Archive for bootstrap

bootstrap(ed) likelihood for ABC

Posted in pictures, Statistics with tags , , , , , , , , on November 6, 2015 by xi'an

AmstabcThis recently arXived paper by Weixuan Zhu , Juan Miguel Marín, and Fabrizio Leisen proposes an alternative to our empirical likelihood ABC paper of 2013, or BCel. Besides the mostly personal appeal for me to report on a Juan Miguel Marín working [in Madrid] on ABC topics, along my friend Jean-Michel Marin!, this paper is another entry on ABC that connects with yet another statistical perspective, namely bootstrap. The proposal, called BCbl, is based on a reference paper by Davison, Hinkley and Worton (1992) which defines a bootstrap likelihood, a notion that relies on a double-bootstrap step to produce a non-parametric estimate of the distribution of a given estimator of the parameter θ. This estimate includes a smooth curve-fitting algorithm step, for which little description is available from the current paper. The bootstrap non-parametric substitute then plays the role of the actual likelihood, with no correction for the substitution just as in our BCel. Both approaches are convergent, with Monte Carlo simulations exhibiting similar or even identical convergence speeds although [unsurprisingly!] no deep theory is available on the comparative advantage.

An important issue from my perspective is that, while the empirical likelihood approach relies on a choice of identifying constraints that strongly impact the numerical value of the likelihood approximation, the bootstrap version starts directly from a subjectively chosen estimator of θ, which may also impact the numerical value of the likelihood approximation. In some ABC settings, finding a primary estimator of θ may be a real issue or a computational burden. Except when using a preliminary ABC step as in semi-automatic ABC. This would be an interesting crash-test for the BCbl proposal! (This would not necessarily increase the computational cost by a large amount.) In addition, I am not sure the method easily extends to larger collections of summary statistics as those used in ABC, in particular because it necessarily relies on non-parametric estimates, only operating in small enough dimensions where smooth curve-fitting algorithms can be used. Critically, the paper only processes examples with a few parameters.

The comparisons between BCel and BCbl that are produced in the paper show some gain towards BCbl. Obviously, it depends on the respective calibrations of the non-parametric methods and of regular ABC, as well as on the available computing time. I find the population genetic example somewhat puzzling: The paper refers to our composite likelihood to set the moment equations. Since this is a pseudo-likelihood, I wonder how the authors do select their parameter estimates in the double-bootstrap experiment. And for the Ising model, it is not straightforward to conceive of a bootstrap algorithm on an Ising model: (a) how does one subsample pixels and (b) what are the validity guarantees for the estimation procedure.

Mathematical underpinnings of Analytics (theory and applications)

Posted in Books, Statistics, University life with tags , , , , , , , , , , , , , , , on September 25, 2015 by xi'an

“Today, a week or two spent reading Jaynes’ book can be a life-changing experience.” (p.8)

I received this book by Peter Grindrod, Mathematical underpinnings of Analytics (theory and applications), from Oxford University Press, quite a while ago. (Not that long ago since the book got published in 2015.) As a book for review for CHANCE. And let it sit on my desk and in my travel bag for the same while as it was unclear to me that it was connected with Statistics and CHANCE. What is [are?!] analytics?! I did not find much of a definition of analytics when I at last opened the book, and even less mentions of statistics or machine-learning, but Wikipedia told me the following:

“Analytics is a multidimensional discipline. There is extensive use of mathematics and statistics, the use of descriptive techniques and predictive models to gain valuable knowledge from data—data analysis. The insights from data are used to recommend action or to guide decision making rooted in business context. Thus, analytics is not so much concerned with individual analyses or analysis steps, but with the entire methodology.”

Barring the absurdity of speaking of a “multidimensional discipline” [and even worse of linking with the mathematical notion of dimension!], this tells me analytics is a mix of data analysis and decision making. Hence relying on (some) statistics. Fine.

“Perhaps in ten years, time, the mathematics of behavioural analytics will be common place: every mathematics department will be doing some of it.”(p.10)

First, and to start with some positive words (!), a book that quotes both Friedrich Nietzsche and Patti Smith cannot get everything wrong! (Of course, including a most likely apocryphal quote from the now late Yogi Berra does not partake from this category!) Second, from a general perspective, I feel the book meanders its way through chapters towards a higher level of statistical consciousness, from graphs to clustering, to hidden Markov models, without precisely mentioning statistics or statistical model, while insisting very much upon Bayesian procedures and Bayesian thinking. Overall, I can relate to most items mentioned in Peter Grindrod’s book, but mostly by first reconstructing the notions behind. While I personally appreciate the distanced and often ironic tone of the book, reflecting upon the author’s experience in retail modelling, I am thus wondering at which audience Mathematical underpinnings of Analytics aims, for a practitioner would have a hard time jumping the gap between the concepts exposed therein and one’s practice, while a theoretician would require more formal and deeper entries on the topics broached by the book. I just doubt this entry will be enough to lead maths departments to adopt behavioural analytics as part of their curriculum… Continue reading

evolution of correlations [award paper]

Posted in Books, pictures, Statistics, University life with tags , , , , , on September 15, 2015 by xi'an

“Many researchers might have observed that the magnitude of a correlation is pretty unstable in small samples.”

