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the interview below, Professor Peter Green discusses his
paper, "Reversible jump Markov chain Monte Carlo
computation and Bayesian model determination," (Biometrika
82[4]: 711-732, December 1995). This paper has been cited 490
times to date and is the fourth most-cited paper in the field
of Mathematics in the ISI
Essential Science Indicators
Web product. Sixteen of Professor Green’s papers in this
field have a total of 823 citations to date. Professor Green
is a member of the Statistics Group in the Department of
Mathematics at the University of Bristol, United Kingdom.
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Why, in your view, is your
paper highly cited?
The paper introduces Reversible jump Markov chain Monte Carlo, a
new simulation-based methodology for fitting statistical models that
have variable-dimension parameters. It caught on simply because
people found this useful in their statistical modelling work, and
indeed even now it must be the most widely used method for treating
these problems. The paper appeared in 1995, just the right time to
make an impact, since the extraordinary rise of interest in
computational Bayesian methods that started in the late ‘80s had
by that time reached just about every area of application of
statistical analysis, and researchers were starting to tackle more
challenging problems. Curiously, the problems that Reversible jump
solves were probably not very widely regarded as interesting before
the paper appeared, but the example applications in the paper, and
early applications of the ideas by other authors, seem to have
generated broader interest in these problems and showed people that
solutions were possible, so the topic became self-reinforcing. Not
very much of this was foreseen when I wrote the paper!
What are the circumstances which led you to your work?
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“Curiously, the problems that Reversible jump solves were probably not very widely regarded as interesting before the paper appeared, but the example applications in the paper, and early applications of the ideas by other authors, seem to have generated broader interest in these problems and showed people that solutions were possible, so the topic became self-reinforcing.”
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In October 1993, Ulf Grenander and Michael Miller presented a
paper on a computer vision problem at a Royal Statistical Society
discussion meeting in London. Papers for such meetings are available
a week or two in advance so that others can prepare contributions
for the discussion, which are eventually published along with the
authors’ rejoinder, and the paper itself. I was studying the paper
the night before the meeting, and was particularly intrigued by the
computational algorithm they developed, involving stochastic
simulation, for inference about objects in microscopy images. It
occurred to me both that (1) this kind of algorithm could equally
well address a huge class of inference problems of which theirs was
just one special example, and (2) the algorithm itself could be made
both more general, and easier to use and understand. I presented
these points in a five-minute slot in the discussion (well, I
probably ran over time by a minute or two).
It didn’t make a very big impact, but I worked on the idea over
the next month or two, verified the mathematical details, attempted
to codify the method into a recipe that was easier to use, and
implemented three fairly substantial examples – to multiple
change-point analysis for point processes, to image segmentation,
and to a partition problem for binomial data. I thought these would
help convince a broad range of readers that the ideas would be
relevant to them. This package formed the paper that I submitted to Biometrika,
which appeared in 1995. By the time the paper had appeared, however,
the work had already being quite widely taken up. The idea was
propagated both through preprint servers, and through conference and
workshop talks; there was a lot of workshop activity in the general
area of stochastic systems in Europe at the time, thanks to the
European Science Foundation funded network on Highly Structured
Stochastic Systems. It was a very exciting time for research in this
area, and I was lucky that my paper was ready at exactly the right
time to get noticed.
Would you describe the significance of this work for your field?
Many problems in statistical inference, in all sorts of field of
application, have the character that "the number of things you
don’t know is one of the things you don’t know." In
microscopy on cell tissue, you want to count the number of cells and
measure the properties of each. In the analysis of genetic data on
quantitative trait loci you want to estimate the number of loci and
the quantitative effect at each. In multiple regression you want to
select relevant variables and estimate effects for the selected
variables. Problems like this are ubiquitous. I was interested in
how you could make simultaneous inference about all unknowns in such
problems, given noisy data. This is a bit of a challenge technically
if you want a statistically rigorous solution, since most
statistical methods deal with vectors of unknowns—parameters—of
fixed dimension. The philosophically appealing way to do this
inference is through fully probabilistic, or Bayesian, analysis.
Bayes theorem tells you the theoretical solution, but you need to
compute it in practice.
Reversible jump Markov chain Monte Carlo is a class of methods
for doing that, which is feasible even in very complex models, and
which is not too cumbersome to set up and use. So the immediate
significance of the work was that people could now solve these
problems. The interesting thing is that secondarily but ultimately
more significantly, people started noticing that these
"variable dimension" problems were all around them, and
started thinking that estimating all the unknowns simultaneously,
and doing other tricks like Bayesian model averaging, might be good
things to do. The methodology embraces both model-choice situations,
where several competing explanations for the data are being
entertained and the analysis needs to simultaneously choose and fit
a model, and settings where there is really a single model which has
a variable dimension parameter—for example, a series
representation of a function where the number of terms is not fixed.
Where has this research gone since the publication of your paper?
Most of the citations of the paper are by researchers who are
implementing the method to solve their own inference problems—the
citations appear in over 160 different journals, in many different
fields. They will often have tailored details of the algorithm to
their specific situation, and so particular idioms and rules of
thumb have emerged over time. Almost all of the papers citing mine
that do not deal with specific applications introduce novel generic
statistical methodology, for example, in model selection or
nonparametric regression, building on Reversible jump. Only a very
few authors have really developed the idea itself and made
significant additions to the simulation methodology. The most
significant papers to mention here are by Tierney (Ann. Appl.
Probab., 1998), Godsill (J. Comput. Graph. Stat., 2001),
by Mira and myself (Biometrika, 2001), by Brooks, Giudici,
and Roberts (J. Roy. Statist. Soc. B, 2003), and by Cappé,
Robert, and Rydén (J. Roy. Statist. Soc. B, 2003).
Where do you see it going 10 years from now?
On the practical side, the methods are not yet easy enough to
use. You can make mistakes setting them up, and even when you do get
it right, the methods usually need to be tuned, and that can be
laborious. So there would be tremendous benefit if the process of
implementation could be automated, and I hope we see work in this
direction. It has started; for example, my Ph.D. student David
Hastie has devised a very nice adaptive/automatic approach based on
normal mixtures.
More fundamentally, there are a host of challenging theoretical
questions about how to summarize inference in complex
variable-dimension models: Reversible jump helps you compute what
you want, but can’t tell you what to compute. I hope my work will
assist researchers in testing out in practical contexts their
developing ideas about these important issues.
Professor Peter J. Green, FRS
Department of Mathematics
University of Bristol
Bristol, UK
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