Shocks sometimes lead to new ideas. One of us was indeed shocked when Robert Wolpert (1995) pointed out that the Raftery et al. (1995) approach of Bayesian synthesis of two independent prior distributions for a parameter was flawed, due to the so-called Borel paradox. The very positive comments he had prepared for the discussion at the Joint Statistical Meetings in 1994 in Toronto were hard to bring forward (Schweder, 1995). The paper under discussion concerned Bayesian estimation of the abundance of bowhead whales off Alaska. The method and resulting estimate had just been accepted by the Scientific Committee of the International Whaling Commission (IWC). The Borel paradox became a central issue at the next IWC meeting, along with associated problems of combining different information sources for the same parameters. A distributional estimate of bowhead abundance in place of the Bayesian posterior was clearly needed. This led to the idea of achieving a distribution from all the confidence intervals obtained by varying the confidence level. It also led to the collaboration of the two authors of the present book, from our paper (Schweder and Hjort, 1996) on the Borel paradox and likelihood synthesis and onwards, via papers tying together the general themes of confidence, likelihood, probability and applications.

Posterior distributions without priors?

Constructing distributions for parameters from the set of all confidence intervals was a new and very good idea, we thought, but it turned out to be not so new after all. Cox (1958) mentions the same idea, we later on learned, and the original discovery of distribution estimators not obtained by a Bayesian calculation from a prior distribution dates back to Fisher (1930) and his fiducial argument. Like most contemporary statisticians we were badly ignorant of the fiducial method, despite its revolutionary character (Neyman, 1934). The method fell into disrepute and neglect because of Fisher's insistence that it could do more than it actually can, and it disappeared from practically all textbooks in statistics and was almost never taught to statisticians during the past fifty years. Fiducial probability was said to be Fisher's biggest blunder. But Efron (1998), among others, expresses hope for a revival of the method, and speculates that Fisher's biggest blunder might be a big hit in our new century.