## Sequentially Constrained Monte Carlo

Posted in Books, Mountains, pictures, Statistics, University life with tags , , , , , , , , , , on November 7, 2014 by xi'an

This newly arXived paper by S. Golchi and D. Campbell from Vancouver (hence the above picture) considers the (quite) interesting problem of simulating from a target distribution defined by a constraint. This is a question that have bothered me for a long while as I could not come up with a satisfactory solution all those years… Namely, when considering a hard constraint on a density, how can we find a sequence of targets that end up with the restricted density? This is of course connected with the zero measure case posted a few months ago. For instance, how do we efficiently simulate a sample from a Student’s t distribution with a fixed sample mean and a fixed sample variance?

“The key component of SMC is the filtering sequence of distributions through which the particles evolve towards the target distribution.” (p.3)

This is indeed the main issue! The paper considers using a sequence of intermediate targets hardening progressively the constraint(s), along with an SMC sampler, but this recommendation remains rather vague and hence I am at loss as to how to make it work when the exact constraint implies a change of measure. The first example is monotone regression where y has mean f(x) and f is monotone. (Everything is unidimensional here.) The sequence is then defined by adding a multiplicative term that is a function of ∂f/∂x, for instance

Φ(τ∂f/∂x),

with τ growing to infinity to make the constraint moving from soft to hard. An interesting introduction, even though the hard constraint does not imply a change of parameter space or of measure. The second example is about estimating the parameters of an ODE, with the constraint being the ODE being satisfied exactly. Again, not exactly what I was looking for. But with an exotic application to deaths from the 1666 Black (Death) plague.

And then the third example is about ABC and the choice of summary statistics! The sequence of constraints is designed to keep observed and simulated summary statistics close enough when the dimension of those summaries increases, which means they are considered simultaneously rather than jointly. (In the sense of Ratmann et al., 2009. That is, with a multidimensional distance.) The model used for the application of the SMC is the dynamic model of Wood (2010, Nature). The outcome of this specific implementation is not that clear compared with alternatives… And again sadly does not deal with the/my zero measure issue.

## projective covariate selection

Posted in Mountains, pictures, Statistics, Travel, University life with tags , , , , , , , , , , , , , , on October 28, 2014 by xi'an

While I was in Warwick, Dan Simpson [newly arrived from Norway on a postdoc position] mentioned to me he had attended a talk by Aki Vehtari in Norway where my early work with Jérôme Dupuis on projective priors was used. He gave me the link to this paper by Peltola, Havulinna, Salomaa and Vehtari that indeed refers to the idea that a prior on a given Euclidean space defines priors by projections on all subspaces, despite the zero measure of all those subspaces. (This notion first appeared in a joint paper with my friend Costas Goutis, who alas died in a diving accident a few months later.) The projection further allowed for a simple expression of the Kullback-Leibler deviance between the corresponding models and for a Pythagorean theorem on the additivity of the deviances between embedded models. The weakest spot of this approach of ours was, in my opinion and unsurprisingly, about deciding when a submodel was too far from the full model. The lack of explanatory power introduced therein had no absolute scale and later discussions led me to think that the bound should depend on the sample size to ensure consistency. (The recent paper by Nott and Leng that was expanding on this projection has now appeared in CSDA.)

“Specifically, the models with subsets of covariates are found by maximizing the similarity of their predictions to this reference as proposed by Dupuis and Robert [12]. Notably, this approach does not require specifying priors for the submodels and one can instead focus on building a good reference model. Dupuis and Robert (2003) suggest choosing the size of the covariate subset based on an acceptable loss of explanatory power compared to the reference model. We examine using cross-validation based estimates of predictive performance as an alternative.” T. Peltola et al.

The paper also connects with the Bayesian Lasso literature, concluding on the horseshoe prior being more informative than the Laplace prior. It applies the selection approach to identify biomarkers with predictive performances in a study of diabetic patients. The authors rank model according to their (log) predictive density at the observed data, using cross-validation to avoid exploiting the data twice. On the MCMC front, the paper implements the NUTS version of HMC with STAN.

