Archive for evidence

Bayesian model averaging in astrophysics

Posted in Books, Statistics, University life with tags , , , , , , , , , , on July 29, 2015 by xi'an

[A 2013 post that somewhat got lost in a pile of postponed entries and referee’s reports…]

In this review paper, now published in Statistical Analysis and Data Mining 6, 3 (2013), David Parkinson and Andrew R. Liddle go over the (Bayesian) model selection and model averaging perspectives. Their argument in favour of model averaging is that model selection via Bayes factors may simply be too inconclusive to favour one model and only one model. While this is a correct perspective, this is about it for the theoretical background provided therein. The authors then move to the computational aspects and the first difficulty is their approximation (6) to the evidence

P(D|M) = E \approx \frac{1}{n} \sum_{i=1}^n L(\theta_i)Pr(\theta_i)\, ,

where they average the likelihood x prior terms over simulations from the posterior, which does not provide a valid (either unbiased or converging) approximation. They surprisingly fail to account for the huge statistical literature on evidence and Bayes factor approximation, incl. Chen, Shao and Ibrahim (2000). Which covers earlier developments like bridge sampling (Gelman and Meng, 1998).

As often the case in astrophysics, at least since 2007, the authors’ description of nested sampling drifts away from perceiving it as a regular Monte Carlo technique, with the same convergence speed n1/2 as other Monte Carlo techniques and the same dependence on dimension. It is certainly not the only simulation method where the produced “samples, as well as contributing to the evidence integral, can also be used as posterior samples.” The authors then move to “population Monte Carlo [which] is an adaptive form of importance sampling designed to give a good estimate of the evidence”, a particularly restrictive description of a generic adaptive importance sampling method (Cappé et al., 2004). The approximation of the evidence (9) based on PMC also seems invalid:

E \approx \frac{1}{n} \sum_{i=1}^n \dfrac{L(\theta_i)}{q(\theta_i)}\, ,

is missing the prior in the numerator. (The switch from θ in Section 3.1 to X in Section 3.4 is  confusing.) Further, the sentence “PMC gives an unbiased estimator of the evidence in a very small number of such iterations” is misleading in that PMC is unbiased at each iteration. Reversible jump is not described at all (the supposedly higher efficiency of this algorithm is far from guaranteed when facing a small number of models, which is the case here, since the moves between models are governed by a random walk and the acceptance probabilities can be quite low).

The second quite unrelated part of the paper covers published applications in astrophysics. Unrelated because the three different methods exposed in the first part are not compared on the same dataset. Model averaging is obviously based on a computational device that explores the posteriors of the different models under comparison (or, rather, averaging), however no recommendation is found in the paper as to efficiently implement the averaging or anything of the kind. In conclusion, I thus find this review somehow anticlimactic.

astronomical evidence

Posted in pictures, Statistics, University life with tags , , , , , , , , , , , , on July 24, 2015 by xi'an

As I have a huge arXiv backlog and an even higher non-arXiv backlog, I cannot be certain I will find time to comment on those three recent and quite exciting postings connecting ABC with astro- and cosmo-statistics [thanks to Ewan for pointing out those to me!]:

on estimating constants…

Posted in pictures, Statistics, Travel, University life with tags , , , , , , , , , on July 21, 2015 by xi'an

While I discussed on the ‘Og in the past the difference I saw between estimating an unknown parameter from a distribution and evaluating a normalising constant, evaluating such constants and hence handling [properly] doubly intractable models is obviously of the utmost importance! For this reason, Nial Friel, Helen Ogden and myself have put together a CRiSM workshop on the topic (with the tongue-in-cheek title of Estimating constants!), to be held at the University of Warwick next April 20-22.

The CRiSM workshop will focus on computational methods for approximating challenging normalising constants found in Monte Carlo, likelihood and Bayesian models. Such methods may be used in a wide range of problems: to compute intractable likelihoods, to find the evidence in Bayesian model selection, and to compute the partition function in Physics. The meeting will bring together different communities working on these related problems, some of which have developed original if little advertised solutions. It will also highlight the novel challenges associated with large data and highly complex models. Besides a dozen invited talks, the schedule will highlight two afternoon poster sessions with speed (2-5mn) oral presentations called ‘Elevator’ talks.

