Estimating convergence of Markov chains

Posted in Statistics by Pierre Jacob on 5 June 2019

Upper bounds as never seen on TV.

Hi all,

Niloy Biswas (PhD student at Harvard) and I have recently arXived a manuscript on the assessment of MCMC convergence (using couplings!). Here I’ll describe the main result, and some experiments (that are not in the current version of the paper) revisiting a 1996 paper by Mary Kathryn Cowles and Jeff Rosenthal entitled “A simulation approach to convergence rates for Markov chain Monte Carlo algorithms“. Code in R producing the figures of this post is available here.




Posted in General, Statistics by nicolaschopin on 4 June 2019

I have released a few months ago a Python package for particle filtering, called particles; you can find it on Github here. You may want to have a look first at the documentation, in particular the tutorials here.

This package has been developed to support our (with Omiros Papaspiliopoulos) forthcoming book called (tentatively): an introduction to Sequential Monte Carlo. It implements all the algorithms discussed in the book; e.g.

  • bootstrap, guided and auxiliary particle filters
  • all standard resampling schemes
  • most particle smoothing algorithms
  • sequential quasi-Monte Carlo
  • PMCMC (PMMH, Particle Gibbs), SMC^2
  • SMC samplers

It also contains all the scripts that were used to perform the numerical experiments discussed in the book.

A random plot taken from the forthcoming book. Can you guess what it represents exactly?

This package is hopefully useful to people with different expectations and level of expertise. For instance, if you just want to run a particle filter for a basic state-space model, you may describe that model as follows:

import particles
from particles import state_space_models as ssm

class ToySSM(ssm.StateSpaceModel):
    def PX0(self):  # Distribution of X_0 
        return dists.Normal()  # X_0 ~ N(0, 1)
    def PX(self, t, xp):  # Distribution of X_t given X_{t-1}
        return dists.Normal(loc=xp)  # X_t ~ N( X_{t-1}, 1)
    def PY(self, t, xp, x):  # Distribution of Y_t given X_t (and X_{t-1}) 
        return dists.Normal(loc=x, scale=self.sigma)  # Y_t ~ N(X_t, sigma^2)

And then simulate data, and run the corresponding bootstrap filter, as follows:

my_model = ToySSM(sigma=0.2)
x, y = my_model.simulate(200)  # sample size is 200

alg = particles.SMC(fk=ssm.Bootstrap(ssm=my_model, data=y), N=200)

On the other hand, if you are an SMC expert, you may re-use only the parts you need; e.g. a resampling scheme:

from particles import resampling 

A = resampling.systematic(W)

Up to now, this package has been tested mostly by my PhD students, and the students of my M2 course on particle filtering at the ENSAE; many thanks to all of them. Since no computer screen has been smashed in the process, I guess I can publicize it a bit more. Please let me know if you have any questions, comments, or feature request. (You may report a bug by raising an issue on the Github page.)

Based on your feedback, I’m planning to write a few more posts in the coming weeks about particles and more generally numerical computation in Python. Stay tuned!

Budget constrained simulations

Posted in Statistics by Pierre Jacob on 5 February 2019


You’ll have to read the post to get what these lines are about.

Hi all,

This post is about some results from “Bias Properties of Budget Constrained Simulations“, by Glynn & Heidelberger and published in Operations Research in 1990. I have found these results extremely useful, and our latest manuscript on unbiased MCMC recalls them in detail. Below I go through some of the results and describe the simulations that lead to the above figure.


Another update on unbiased smoothing

Posted in General by Pierre Jacob on 19 January 2019


Hi all,

This is a short update on my research on unbiased smoothing with coupled conditional particle filters. In a previous post I naively explained that I was done with the project since the article was accepted for publication in a journal.

However, a bug was found in the code thanks to a very careful reader. So I’ve fixed the code, re-launched all the simulations, and updated the arXiv version straight away; that was on September 5, 2018. The conclusions of the article are unchanged, thankfully; in fact, the text of the article is exactly the same, only the numerical values in the tables and the figures are different. Phew!

Since then the version with the buggy results was retracted from the journal, and hopefully, the updated version will be published soon. In the meantime the arXiv version is up-to-date.