## Basil the chipmunk (#2)

Posted in Kids, Mountains, pictures, Travel with tags , , , , , on September 20, 2014 by xi'an

## up [and down] Pöstlingberg

Posted in Mountains, pictures, Running, Travel, University life with tags , , , , , , , , , on September 18, 2014 by xi'an

Early morning today, following my Linz guests’ advice, I went running towards the top of Pöstlingberg, a hill 250m over Linz and the Danube river. A perfect beacon thus avoiding wrong turns and extra-mileage, but still a wee climb on a steep path for the last part. The reward of the view from the top was definitely worth the [mild] effort and I even had enough time to enjoy a good Austrian breakfast before my ABC talk

## talk in Linz [first slide]

Posted in Mountains, pictures, Running, University life with tags , , , , , , , , , on September 17, 2014 by xi'an

## available dark [book review]

Posted in Books, Mountains, Travel with tags , , , , , , on September 13, 2014 by xi'an

“Paved roads had long ago surrended to gravel tracks that disappeared into a desert of snow covered lava. Black spires like a forest of charred trees blotted out the stars near the horizon.”

This is the last book I read from my Amazon package: Available Dark by Elizabeth Hand. I cannot remember how I came to order it… Maybe a confusion with another fantasy author like Elizabeth Moon? Or simply because the story was taking place between MaineFinland and Iceland?! Anyway, I read the book within two days during a short hiking trip to the volcano region of Central France. The plot has indeed a mesmerizing quality that made me keep reading further and further at ungodly hours. (With the help of an US jetlag.) It is original and intense enough to overcome the major difficulty that the central character, Cas, is far from sympathetic, from specialising in corpse photography to being almost constantly on drugs. But the construction of the plot and the introduction of the characters, always seen from Cas’ viewpoint, are well-done, even though the ending is both precipitated and unrealistic. Too many coincidences. The original setup of this novel is the Finnish black metal scene, with its undercurrents of satanism, ritual murders, and church burnings. Rather accurate judging from the wikipedia page on the topic! What I appreciated most was the description of the first impression of Iceland on Cas, when she landed from Helsinki. “The trip to Reykjavik [from the airport] was like a bus tour through Mordor. Black lava fields, an endless waste broken here and there by ruined machinery or a building of stained corrugated metal.” So I may consider reading another novel in the series in a near future…

## my life as a mixture [BAYSM 2014, Wien]

Posted in Books, Kids, Mountains, pictures, Statistics, Travel, University life with tags , , , , , , , , , , on September 12, 2014 by xi'an

Next week I am giving a talk at BAYSM in Vienna. BAYSM is the Bayesian Young Statisticians meeting so one may wonder why, but with Chris Holmes and Mike West, we got invited as more… erm… senior speakers! So I decided to give a definitely senior talk on a thread pursued throughout my career so far, namely mixtures. Plus it also relates to works of the other senior speakers. Here is the abstract for the talk:

Mixtures of distributions are fascinating objects for statisticians in that they both constitute a straightforward extension of standard distributions and offer a complex benchmark for evaluating statistical procedures, with a likelihood both computable in a linear time and enjoying an exponential number of local models (and sometimes infinite modes). This fruitful playground appeals in particular to Bayesians as it constitutes an easily understood challenge to the use of improper priors and of objective Bayes solutions. This talk will review some ancient and some more recent works of mine on mixtures of distributions, from the 1990 Gibbs sampler to the 2000 label switching and to later studies of Bayes factor approximations, nested sampling performances, improper priors, improved importance samplers, ABC, and a inverse perspective on the Bayesian approach to testing of hypotheses.

I am very grateful to the scientific committee for this invitation, as it will give me the opportunity to meet the new generation, learn from them and in addition discover Vienna where I have never been, despite several visits to Austria. Including its top, the Großglockner. I will also give a seminar in Linz the day before. In the Institut für Angewandte Statistik.