Bayesian inference for the parameter of a uniform distribution

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Subhash Lele writes:

I was wondering if you might know some good references to Bayesian treatment of parameter estimation for U(0,b) type distributions. I am looking for cases where the parameter is on the boundary. I would appreciate any help and advice you could provide. I am, in particular, looking for an MCMC (preferably in WinBUGS) based approach. I figured out the WinBUGS part but I am still curious about the theoretical papers, asymptotics etc.

I actually can’t think of any examples! But maybe you, the readers, can.

We also should think of the best way to implement this model in Stan. We like to transform to avoid hard boundary constraints, but it seems a bit tacky to do a data-based transformation (which itself would not work if there are latent variables).

P.S. I actually saw Lele speak at a statistics conference around 20 years ago. There was a lively exchange between Lele and an older guy who was working on similar problems using a different method. The other guy couldn’t stand what Lele was doing and was upset that at the conference organizers for not disavowing Lele’s talk. I don’t remember all the details but there was some real fury. In retrospect it all seems pretty silly but I imagine it was upsetting to Lele at the time. I speak by analogy to my own very disturbing experience having my research loudly denounced by people who worked on similar problems and seemed to think I was a complete idiot. (I’m not speaking, by the way, of the people who didn’t want to tenure me at Berkeley. Oddly enough, one of them told me they all agreed I was “brilliant.” Which was odd in its own way.)

9 thoughts on “Bayesian inference for the parameter of a uniform distribution”

  3. You could ask Prof. Bickel directly, but my impression is that they thought I was brilliant but that my work was unsound. I don’t think they had a problem with my research on monitoring the convergence of iterative simulations, but as non-Bayesians they didn’t have much appreciation for the importance of statistical computation as a research area on its own. Their main problem seemed to be with my applied work: they didn’t like that I used hierarchical models to data in problems such as voting and toxicology. Part of this, I believe, was a generalized suspicion of my attitude that Bayesian methods could work for such a disparate set of applications, and part of it was a discomfort with any method that made assumptions. Of course they used methods with assumptions too, but at least they felt bad about it, and a major part of Berkeley statistics research at the time was to work to reduce dependence on assumptions. I think they really didn’t like that I was happy with Bayesian methods and that I thought my assumptions could be checked. This was such a different approach from theirs that they were happy to find an excuse to get rid of me!

  4. The discrete analogue of U(1,b) is the distribution of serial numbers of objects numbered 1 up. Estimation of b is then natural as estimation of the most interesting property, how many objects are there? Jeffreys had a discussion in his “Theory of probability”. Jaynes claimed in his posthumously published book that non-Bayesians have ignored this problem, but that is incorrect: e.g. there was a note by Leo Goodman sometime in the early 1950s. In WWII estimating enemy strength in hardware by looking at the serial numbers of objects destroyed or captured was a common problem.

    I’m not clear how important it is the original poster that the distribution be continuous.

  5. Lele is very anti-Bayesian (at least in print), including his over-the-top article in Ecological Applications “Bayesian methods for hierarchical models: Are ecologists making a Faustian bargain.” I don’t know if that could be important background to the previous exchange or not. I have not met him in person, but I find the ecological articles he’s written attacking Bayesian inference (while there is some legitimacy since many consumers of hierarchical Bayes in ecology don’t know what a likelihood is yet are attempt to fit very complex HB models) irritating.

    • I disagree with Lele on that issue—in fact, I have a partly-written paper that responds to various anti-Bayesian attitudes, including Lele’s. On the other hand, in practice you can get the benefit of Bayes in other ways—as Rubin once said, any Bayesian method can be reformulated as “non-Bayesian” if you just describe the procedure and don’t label it as probabilistic. In any case, the conference exchange from 1991 or so was about spatial modeling but I don’t recall any discussion of Bayes there, one way or the other.

    • In writing in Ecological Applications, Lele was writing for a scientific audience, and further one that is generally unfamiliar with conceptual debates between frequentists and Bayesians, and that includes a lot of people who do statistics in a cookbook, just-give-me-the-answer fashion (which they mistakenly think of as ‘pragmatism’). In that article, Lele was in my view doing a very useful service to the field, raising issues that either have not occurred to many ecologists, or else been quickly dismissed by ecologists as irrelevant, how-many-angels-can-dance-on-the-head-of-a-pin-type stuff. Take it from an ecologist: given the audience he was writing for, his choice of style was appropriate, if only as a device to get readers’ attention.

  6. Funny that you mention Lele and the disagreement with another person working on the same topic: I blogged twice here and there about the strong similarity between our 2002 SAME algorithm and his 2007 data cloning algorithm…

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