when you take the effort to meet ∗We are grateful to David Cox for his challenge questions and for his many thoughtful comments on draft versions of this manuscript. We thank Robin Gong, Xiao-Li Meng, and Madeleine Straubel for their organization of the Fourth BFF conference. We also thank the reviewers for their prompt and helpful comments. This research was partially supported by the Natural Sciences and Engineering Research Council of Canada. MSC 2010 subject classifications: Primary 62A01; secondary 62F99

When should modes of inference disagree? Some simple but challenging examples