While 2016 is going to be quite busy with all kinds of meetings (MCMSkv, ISBA 2016, the CIRM Statistics month, AISTATS 2016, …), this should be an exciting two-day workshop, given the on-going activity in this area, and I thus suggest interested readers to mark the dates in their diary. I will obviously keep you posted about registration and accommodation when those entries are available.

reading classics (The End)

Posted in Books, Kids, Statistics, University life with tags , , , , , , , , , on February 24, 2015 by xi'an

La Défense from Paris-Dauphine, Nov. 15, 2012Today was the final session of our Reading Classics Seminar for the academic year 2014-2015. I have not reported on this seminar much so far because it has had starting problems, namely hardly any student present on the first classes and therefore several re-starts until we reached a small group of interested students. And this is truly The End for this enjoyable experiment as this is the final year for my TSI Master at Paris-Dauphine, as it will become integrated within the new MASH Master next year.

As a last presentation for the entire series, my student picked John Skilling’s Nested Sampling, not that it was in my list of “classics”, but he had worked on the paper in a summer project and was thus reasonably fluent with the topic. As he did a good enough job (!), here are his slides.

Some of the questions that came to me during the talk were on how to run nested sampling sequentially, both in the data and in the number of simulated points, and on incorporating more deterministic moves in order to remove some of the Monte Carlo variability. I was about to ask about (!) the Hamiltonian version of nested sampling but then he mentioned his last summer internship on this very topic! I also realised during that talk that the formula (for positive random variables)

\int_0^\infty(1-F(x))\text{d}x = \mathbb{E}_F[X]

does not require absolute continuity of the distribution F.

ABC by population annealing

Posted in Statistics, University life with tags , , , , , , , , on January 6, 2015 by xi'an

The paper “Bayesian Parameter Inference and Model Selection by Population Annealing in System Biology” by Yohei Murakami got published in PLoS One last August but I only became aware of it when ResearchGate pointed it out to me [by mentioning one of our ABC papers was quoted there].

“We are recommended to try a number of annealing schedules to check the influence of the schedules on the simulated data (…) As a whole, the simulations with the posterior parameter ensemble could, not only reproduce the data used for parameter inference, but also capture and predict the data which was not used for parameter inference.”

Population annealing is a notion introduced by Y Iba, the very same IBA who introduced the notion of population Monte Carlo that we studied in subsequent papers. It reproduces the setting found in many particle filter papers of a sequence of (annealed or rather tempered) targets ranging from an easy (i.e., almost flat) target to the genuine target, and of an update of a particle set by MCMC moves and reweighing. I actually have trouble perceiving the difference with other sequential Monte Carlo schemes as those exposed in Del Moral, Doucet and Jasra (2006, Series B). And the same is true of the ABC extension covered in this paper. (Where the annealed intermediate targets correspond to larger tolerances.) This sounds like a traditional ABC-SMC algorithm. Without the adaptive scheme on the tolerance ε found e.g. in Del Moral et al., since the sequence is set in advance. [However, the discussion about the implementation includes the above quote that suggests a vague form of cross-validated tolerance construction]. The approximation of the marginal likelihood also sounds standard, the marginal being approximated by the proportion of accepted pseudo-samples. Or more exactly by the sum of the SMC weights at the end of the annealing simulation. This actually raises several questions: (a) this estimator is always between 0 and 1, while the marginal likelihood is not restricted [but this is due to a missing 1/ε in the likelihood estimate that cancels from both numerator and denominator]; (b) seeing the kernel as a non-parametric estimate of the likelihood led me to wonder why different ε could not be used in different models, in that the pseudo-data used for each model under comparison differs. If we were in a genuine non-parametric setting the bandwidth would be derived from the pseudo-data.

“Thus, Bayesian model selection by population annealing is valid.”

The discussion about the use of ABC population annealing somewhat misses the point of using ABC, which is to approximate the genuine posterior distribution, to wit the above quote: that the ABC Bayes factors favour the correct model in the simulation does not tell anything about the degree of approximation wrt the original Bayes factor. [The issue of non-consistent Bayes factors does not apply here as there is no summary statistic applied to the few observations in the data.] Further, the magnitude of the variability of the values of this Bayes factor as ε varies, from 1.3 to 9.6, mostly indicates that the numerical value is difficult to trust. (I also fail to explain the huge jump in Monte Carlo variability from 0.09 to 1.17 in Table 1.) That this form of ABC-SMC improves upon the basic ABC rejection approach is clear. However it needs to build some self-control to avoid arbitrary calibration steps and reduce the instability of the final estimates.