Another take on the Hyvärinen score for model comparison

Posted in Statistics by Stephane Shao on 22 September 2018


Exact log-Bayes factors (log-BF) and H-factors (HF) of M1 against M2, computed for 100 independent samples (thin solid lines) of 1000 observations generated as i.i.d. N(1,1), under three increasingly vague priors for θ1.

In a former post, Pierre wrote about Bayesian model comparison and the limitations of Bayes factors in the presence of vague priors. Here we are, one year later, and I am happy to announce that our joint work with Jie Ding and Vahid Tarokh has been recently accepted for publication. As way of celebrating, allow me to give you another take on the matter.


Final update on unbiased smoothing

Posted in Statistics by Pierre Jacob on 27 August 2018


Two coupled chains, marginally following a conditional particle filter algorithm with ancestor sampling, and meeting in 20 iterations. Script here.


Two years ago I blogged about couplings of conditional particle filters for smoothing.  The paper with Fredrik Lindsten and Thomas Schön has just been accepted for publication at JASA, and the arXiv version and github repository are hopefully in their final forms. Here I’ll mention a few recent developments and follow-up articles by other researchers.


Couplings of Normal variables

Posted in R, Statistics by Pierre Jacob on 24 August 2018



Just to play a bit with the gganimate package, and to celebrate National Coupling Day, the above plot shows different couplings of two univariate Normal distributions, Normal(0,1) and Normal(2,1). That is, each point is a pair (x,y) where x follows a Normal(0,1) and y follows a Normal(2,1). Below I’ll recall briefly how each coupling operates, in the Normal case. The code is available at the end of the post.


Different ways of using MCMC algorithms

Posted in General, Statistics by Pierre Jacob on 19 July 2018



This post is about different ways of using Markov chain Monte Carlo (MCMC) algorithms for numerical integration or sampling. It can be a hard job to design an MCMC algorithm for a given target distribution. Once it’s finally implemented, it gives a way of sampling a new point X’ given an existing point X. From there, the algorithm can be used in various ways to construct estimators of integrals/distribution of interest. Some ways are more amenable to parallel computing than others. I give some examples with references below.


Bayesian workshop in Grenoble, September 6-7

Posted in General, Seminar/Conference by Julyan Arbel on 23 May 2018


We are organising a two-day Bayesian workshop in Grenoble in September 6-7, 2018. It will be the second edition of the Italian-French statistics seminar (link to first edition), titled this year: Bayesian learning theory for complex data modeling. The workshop will give to young statisticians the opportunity to learn from and interact with highly qualified senior researchers in probability, theoretical and applied statistics, with a particular focus on Bayesian methods.

Anyone interested in this field is welcome. There will be two junior sessions and a poster session with a call for abstract open until June 30. A particular focus will be given to researchers in the early stage of their career, or currently studying for a PhD, MSc or BSc. The junior session is supported by ISBA through travel awards.

There will be a social dinner on September 6, and a hike organised in the mountains on September 8.

Confirmed invited speakers

• Simon Barthelmé, Gipsa-lab, Grenoble, France
• Arnoldo Frigessi, University of Oslo, Norway
• Benjamin Guedj, Inria Lille – Nord Europe, France
• Alessandra Guglielmi, Politecnico di Milano, Italy
• Antonio Lijoi, University Bocconi, Milan, Italy
• Bernardo Nipoti, Trinity College Dublin, Ireland
• Sonia Petrone, University Bocconi, Milan, Italy

Important Dates:

• June 30, 2018: Abstract submission closes
• July 20, 2018: Notification on abstract acceptance
• August 25, 2018: Registration closes

More details and how to register:

We look forward to seeing you in Grenoble.



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Scaling of MCMC with dimension (experiments)

Posted in Statistics by Pierre Jacob on 15 May 2018


Taken from Wikipedia’s article on dimension.

Hi all,

In this post, I’ll go through numerical experiments illustrating the scaling of some MCMC algorithms with respect to the dimension. I will focus on a simple setting, to illustrate some theoretical results developed by Gareth Roberts, Andrew Stuart, Alexandros Beskos, Jeff Rosenthal and many of their co-authors over many years, for instance here for random walk Metropolis-Hastings (RWMH, and see here more recently), here for Modified Adjusted Langevin Algorithm (MALA), here for Hamiltonian Monte Carlo (HMC).


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