Archive for the University life Category

Tractable Fully Bayesian inference via convex optimization and optimal transport theory

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

IMG_0294“Recently, El Moselhy et al. proposed a method to construct a map that pushed forward the prior measure to the posterior measure, casting Bayesian inference as an optimal transport problem. Namely, the constructed map transforms a random variable distributed according to the prior into another random variable distributed according to the posterior. This approach is conceptually different from previous methods, including sampling and approximation methods.”

Yesterday, Kim et al. arXived a paper with the above title, linking transport theory with Bayesian inference. Rather strangely, they motivate the transport theory with Galton’s quincunx, when the apparatus is a discrete version of the inverse cdf transform… Of course, in higher dimensions, there is no longer a straightforward transform and the paper shows (or recalls) that there exists a unique solution with positive Jacobian for log-concave posteriors. For instance, log-concave priors and likelihoods. This solution remains however a virtual notion in practice and an approximation is constructed via a (finite) functional polynomial basis. And minimising an empirical version of the Kullback-Leibler distance.

I am somewhat uncertain as to how and why apply such a transform to simulations from the prior (which thus has to be proper). Producing simulations from the posterior certainly is a traditional way to approximate Bayesian inference and this is thus one approach to this simulation. However, the discussion of the advantage of this approach over, say, MCMC, is quite limited. There is no comparison with alternative simulation or non-simulation methods and the computing time for the transport function derivation. And on the impact of the dimension of the parameter space on the computing time. In connection with recent discussions on probabilistic numerics and super-optimal convergence rates, Given that it relies on simulations, I doubt optimal transport can do better than O(√n) rates. One side remark about deriving posterior credible regions from (HPD)  prior credible regions: there is no reason the resulting region is optimal in volume (HPD) given that the transform is non-linear.

importance sampling with multiple MCMC sequences

Posted in Mountains, pictures, Statistics, Travel, University life with tags , , , , , , , , , , on October 2, 2015 by xi'an

Vivek Roy, Aixian Tan and James Flegal arXived a new paper, Estimating standard errors for importance sampling estimators with multiple Markov chains, where they obtain a central limit theorem and hence standard error estimates when using several MCMC chains to simulate from a mixture distribution as an importance sampling function. Just before I boarded my plane from Amsterdam to Calgary, which gave me the opportunity to read it completely (along with half a dozen other papers, since it is a long flight!) I first thought it was connecting to our AMIS algorithm (on which convergence Vivek spent a few frustrating weeks when he visited me at the end of his PhD), because of the mixture structure. This is actually altogether different, in that a mixture is made of unnormalised complex enough densities, to act as an importance sampler, and that, due to this complexity, the components can only be simulated via separate MCMC algorithms. Behind this characterisation lurks the challenging problem of estimating multiple normalising constants. The paper adopts the resolution by reverse logistic regression advocated in Charlie Geyer’s famous 1994 unpublished technical report. Beside the technical difficulties in establishing a CLT in this convoluted setup, the notion of mixing importance sampling and different Markov chains is quite appealing, especially in the domain of “tall” data and of splitting the likelihood in several or even many bits, since the mixture contains most of the information provided by the true posterior and can be corrected by an importance sampling step. In this very setting, I also think more adaptive schemes could be found to determine (estimate?!) the optimal weights of the mixture components.

a simulated annealing approach to Bayesian inference

Posted in Books, pictures, Statistics, University life with tags , , , , , , , , , , on October 1, 2015 by xi'an

Paris/Zürich, Oct. 3, 2011 A misleading title if any! Carlos Albert arXived a paper with this title this morning and I rushed to read it. Because it sounded like Bayesian analysis could be expressed as a special form of simulated annealing. But it happens to be a rather technical sequel [“that complies with physics standards”] to another paper I had missed, A simulated annealing approach to ABC, by Carlos Albert, Hans Künsch, and Andreas Scheidegger. Paper that appeared in Statistics and Computing last year, and which is most interesting!

“These update steps are associated with a flow of entropy from the system (the ensemble of particles in the product space of parameters and outputs) to the environment. Part of this flow is due to the decrease of entropy in the system when it transforms from the prior to the posterior state and constitutes the well-invested part of computation. Since the process happens in finite time, inevitably, additional entropy is produced. This entropy production is used as a measure of the wasted computation and minimized, as previously suggested for adaptive simulated annealing” (p.3)

The notion behind this simulated annealing intrusion into the ABC world is that the choice of the tolerance can be adapted along iterations according to a simulated annealing schedule. Both papers make use of thermodynamics notions that are completely foreign to me, like endoreversibility, but aim at minimising the “entropy production of the system, which is a measure for the waste of computation”. The central innovation is to introduce an augmented target on (θ,x) that is


where ε is the tolerance, while ρ(x,y) is a measure of distance to the actual observations, and to treat ε as an annealing temperature. In an ABC-MCMC implementation, the acceptance probability of a random walk proposal (θ’,x’) is then


Under some regularity constraints, the sequence of targets converges to


if ε decreases slowly enough to zero. While the representation of ABC-MCMC through kernels other than the Heaviside function can be found in the earlier ABC literature, the embedding of tolerance updating within the modern theory of simulated annealing is rather exciting.

Furthermore, we will present an adaptive schedule that attempts convergence to the correct posterior while minimizing the required simulations from the likelihood. Both the jump distribution in parameter space and the tolerance are adapted using mean fields of the ensemble.” (p.2)

What I cannot infer from a rather quick perusal of the papers is whether or not the implementation gets into the way of the all-inclusive theory. For instance, how can the Markov chain keep moving as the tolerance gets to zero? Even with a particle population and a sequential Monte Carlo implementation, it is unclear why the proposal scale factor [as in equation (34)] does not collapse to zero in order to ensure a non-zero acceptance rate. In the published paper, the authors used the same toy mixture example as ours [from Sisson et al., 2007], where we earned the award of the “incredibly ugly squalid picture”, with improvements in the effective sample size, but this remains a toy example. (Hopefully a post to be continued in more depth…)

Je reviendrai à Montréal [NIPS 2015]

Posted in pictures, Statistics, Travel, University life with tags , , , , , , , , , , , , , , on September 30, 2015 by xi'an

I will be back in Montréal, as the song by Robert Charlebois goes, for the NIPS 2015 meeting there, more precisely for the workshops of December 11 and 12, 2015, on probabilistic numerics and ABC [à Montréal]. I was invited to give the first talk by the organisers of the NIPS workshop on probabilistic numerics, presumably to present a contrapuntal perspective on this mix of Bayesian inference with numerical issues, following my somewhat critical posts on the topic. And I also plan to attend some lectures in the (second) NIPS workshop on ABC methods. Which does not leave much free space for yet another workshop on Approximate Bayesian Inference! The day after, while I am flying back to London, there will be a workshop on scalable Monte Carlo. All workshops are calling for contributed papers to be presented during central poster sessions. To be submitted to and to and to aabi2015. Before October 16.

Funny enough, I got a joking email from Brad, bemoaning my traitorous participation to the workshop on probabilistic numerics because of its “anti-MCMC” agenda, reflected in the summary:

“Integration is the central numerical operation required for Bayesian machine learning (in the form of marginalization and conditioning). Sampling algorithms still abound in this area, although it has long been known that Monte Carlo methods are fundamentally sub-optimal. The challenges for the development of better performing integration methods are mostly algorithmic. Moreover, recent algorithms have begun to outperform MCMC and its siblings, in wall-clock time, on realistic problems from machine learning.

The workshop will review the existing, by now quite strong, theoretical case against the use of random numbers for integration, discuss recent algorithmic developments, relationships between conceptual approaches, and highlight central research challenges going forward.”

Position that I hope to water down in my talk! In any case,

Je veux revoir le long désert
Des rues qui n’en finissent pas
Qui vont jusqu’au bout de l’hiver
Sans qu’il y ait trace de pas

a weekend in Banff

Posted in Mountains, pictures, Running, Statistics, Travel, University life with tags , , , , , , on September 26, 2015 by xi'an

Yesterday, I flew from Amsterdam to Calgary to attend the Canadian Statistical Sciences Institute Leadership Retreat (15w2214) at the Banff International Research Station (BIRS). The point of this meeting is to brainstorm towards building a research policy and strategy for the newly created CANSSI. To which scientific advisory committee I joined last semester. I just hope my brain will remain functional enough to contribute to the discussion, if not to storm, despite the eight hour time lag! (The drive from Calgary to Banff was beautiful, with flashy yellow bursting from the green landscape: Fall is coming.)

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

mixtures, Heremite polynomials, and ideals

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

mixture estimation from Bayesian Core (c.) Marin-Robert, 2007A 3 page note that got arXived today is [University of Colorado?!] Andrew Clark’s “Expanding the Computation of Mixture Models by the use of Hermite Polynomials and Ideals“. With a typo on Hermite‘s name in the pdf title. The whole point of the note is to demonstrate that mixtures of different types of distributions (like t and Gaussian) are manageable.  A truly stupendous result… As if no one had ever mixed different distributions before.

“Using Hermite polynomials and computing ideals allows the investigator to mix distributions from distinct families.”

The second point of the paper is to derive the mixture weights from an algebraic equation based on the Hermite polynomials of the components, which implies that the components and the mixture distribution itself are already known. Which thus does not seem particularly relevant for mixture estimation…


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