Unbiased Hamiltonian Monte Carlo with couplings

Posted in Statistics by Pierre Jacob on 17 September 2017

Two Hamiltonian Monte Carlo chains, casually exploring a target distribution while contracting.

With Jeremy Heng we have recently arXived a paper describing how to remove the burn-in bias of Hamiltonian Monte Carlo (HMC). This follows a recent work on unbiased MCMC estimators in general on which I blogged here. The case of HMC requires a specific yet very simple coupling. A direct consequence of this work is that Hamiltonian Monte Carlo can be massively parallelized: instead of running one chain for many iterations, one can run short coupled chains independently in parallel. The proposed estimators are consistent in the limit of the number of parallel replicates. This is appealing as the number of available processors increases much faster than clock speed, over recent years and for the years to come, for a number of reasons explained e.g. here.



Statistical learning in models made of modules

Posted in General, Statistics by Pierre Jacob on 9 September 2017



Graph of variables in a model made of two modules: the first with parameter theta1 and data Y1, and the second with parameter theta2 and data Y2, defined conditionally upon theta1.



With Lawrence Murray, Chris Holmes and Christian Robert, we have recently arXived a paper entitled “Better together? Statistical learning in models made of modules”. Christian blogged about it already. The context is the following: parameters of a first model appear as inputs in another model. The question is whether to consider a “joint model approach”, where all parameters are estimated simultaneously with all of the data. Or if one should instead follow a “modular approach”, where the first parameters are estimated with the first model only, ignoring the second model. Examples of modular approaches include the “cut distribution“, or “two-step estimators” (e.g. Chapter 6 of Newey & McFadden (1994)). In many fields, modular approaches are preferred, because the second model is suspected of being more misspecified than the first one. Misspecification of the second model can “contaminate” the joint model, with dire consequences on inference, as described e.g. in Bayarri, Berger & Liu (2009). Other reasons include computational constraints and the lack of simultaneous availability of all models and associated data. In the paper, we try to make sense of the defects of the joint model approach and we propose a principled, quantitative way of choosing between joint and modular approaches.


Sampling from a maximal coupling

Posted in Statistics by Pierre Jacob on 6 September 2017



Sample from a maximal coupling of two Normal distributions, X ~ N(0.5,0.82) and Y ~ N(-0.5,0.22).


In a recent work on parallel computation for MCMC, and also in another one, and in fact also in an earlier one, my co-authors and I use a simple yet very powerful object that is standard in Probability but not so well-known in Statistics: the maximal coupling. Here I’ll describe what this is and an algorithm to sample from such couplings.


Update on inference with Wasserstein distances

Posted in Statistics by Pierre Jacob on 15 August 2017

You have to read the arXiv report to understand this figure. There’s no way around it.

Hi again,

As described in an earlier postEspen BerntonMathieu Gerber and Christian P. Robert and I are exploring Wasserstein distances for parameter inference in generative models. Generally, ABC and indirect inference are fun to play with, as they make the user think about useful distances between data sets (i.i.d. or not), which is sort of implicit in classical likelihood-based approaches. Thinking about distances between data sets can be a helpful and healthy exercise, even if not always necessary for inference. Viewing data sets as empirical distributions leads to considering the Wasserstein distance, and we try to demonstrate in the paper that it leads to an appealing inferential toolbox.

In passing, the first author Espen Bernton will be visiting Marco Cuturi,  Christian Robert, Nicolas Chopin and others in Paris from September to January; get in touch with him if you’re over there!

We have just updated the arXiv version of the paper, and the main modifications are as follows.


Unbiased MCMC with couplings

Posted in Statistics by Pierre Jacob on 14 August 2017



Two chains meeting at time 10, and staying faithful forever. ❤



With John O’Leary and Yves Atchadé , we have just arXived our work on removing the bias of MCMC estimators. Here I’ll explain what this bias is about, and the benefits of removing it.


Particle methods in Statistics

Posted in General, Statistics by Pierre Jacob on 30 June 2017
Welding it together

A statistician sampling from a posterior distribution with particle methods

Hi there,

In this post, just in time for the summer, I propose a reading list for people interested in discovering the fascinating world of particle methods, aka sequential Monte Carlo methods, and their use in statistics. I also take the opportunity to advertise the SMC workshop in Uppsala (30 Aug – 1 Sept), which features an amazing list of speakers, including my postdoctoral collaborator Jeremy Heng:


Likelihood calculation for the g-and-k distribution

Posted in R, Statistics by Pierre Jacob on 11 June 2017



Histogram of 1e5 samples from the g-and-k distribution, and overlaid probability density function