On the statsblog aggregator, I spotted an entry that eventually led me to this post about the best paper award for the evolution of correlation, a paper published in the Journal of Research in Personality. A journal not particularly well-known for its statistical methodology input. The main message of the paper is that, while the empirical correlation is highly varying for small n’s, an interval (or corridor of stability!) can be constructed so that a Z-transform of the correlation does not vary away from the true value by more than a chosen quantity like 0.1. And the point of stability is then defined as the sample size after which the trajectory of the estimate does not leave the corridor… Both corridor and point depending on the true and unknown value of the correlation parameter by the by. Which implies resorting to bootstrap to assess the distribution of this point of stability. And deduce quantiles that can be used for… For what exactly?! Setting the necessary sample size? But this requires a preliminary run to assess the possible value of the true correlation ρ. The paper concludes that “for typical research scenarios reasonable trade-offs between accuracy and confidence start to be achieved when n approaches 250”. This figure was achieved by a bootstrap study on a bivariate Gaussian population with 10⁶ datapoints, yes indeed 10⁶!, and bootstrap samples of maximal size 10³. All in all, while I am at a loss as to why the Journal of Research in Personality would promote the estimation of a correlation coefficient with 250 datapoints, there is nothing fundamentally wrong with the paper (!), except for this recommendation of the 250 datapoint, as the figure stems from a specific setting with particular calibrations and cannot be expected to apply in every and all cases.

bespprActually, the graph in the paper was the thing that first attracted my attention because it looks very much like the bootstrap examples I show my third year students to demonstrate the appeal of bootstrap. Which is not particularly useful in the current case. A quick simulation on 100 samples of size 300 showed [above] that Monte Carlo simulations produce a tighter confidence band than the one created by bootstrap, in the Gaussian case. Continue reading

Conditional love [guest post]

Posted in Books, Kids, Statistics, University life with tags , , , , , , , , , , , , , , , , , , , , on August 4, 2015 by xi'an

[When Dan Simpson told me he was reading Terenin’s and Draper’s latest arXival in a nice Bath pub—and not a nice bath tub!—, I asked him for a blog entry and he agreed. Here is his piece, read at your own risk! If you remember to skip the part about Céline Dion, you should enjoy it very much!!!]

Probability has traditionally been described, as per Kolmogorov and his ardent follower Katy Perry, unconditionally. This is, of course, excellent for those of us who really like measure theory, as the maths is identical. Unfortunately mathematical convenience is not necessarily enough and a large part of the applied statistical community is working with Bayesian methods. These are unavoidably conditional and, as such, it is natural to ask if there is a fundamentally conditional basis for probability.

Bruno de Finetti—and later Richard Cox and Edwin Jaynes—considered conditional bases for Bayesian probability that are, unfortunately, incomplete. The critical problem is that they mainly consider finite state spaces and construct finitely additive systems of conditional probability. For a variety of reasons, neither of these restrictions hold much truck in the modern world of statistics.

In a recently arXiv’d paper, Alexander Terenin and David Draper devise a set of axioms that make the Cox-Jaynes system of conditional probability rigorous. Furthermore, they show that the complete set of Kolmogorov axioms (including countable additivity) can be derived as theorems from their axioms by conditioning on the entire sample space.

This is a deep and fundamental paper, which unfortunately means that I most probably do not grasp it’s complexities (especially as, for some reason, I keep reading it in pubs!). However I’m going to have a shot at having some thoughts on it, because I feel like it’s the sort of paper one should have thoughts on. Continue reading

approximate approximate Bayesian computation [not a typo!]

Posted in Books, Statistics, University life with tags , , , , , on January 12, 2015 by xi'an

“Our approach in handling the model uncertainty has some resemblance to statistical ‘‘emulators’’ (Kennedy and O’Hagan, 2001), approximative methods used to express the model uncertainty when simulating data under a mechanistic model is computationally intensive. However, emulators are often motivated in the context of Gaussian processes, where the uncertainty in the model space can be reasonably well modeled by a normal distribution.”

Pierre Pudlo pointed out to me the paper AABC: Approximate approximate Bayesian computation for inference in population-genetic models by Buzbas and Rosenberg that just appeared in the first 2015 issue of Theoretical Population Biology. Despite the claim made above, including a confusion on the nature of Gaussian processes, I am rather reserved about the appeal of this AA rated ABC…

“When likelihood functions are computationally intractable, likelihood-based inference is a challenging problem that has received considerable attention in the literature (Robert and Casella, 2004).”

