Archive for simulation

Pre-processing for approximate Bayesian computation in image analysis

Posted in R, Statistics, University life with tags , , , , , , , , , , , , , on March 21, 2014 by xi'an

ridge6With Matt Moores and Kerrie Mengersen, from QUT, we wrote this short paper just in time for the MCMSki IV Special Issue of Statistics & Computing. And arXived it, as well. The global idea is to cut down on the cost of running an ABC experiment by removing the simulation of a humongous state-space vector, as in Potts and hidden Potts model, and replacing it by an approximate simulation of the 1-d sufficient (summary) statistics. In that case, we used a division of the 1-d parameter interval to simulate the distribution of the sufficient statistic for each of those parameter values and to compute the expectation and variance of the sufficient statistic. Then the conditional distribution of the sufficient statistic is approximated by a Gaussian with these two parameters. And those Gaussian approximations substitute for the true distributions within an ABC-SMC algorithm à la Del Moral, Doucet and Jasra (2012).

residuals

Across 20 125 × 125 pixels simulated images, Matt’s algorithm took an average of 21 minutes per image for between 39 and 70 SMC iterations, while resorting to pseudo-data and deriving the genuine sufficient statistic took an average of 46.5 hours for 44 to 85 SMC iterations. On a realistic Landsat image, with a total of 978,380 pixels, the precomputation of the mapping function took 50 minutes, while the total CPU time on 16 parallel threads was 10 hours 38 minutes. By comparison, it took 97 hours for 10,000 MCMC iterations on this image, with a poor effective sample size of 390 values. Regular SMC-ABC algorithms cannot handle this scale: It takes 89 hours to perform a single SMC iteration! (Note that path sampling also operates in this framework, thanks to the same precomputation: in that case it took 2.5 hours for 10⁵ iterations, with an effective sample size of 10⁴…)

Since my student’s paper on Seaman et al (2012) got promptly rejected by TAS for quoting too extensively from my post, we decided to include me as an extra author and submitted the paper to this special issue as well.

Approximate Integrated Likelihood via ABC methods

Posted in Books, Statistics, University life with tags , , , , , , , , on March 13, 2014 by xi'an

My PhD student Clara Grazian just arXived this joint work with Brunero Liseo on using ABC for marginal density estimation. The idea in this paper is to produce an integrated likelihood approximation in intractable problems via the ratio

L(\psi|x)\propto \dfrac{\pi(\psi|x)}{\pi(\psi)}

both terms in the ratio being estimated from simulations,

\hat L(\psi|x) \propto \dfrac{\hat\pi^\text{ABC}(\psi|x)}{\hat\pi(\psi)}

(with possible closed form for the denominator). Although most of the examples processed in the paper (Poisson means ratio, Neyman-Scott’s problem, g-&-k quantile distribution, semi-parametric regression) rely on summary statistics, hence de facto replacing the numerator above with a pseudo-posterior conditional on those summaries, the approximation remains accurate (for those examples). In the g-&-k quantile example, Clara and Brunero compare our ABC-MCMC algorithm with the one of Allingham et al. (2009, Statistics & Computing): the later does better by not replicating values in the Markov chain but instead proposing a new value until it is accepted by the usual Metropolis step. (Although I did not spend much time on this issue, I cannot see how both approaches could be simultaneously correct. Even though the outcomes do not look very different.) As noted by the authors, “the main drawback of the present approach is that it requires the use of proper priors”, unless the marginalisation of the prior can be done analytically. (This is an interesting computational problem: how to provide an efficient approximation to a marginal density of a σ-finite measure, assuming this density exists.)

Clara will give a talk at CREST-ENSAE today about this work, in the Bayes in Paris seminar: 2pm in room 18.

Advances in Scalable Bayesian Computation [group photo]

Posted in Kids, Mountains, pictures, Statistics, Travel, University life with tags , , , , , , , , on March 8, 2014 by xi'an

Foundations of Statistical Algorithms [book review]

Posted in Statistics, University life, Books, Linux, R with tags , , , , , , , , , , , , , on February 28, 2014 by xi'an

There is computational statistics and there is statistical computing. And then there is statistical algorithmic. Not the same thing, by far. This 2014 book by Weihs, Mersman and Ligges, from TU Dortmund, the later being also a member of the R Core team, stands at one end of this wide spectrum of techniques required by modern statistical analysis. In short, it provides the necessary skills to construct statistical algorithms and hence to contribute to statistical computing. And I wish I had the luxury to teach from Foundations of Statistical Algorithms to my graduate students, if only we could afford an extra yearly course…

“Our aim is to enable the reader (…) to quickly understand the main ideas of modern numerical algorithms [rather] than having to memorize the current, and soon to be outdated, set of popular algorithms from computational statistics.”(p.1)

The book is built around the above aim, first presenting the reasons why computers can produce answers different from what we want, using least squares as a mean to check for (in)stability, then second establishing the ground forFishman Monte Carlo methods by discussing (pseudo-)random generation, including MCMC algorithms, before moving in third to bootstrap and resampling techniques, and  concluding with parallelisation and scalability. The text is highly structured, with frequent summaries, a division of chapters all the way down to sub-sub-sub-sections, an R implementation section in each chapter, and a few exercises. Continue reading

Nonlinear Time Series just appeared

Posted in Books, R, Statistics, University life with tags , , , , , , , , , , , , , , , on February 26, 2014 by xi'an

My friends Randal Douc and Éric Moulines just published this new time series book with David Stoffer. (David also wrote Time Series Analysis and its Applications with Robert Shumway a year ago.) The books reflects well on the research of Randal and Éric over the past decade, namely convergence results on Markov chains for validating both inference in nonlinear time series and algorithms applied to those objects. The later includes MCMC, pMCMC, sequential Monte Carlo, particle filters, and the EM algorithm. While I am too close to the authors to write a balanced review for CHANCE (the book is under review by another researcher, before you ask!), I think this is an important book that reflects the state of the art in the rigorous study of those models. Obviously, the mathematical rigour advocated by the authors makes Nonlinear Time Series a rather advanced book (despite the authors’ reassuring statement that “nothing excessively deep is used”) more adequate for PhD students and researchers than starting graduates (and definitely not advised for self-study), but the availability of the R code (on the highly personal page of David Stoffer) comes to balance the mathematical bent of the book in the first and third parts. A great reference book!

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