Mea Culpa

Posted in Statistics with tags , , , , , , , , , , , on April 10, 2020 by xi'an

[A quote from Jaynes about improper priors that I had missed in his book, Probability Theory.]

For many years, the present writer was caught in this error just as badly as anybody else, because Bayesian calculations with improper priors continued to give just the reasonable and clearly correct results that common sense demanded. So warnings about improper priors went unheeded; just that psychological phenomenon. Finally, it was the marginalization paradox that forced recognition that we had only been lucky in our choice of problems. If we wish to consider an improper prior, the only correct way of doing it is to approach it as a well-defined limit of a sequence of proper priors. If the correct limiting procedure should yield an improper posterior pdf for some parameter α, then probability theory is telling us that the prior information and data are too meager to permit any inferences about α. Then the only remedy is to seek more data or more prior information; probability theory does not guarantee in advance that it will lead us to a useful answer to every conceivable question.Generally, the posterior pdf is better behaved than the prior because of the extra information in the likelihood function, and the correct limiting procedure yields a useful posterior pdf that is analytically simpler than any from a proper prior. The most universally useful results of Bayesian analysis obtained in the past are of this type, because they tended to be rather simple problems, in which the data were indeed so much more informative than the prior information that an improper prior gave a reasonable approximation – good enough for all practical purposes – to the strictly correct results (the two results agreed typically to six or more significant figures).

In the future, however, we cannot expect this to continue because the field is turning to more complex problems in which the prior information is essential and the solution is found by computer. In these cases it would be quite wrong to think of passing to an improper prior. That would lead usually to computer crashes; and, even if a crash is avoided, the conclusions would still be, almost always, quantitatively wrong. But, since likelihood functions are bounded, the analytical solution with proper priors is always guaranteed to converge properly to finite results; therefore it is always possible to write a computer program in such a way (avoid underflow, etc.) that it cannot crash when given proper priors. So, even if the criticisms of improper priors on grounds of marginalization were unjustified,it remains true that in the future we shall be concerned necessarily with proper priors.

seeking the error in nested sampling

Posted in pictures, Statistics, Travel with tags , , , , , , on April 13, 2017 by xi'an

A newly arXived paper on the error in nested sampling, written by Higson and co-authors, and read in Berlin, looks at the difficult task of evaluating the sampling error of nested sampling. The conclusion is essentially negative in that the authors recommend multiple runs of the method to assess the magnitude of the variability of the output by bootstrap, i.e. to call for the most empirical approach…

The core of this difficulty lies in the half-plug-in, half-quadrature, half-Monte Carlo (!) feature of the nested sampling algorithm, in that (i) the truncation of the unit interval is based on a expectation of the mass of each shell (i.e., the zone between two consecutive isoclines of the likelihood, (ii) the evidence estimator is a quadrature formula, and (iii) the level of the likelihood at the truncation is replaced with a simulated value that is not even unbiased (and correlated with the previous value in the case of an MCMC implementation). As discussed in our paper with Nicolas, the error in the evidence approximation is of the same order as other Monte Carlo methods in that it gets down like the square root of the number of terms at each iteration. Contrary to earlier intuitions that focussed on the error due to the quadrature.

But the situation is much less understood when the resulting sample is used for estimation of quantities related with the posterior distribution. With no clear approach to assess and even less correct the resulting error, since it is not solely a Monte Carlo error. As noted by the authors, the quadrature approximation to the univariate integral replaces the unknown prior weight of a shell with its Beta order statistic expectation and the average of the likelihood over the shell with a single (uniform???) realisation. Or the mean value of a transform of the parameter with a single (biased) realisation. Since most posterior expectations can be represented as integrals over likelihood levels of the average value over an iso-likelihood contour. The approach advocated in the paper involved multiple threads of an “unwoven nested sampling run”, which means launching n nested sampling runs with one living term from the n currents living points in the current nested sample. (Those threads may then later be recombined into a single nested sample.) This is the starting point to a nested flavour of bootstrapping, where threads are sampled with replacement, from which confidence intervals and error estimates can be constructed. (The original notion appears in Skilling’s 2006 paper, but I missed it.)

The above graphic is an attempt within the paper at representing the (marginal) posterior of a transform f(θ). That I do not fully understand… The notations are rather horrendous as X is not the data but the prior probability for the likelihood to be above a given bound which is actually the corresponding quantile. (There is no symbol for data and £ is used for the likelihood function as well as realisations of the likelihood function…) A vertical slice on the central panel gives the posterior distribution of f(θ) given the event that the likelihood is in the corresponding upper tail. Or given the corresponding shell (?).

seminar im München, am Max-Planck-Institut für Astrophysik

Posted in Statistics, Travel, University life with tags , , , , , , , , , , , , on October 15, 2015 by xi'an

On Friday, I give a talk in München on ABC model choice. At the Max-Planck Institute for Astrophysics. As coincidence go, I happen to talk the week after John Skilling gave a seminar there. On Bayesian tomography, not on nested sampling. And the conference organisers put the cover of the book Think Bayes: Bayesian Statistics Made Simple, written by Allen Downey, a book I reviewed yesterday night for CHANCE (soon to appear on the ‘Og!) [not that I understand the connection with the Max-Planck Institute or with my talk!, warum nicht?!] The slides are the same as in Oxford for SPA 2015: