Archive for Bayesian asymptotics

probably ABC [and provably robust]

Posted in Books, pictures, Statistics, Travel with tags , , , , , , , , on August 8, 2017 by xi'an

Two weeks ago, James Ridgway (formerly CREST) arXived a paper on misspecification and ABC, a topic on which David Frazier, Judith Rousseau and I have been working for a while now [and soon to be arXived as well].  Paper that I re-read on a flight to Amsterdam [hence the above picture], written as a continuation of our earlier paper with David, Gael, and Judith. One specificity of the paper is to use an exponential distribution on the distance between the observed and simulated sample within the ABC distribution. Which reminds me of the resolution by Bissiri, Holmes, and Walker (2016) of the intractability of the likelihood function. James’ paper contains oracle inequalities between the ABC approximation and the genuine distribution of the summary statistics, like a bound on the distance between the expectations of the summary statistics under both models. Which writes down as a sum of a model bias, of two divergences between empirical and theoretical averages, on smoothness penalties, and on a prior impact term. And a similar bound on the distance between the expected distance to the oracle estimator of θ under the ABC distribution [and a Lipschitz type assumption also found in our paper]. Which first sounded weird [to me] as I would have expected the true posterior, until it dawned on me that the ABC distribution is the one used for the estimation [a passing strike of over-Bayesianism!]. While the oracle bound could have been used directly to discuss the rate of convergence of the exponential rate λ to zero [with the sample size n], James goes into the interesting alternative direction of setting a prior on λ, an idea that dates back to Olivier Catoni and Peter Grünwald. Or rather a pseudo-posterior on λ, a common occurrence in the PAC-Bayesian literature. In one of his results, James obtains a dependence of λ on the dimension m of the summary [as well as the root dependence on the sample size n], which seems to contradict our earlier independence result, until one realises this scale parameter is associated with a distance variable, itself scaled in m.

The paper also contains a non-parametric part, where the parameter θ is the unknown distribution of the data and the summary the data itself. Which is quite surprising as I did not deem it possible to handle non-parametrics with ABC. Especially in a misspecified setting (although I have trouble perceiving what this really means).

“We can use most of the Monte Carlo toolbox available in this context.”

The theoretical parts are a bit heavy on notations and hard to read [as a vacation morning read at least!]. They are followed by a Monte Carlo implementation using SMC-ABC.  And pseudo-marginals [at least formally as I do not see how the specific features of pseudo-marginals are more that an augmented representation here]. And adaptive multiple pseudo-samples that reminded me of the Biometrika paper of Anthony Lee and Krys Latuszynski (Warwick). Therefore using indeed most of the toolbox!

CORE talk at Louvain-la-Neuve

Posted in Statistics with tags , , , , , , , on March 16, 2017 by xi'an

Tomorrow, I will give a talk at the seminar for econometrics and finance of CORE, in Louvain-la-Neuve, Belgium. Here are my slides, recycled from several earlier talks and from Judith’s slides in Banff:

 

mixture models with a prior on the number of components

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

mixdir

“From a Bayesian perspective, perhaps the most natural approach is to treat the numberof components like any other unknown parameter and put a prior on it.”

Another mixture paper on arXiv! Indeed, Jeffrey Miller and Matthew Harrison recently arXived a paper on estimating the number of components in a mixture model, comparing the parametric with the non-parametric Dirichlet prior approaches. Since priors can be chosen towards agreement between those. This is an obviously interesting issue, as they are often opposed in modelling debates. The above graph shows a crystal clear agreement between finite component mixture modelling and Dirichlet process modelling. The same happens for classification.  However, Dirichlet process priors do not return an estimate of the number of components, which may be considered a drawback if one considers this is an identifiable quantity in a mixture model… But the paper stresses that the number of estimated clusters under the Dirichlet process modelling tends to be larger than the number of components in the finite case. Hence that the Dirichlet process mixture modelling is not consistent in that respect, producing parasite extra clusters…

In the parametric modelling, the authors assume the same scale is used in all Dirichlet priors, that is, for all values of k, the number of components. Which means an incoherence when marginalising from k to (k-p) components. Mild incoherence, in fact, as the parameters of the different models do not have to share the same priors. And, as shown by Proposition 3.3 in the paper, this does not prevent coherence in the marginal distribution of the latent variables. The authors also draw a comparison between the distribution of the partition in the finite mixture case and the Chinese restaurant process associated with the partition in the infinite case. A further analogy is that the finite case allows for a stick breaking representation. A noteworthy difference between both modellings is about the size of the partitions

\mathbb{P}(s_1,\ldots,s_k)\propto\prod_{j=1}^k s_j^{-\gamma}\quad\text{versus}\quad\mathbb{P}(s_1,\ldots,s_k)\propto\prod_{j=1}^k s_j^{-1}

in the finite (homogeneous partitions) and infinite (extreme partitions) cases.

An interesting entry into the connections between “regular” mixture modelling and Dirichlet mixture models. Maybe not ultimately surprising given the past studies by Peter Green and Sylvia Richardson of both approaches (1997 in Series B and 2001 in JASA).

model selection by likelihood-free Bayesian methods

Posted in Books, pictures, Running, Statistics, University life with tags , , , , , , on May 29, 2014 by xi'an

Just glanced at the introduction of this arXived paper over breakfast, back from my morning run: the exact title is “Model Selection for Likelihood-free Bayesian Methods Based on Moment Conditions: Theory and Numerical Examples” by Cheng Li and Wenxin Jiang. (The paper is 81 pages long.) I selected the paper for its title as it connected with an interrogation of ours on the manner to extend our empirical likelihood [A]BC work to model choice. We looked at this issue with Kerrie Mengersen and Judith Rousseau the last time Kerrie visited Paris but could not spot a satisfying entry… The current paper is of a theoretical nature, considering a moment defined model

\mathbb{E}[g(D,\theta)]=0,

where D denotes the data, as the dimension p of the parameter θ grows with n, the sample size. The approximate model is derived from a prior on the parameter θ and of a Gaussian quasi-likelihood on the moment estimating function g(D,θ). Examples include single index longitudinal data, quantile regression and partial correlation selection. The model selection setting is one of variable selection, resulting in 2p models to compare, with p growing to infinity… Which makes the practical implementation rather delicate to conceive. And the probability one of hitting the right model a fairly asymptotic concept. (At least after a cursory read from my breakfast table!)

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.

my week at War[wick]

Posted in pictures, Running, Statistics, Travel, Uncategorized with tags , , , , , , , , , on February 1, 2014 by xi'an

This was a most busy and profitable week in Warwick as, in addition to meeting with local researchers and students on a wide range of questions and projects, giving an extended seminar to MASDOC students, attending as many seminars as humanly possible (!), and preparing a 5k race by running in the Warwickshire countryside (in the dark and in the rain), I received the visits of Kerrie Mengersen, Judith Rousseau and Jean-Michel Marin, with whom I made some progress on papers we are writing together. In particular, Jean-Michel and I wrote the skeleton of a paper we (still) plan to submit to COLT 2014 next week. And Judith, Kerrie and I drafted new if paradoxical aconnections between empirical likelihood and model selection. Jean-Michel and Judith also gave talks at the CRiSM seminar, Jean-Michel presenting the latest developments on the convergence of our AMIS algorithm, Judith summarising several papers on the analysis of empirical Bayes methods in non-parametric settings.