“The weighting function is set to be large value when the observed data and the simulated data are ‘‘close’’, small value when they are ‘‘distant’’, and constant when they are ‘‘equal’’.”

The above quote is somewhat surprising as the estimated likelihood f(xobs|xobs,θ) is naturally constant when xobs=xsim… I also failed to understand how the model intervened in the indicator function used as a default ABC kernel

importance sampling schemes for evidence approximation [revised]

Posted in Statistics, University life with tags , , , , , , , on November 18, 2014 by xi'an

After a rather intense period of new simulations and versions, Juong Een (Kate) Lee and I have now resubmitted our paper on (some) importance sampling schemes for evidence approximation in mixture models to Bayesian Analysis. There is no fundamental change in the new version but rather a more detailed description of what those importance schemes mean in practice. The original idea in the paper is to improve upon the Rao-Blackwellisation solution proposed by Berkoff et al. (2002) and later by Marin et al. (2005) to avoid the impact of label switching on Chib’s formula. The Rao-Blackwellisation consists in averaging over all permutations of the labels while the improvement relies on the elimination of useless permutations, namely those that produce a negligible conditional density in Chib’s (candidate’s) formula. While the improvement implies truncated the overall sum and hence induces a potential bias (which was the concern of one referee), the determination of the irrelevant permutations after relabelling next to a single mode does not appear to cause any bias, while reducing the computational overload. Referees also made us aware of many recent proposals that conduct to different evidence approximations, albeit not directly related with our purpose. (One was Rodrigues and Walker, 2014, discussed and commented in a recent post.)

independent component analysis and p-values

Posted in pictures, Running, Statistics, Travel, University life with tags , , , , , , , , , , , , , on September 8, 2014 by xi'an

WariseLast morning at the neuroscience workshop Jean-François Cardoso presented independent component analysis though a highly pedagogical and enjoyable tutorial that stressed the geometric meaning of the approach, summarised by the notion that the (ICA) decomposition

X=AS

of the data X seeks both independence between the columns of S and non-Gaussianity. That is, getting as away from Gaussianity as possible. The geometric bits came from looking at the Kullback-Leibler decomposition of the log likelihood

-\mathbb{E}[\log L(\theta|X)] = KL(P,Q_\theta) + \mathfrak{E}(P)

where the expectation is computed under the true distribution P of the data X. And Qθ is the hypothesised distribution. A fine property of this decomposition is a statistical version of Pythagoreas’ theorem, namely that when the family of Qθ‘s is an exponential family, the Kullback-Leibler distance decomposes into

KL(P,Q_\theta) = KL(P,Q_{\theta^0}) + KL(Q_{\theta^0},Q_\theta)

where θ⁰ is the expected maximum likelihood estimator of θ. (We also noticed this possibility of a decomposition in our Kullback-projection variable-selection paper with Jérôme Dupuis.) The talk by Aapo Hyvärinen this morning was related to Jean-François’ in that it used ICA all the way to a three-level representation if oriented towards natural vision modelling in connection with his book and the paper on unormalised models recently discussed on the ‘Og.

On the afternoon, Eric-Jan Wagenmaker [who persistently and rationally fight the (ab)use of p-values and who frequently figures on Andrew’s blog] gave a warning tutorial talk about the dangers of trusting p-values and going fishing for significance in existing studies, much in the spirit of Andrew’s blog (except for the defence of Bayes factors). Arguing in favour of preregistration. The talk was full of illustrations from psychology. And included the line that ESP testing is the jester of academia, meaning that testing for whatever form of ESP should be encouraged as a way to check testing procedures. If a procedure finds a significant departure from the null in this setting, there is something wrong with it! I was then reminded that Eric-Jan was one of the authors having analysed Bem’s controversial (!) paper on the “anomalous processes of information or energy transfer that are currently unexplained in terms of known physical or biological mechanisms”… (And of the shocking talk by Jessica Utts on the same topic I attended in Australia two years ago.)

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