An example often used in the ABC literature is the g-and-k distribution (e.g. reference [1] below), which is defined through the inverse of its cumulative distribution function (cdf). It is easy to simulate from such distributions by drawing uniform variables and applying the inverse cdf to them. However, since there is no closed-form formula for the probability density function (pdf) of the g-and-k distribution, the likelihood is often considered intractable. It has been noted in [2] that one can still numerically compute the pdf, by 1) numerically inverting the quantile function to get the cdf, and 2)  numerically differentiating the cdf, using finite differences, for instance. As it happens, this is very easy to implement, and I coded up an R tutorial at:

for anyone interested. This is part of the winference package that goes with our tech report on ABC with the Wasserstein distance  (joint work with Espen Bernton, Mathieu Gerber and Christian Robert, to be updated very soon!). This enables standard MCMC algorithms for the g-and-k example. It is also very easy to compute the likelihood for the multivariate extension of [3], since it only involves a fixed number of one-dimensional numerical inversions and differentiations (as opposed to a multivariate inversion).

Surprisingly, most of the papers that present the g-and-k example do not compare their ABC approximations to the posterior; instead, they typically compare the proposed ABC approach to existing ones. Similarly, the so-called Ricker model is commonly used in the ABC literature, and its posterior can be tackled efficiently using particle MCMC methods; as well as the M/G/1 model, which can be tackled either with particle MCMC methods or with tailor-made MCMC approaches such as [4].

These examples can still have great pedagogical value in ABC papers, but it would perhaps be nice to see more comparisons to the ground truth when it’s available; ground truth here being the actual posterior distribution.

  1. Fearnhead, P. and Prangle, D. (2012) Constructing summary statistics for approximate Bayesian computation: semi-automatic approximate Bayesian computation. Journal of the Royal Statistical Society: Series B, 74, 419–474.
  2. Rayner, G. D. and MacGillivray, H. L. (2002) Numerical maximum likelihood estimation for the g-and-k and generalized g-and-h distributions. Statistics and Computing, 12, 57–75.
  3. Drovandi, C. C. and Pettitt, A. N. (2011) Likelihood-free Bayesian estimation of multivari- ate quantile distributions. Computational Statistics & Data Analysis, 55, 2541–2556.
  4. Shestopaloff, A. Y. and Neal, R. M. (2014) On Bayesian inference for the M/G/1 queue with efficient MCMC sampling. arXiv preprint arXiv:1401.5548.

ABC in Banff

Posted in General, Seminar/Conference, Statistics by Pierre Jacob on 6 March 2017

Banff, also known as not the worst location for a scientific meeting.

Hi all,

Last week I attended a wonderful meeting on Approximate Bayesian Computation in Banff, which gathered a nice crowd of ABC users and enthusiasts, including lots of people outside of computational stats, whom I wouldn’t have met otherwise. Christian blogged about it there. My talk on Inference with Wasserstein distances is available as a video here (joint work with Espen Bernton, Mathieu Gerber and Christian Robert, the paper is here). In this post, I’ll summarize a few (personal) points and questions on ABC methods, after recalling the basics of ABC (ahem).


Statistical inference with the Wasserstein distance

Posted in Statistics by Pierre Jacob on 27 January 2017
NGS Picture ID:1440112

Nature transporting piles of sand.

Hi! It’s been too long!

In a recent arXiv entryEspen Bernton, Mathieu Gerber and Christian P. Robert and I explore the use of the Wasserstein distance to perform parameter inference in generative models. A by-product is an ABC-type approach that bypasses the choice of summary statistics. Instead, one chooses a metric on the observation space. Our work fits in the minimum distance estimation framework and is particularly related to “On minimum Kantorovich distance estimators”, by Bassetti, Bodini and Regazzini. A recent and very related paper is “Wasserstein training of restricted Boltzmann machines“, by Montavon, Müller and Cuturi, who have similar objectives but are not considering purely generative models. Similarly to that paper, we make heavy use of recent breakthroughs in numerical methods to approximate Wasserstein distances, breakthroughs which were not available to Bassetti, Bodini and Regazzini in 2006.

Here I’ll describe the main ideas in a simple setting.  If you’re excited about ABCasymptotic properties of minimum Wasserstein estimators, Hilbert space-filling curves, delay reconstructions and Takens’ theorem, or SMC samplers with r-hit kernels, check our paper!


Coupling of particle filters: smoothing

Posted in Statistics by Pierre Jacob on 20 July 2016



Two trajectories made for each other.


Hi again!

In this post, I’ll explain the new smoother introduced in our paper Coupling of Particle Filters with Fredrik Lindsten and Thomas B. Schön from Uppsala University. Smoothing refers to the task of estimating a latent process x_{0:T} = (x_0,\ldots, x_T) of length T, given noisy measurements of it, y_{1:T} = (y_0,\ldots, y_T); the smoothing distribution refers to p(dx_{0:T}|y_{1:T}). The setting is state-space models (what else?!), with a fixed parameter assumed to have been previously estimated.


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