The ABC approach suggested therein is doubly approximate in that simulation from the sampling distribution is replaced with simulation from a substitute cheaper model. After a learning stage using the costly sampling distribution. While there is convergence of the approximation to the genuine ABC posterior under infinite sample and Monte Carlo sample sizes, there is no correction for this approximation and I am puzzled by its construction. It seems (see p.34) that the cheaper model is build by a sort of weighted bootstrap: given a parameter simulated from the prior, weights based on its distance to a reference table are constructed and then used to create a pseudo-sample by weighted sampling from the original pseudo-samples. Rather than using a continuous kernel centred on those original pseudo-samples, as would be the suggestion for a non-parametric regression. Each pseudo-sample is accepted only when a distance between the summary statistics is small enough. This bootstrap flavour is counter-intuitive in that it requires a large enough sample from the true  sampling distribution to operate with some confidence… I also wonder at what happens when the data is not iid.  (I added the quote above as another source of puzzlement, since the book is about cases when the likelihood is manageable.)

a bootstrap likelihood approach to Bayesian computation

Posted in Books, R, Statistics, University life with tags , , , , , , , , on October 16, 2014 by xi'an

This paper by Weixuan Zhu, Juan Miguel Marín [from Carlos III in Madrid, not to be confused with Jean-Michel Marin, from Montpellier!], and Fabrizio Leisen proposes an alternative to our 2013 PNAS paper with Kerrie Mengersen and Pierre Pudlo on empirical likelihood ABC, or BCel. The alternative is based on Davison, Hinkley and Worton’s (1992) bootstrap likelihood, which relies on a double-bootstrap to produce a non-parametric estimate of the distribution of a given estimator of the parameter θ. Including a smooth curve-fitting algorithm step, for which not much description is available from the paper.

“…in contrast with the empirical likelihood method, the bootstrap likelihood doesn’t require any set of subjective constrains taking advantage from the bootstrap methodology. This makes the algorithm an automatic and reliable procedure where only a few parameters need to be specified.”

The spirit is indeed quite similar to ours in that a non-parametric substitute plays the role of the actual likelihood, with no correction for the substitution. Both approaches are convergent, with similar or identical convergence speeds. While the empirical likelihood relies on a choice of parameter identifying constraints, the bootstrap version starts directly from the [subjectively] chosen estimator of θ. For it indeed needs to be chosen. And computed.

“Another benefit of using the bootstrap likelihood (…) is that the construction of bootstrap likelihood could be done once and not at every iteration as the empirical likelihood. This leads to significant improvement in the computing time when different priors are compared.”

This is an improvement that could apply to the empirical likelihood approach, as well, once a large enough collection of likelihood values has been gathered. But only in small enough dimensions where smooth curve-fitting algorithms can operate. The same criticism applying to the derivation of a non-parametric density estimate for the distribution of the estimator of θ. Critically, the paper only processes examples with a few parameters.

In the comparisons between BCel and BCbl that are produced in the paper, the gain is indeed towards BCbl. Since this paper is mostly based on examples and illustrations, not unlike ours, I would like to see more details on the calibration of the non-parametric methods and of regular ABC, as well as on the computing time. And the variability of both methods on more than a single Monte Carlo experiment.

I am however uncertain as to how the authors process the population genetic example. They refer to the composite likelihood used in our paper to set the moment equations. Since this is not the true likelihood, how do the authors select their parameter estimates in the double-bootstrap experiment? The inclusion of Crakel’s and Flegal’s (2013) bivariate Beta, is somewhat superfluous as this example sounds to me like an artificial setting.

In the case of the Ising model, maybe the pre-processing step in our paper with Matt Moores could be compared with the other algorithms. In terms of BCbl, how does the bootstrap operate on an Ising model, i.e. (a) how does one subsample pixels and (b)what are the validity guarantees?

A test that would be of interest is to start from a standard ABC solution and use this solution as the reference estimator of θ, then proceeding to apply BCbl for that estimator. Given that the reference table would have to be produced only once, this would not necessarily increase the computational cost by a large amount…

Statistics slides (3)

Posted in Books, Kids, Statistics, University life with tags , , , , , , , , , , on October 9, 2014 by xi'an

La Défense from Paris-Dauphine, Nov. 15, 2012Here is the third set of slides for my third year statistics course. Nothing out of the ordinary, but the opportunity to link statistics and simulation for students not yet exposed to Monte Carlo methods. (No ABC yet, but who knows?, I may use ABC as an entry to Bayesian statistics, following Don Rubin’s example! Surprising typo on the Project Euclid page for this 1984 paper, by the way…) On Monday, I had the pleasant surprise to see Shravan Vasishth in the audience, as he is visiting Université Denis Diderot (Paris 7) this month.


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