and Happy New Year! This post is about some statistical inferences that one can do using as “data” the output of MCMC algorithms. Consider the trace plot above. It has been generated by Metropolis–Hastings using a Normal random walk proposal, with a standard deviation “sigma”, on a certain target. Suppose that you are given a function that evaluates the pdf of that target. Can you retrieve the value of sigma used to generate that chain?

Rémi Bardenet and I are starting a new course on Bayesian machine learning at the MVA master (Mathématiques, Vision et Apprentissage) at ENS Paris-Saclay. Details on the syllabus can be found on the MVA webpage and on this Github repository. In this post, I shortly describe what motivated us for proposing this course and provide results of topic modeling of recent machine learning (conference) papers mentioning Bayesian in their abstract.Continue reading “Course on Bayesian machine learning in Paris”

At the dawn of 2020, in case anyone in the stat/ML community is not aware yet of Francis Bach’s blog started last year: this is a great place to learn about general tricks in machine learning explained with easy words. This month’s post The sum of a geometric series is all you need! shows how ubiquitous geometric series are in stochastic gradient descent, among others. In this post, I describe just another situation where the sum of a geometric series can be useful in statistics.

Turning a sum of expectations into a single expectation

I also found the sum of geometric series useful for turning a sum of expectations into a single expectation, by the linearity of expectation. More specifically, for a random variable compactly supported on ,

I’m glad to write about my habilitation entitled Bayesian statistical learning and applications I defended yesterday at Inria Grenoble. This Habilitation à Diriger des Recherches (HDR) is the highest degree issued through a university examination in France. If I am to believe official texts, the HDR recognizes a candidate’s “high scientific level, the originality of approach in a field of science, ability to master a research strategy in a sufficiently broad scientific or technological field and ability to supervise young researchers”. No less! In French academia, HDR is a necessary condition for supervising a Ph.D., being a reviewer for a Ph.D., and applying to prof positions. Part of the process consists of writing a manuscript summarizing the research done since the Ph.D. Mine is organized in three parts:

Chapter 1. Bayesian Nonparametric Mixture Modeling. This chapter is devoted to mixture models that are central to Bayesian nonparametrics. It focusses on BNP mixture models for (i) survival analysis, (ii) image segmentation and (ii) ecotoxicological applications.

Chapter 2. Approximate Bayesian Inference. This chapter is concerned in large parts with computational aspects of Bayesian inference. It describes: (i) conditional approaches in the form of truncation-based approximations for the Pitman–Yor process and for completely random measures, (ii) a marginal approach based on approximations of the predictive distribution of Gibbs-type processes, (iii) an approximate Bayesian computation (ABC) algorithm using the energy distance as data discrepancy.

Chapter 3. Distributional Properties of Statistical and Machine Learning Models. This chapter is concerned with general distributional properties of statistical and machine learning models, including (i) sub-Gaussian and sub-Weibull properties, (ii) a prior analysis of Bayesian neural networks and (iii) theoretical properties of asymmetric copulas.Continue reading “French Habilitation: Bayesian statistical learning and applications”

The paper “unbiased Markov chain Monte Carlo with couplings” co-written with John O’Leary and Yves Atchadé has been accepted as a read paper in JRSS: Series B, to be presented on December 11 at the Royal Statistical Society. Comments can be submitted (400 words max) until two weeks after, that is December 28; see the instructions here. The main contribution is on the removal of the burn-in bias of Markov chain Monte Carlo (MCMC) estimators, using coupled chains. We argue that 1) the burn-bias can be removed for a controllable loss of efficiency, and 2) this can be done in many settings, for low or high-dimensional, discrete or continuous target distributions, provided that the underlying MCMC algorithm works well enough. This might have consequences on the use of parallel processors for MCMC, but also on the estimation of Monte Carlo errors. We believe that there are a lot of open questions related to our work, on the theoretical, methodological and applied sides, and hope that the discussion will be interesting.

All the ingredients of our approach were already in the literature, from coupling of Markov chains to debiasing techniques, to the analysis of budget-constrained simulations, but we also propose multiple contributions. Below, I summarize the main methodological contributions of the paper, the way I see it at least, and I try to contrast our approach with those of Glynn & Rhee (2014) and Agapiou, Roberts & Vollmer (2018).

which shows, for two parameters of the model, the cumulative distribution function (CDF) under standard Bayes (blue thin line) and under BayesBag (wider red line). BayesBag results in distributions on the parameter space that are more “spread out” than standard Bayes.

Basically the R script starts like a standard script that would use rstan for inference; it runs the default algorithm of Stan for a little while, then extracts some info from the “stanfit” object. With these, a pure R implementation of TV upper bounds for a naive HMC algorithm follows, that relies on functions called “stan_logtarget” and “stan_gradlogtarget” to evaluate the target log-pdf and its gradient.

The script takes a few minutes to run in total. Some time is first needed to compile the Stan code, and to run Stan for a few steps. Then some time spent towards the end of the script to generate 250 independent meeting times with a lag of 500 between the chains; the exact run time will of course depend a lot on your number of available processors (on my machine it takes around one minute). The script produces this plot:

This plot suggests that vanilla HMC as implemented in the script converges in less than 1000 iterations to its stationary distribution. This is probably quite conservative, but it’s still usable.

In passing, upon profiling the code of the function that generates each meeting time, it appears that half of the time is spent in Stan‘s “grad_log_prob” function (which computes the gradient of the log pdf of the target). This implies that not that much efficiency is lost in the fact that the algorithms are coded in pure R, at least for this model.

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.

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)
alg.run()

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!

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.