Summary. Markov chain Monte Carlo and sequential Monte Carlo methods have emerged as the two main tools to sample from high dimensional probability distributions. Although asymptotic convergence of Markov chain Monte Carlo algorithms is ensured under weak assumptions, the performance of these algorithms is unreliable when the proposal distributions that are used to explore the space are poorly chosen and/or if highly correlated variables are updated independently. We show here how it is possible to build efficient high dimensional proposal distributions by using sequential Monte Carlo methods. This allows us not only to improve over standard Markov chain Monte Carlo schemes but also to make Bayesian inference feasible for a large class of statistical models where this was not previously so. We demonstrate these algorithms on a non-linear state space model and a Levy-driven stochastic volatility model.

https://rss.onlinelibrary.wiley.com/doi/pdf/10.1111/j.1467-9868.2009.00736.x

Particle Markov chain Monte Carlo methods

The random forest algorithm, proposed by L. Breiman in 2001, has been extremely successful as a general-purpose classification and regression method. The approach, which combines several randomized decision trees and aggregates their predictions by averaging, has shown excellent performance in settings where the number of variables is much larger than the number of observations. Moreover, it is versatile enough to be applied to large-scale problems, is easily adapted to various ad hoc learning tasks, and returns measures of variable importance. The present article reviews the most recent theoretical and methodological developments for random forests. Emphasis is placed on the mathematical forces driving the algorithm, with special attention given to the selection of parameters, the resampling mechanism, and variable importance measures. This review is intended to provide non-experts easy access to the main ideas.

/pdf/a-random-forest-guided-tour-3vnpr6ncpy.pdf

A random forest guided tour

Many scientific and engineering challenges—ranging from personalized medicine to customized marketing recommendations—require an understanding of treatment effect heterogeneity. In this paper, we develop a non-parametric causal forest for estimating heterogeneous treatment effects that extends Breiman's widely used random forest algorithm. In the potential outcomes framework with unconfoundedness, we show that causal forests are pointwise consistent for the true treatment effect, and have an asymptotically Gaussian and centered sampling distribution. We also discuss a practical method for constructing asymptotic confidence intervals for the true treatment effect that are centered at the causal forest estimates. Our theoretical results rely on a generic Gaussian theory for a large family of random forest algorithms. To our knowledge, this is the first set of results that allows any type of random forest, including classification and regression forests, to be used for provably valid statistical infe...

Estimation and Inference of Heterogeneous Treatment Effects using Random Forests

The random forest algorithm, proposed by L. Breiman in 2001, has been extremely successful as a general-purpose classification and regression method. The approach, which combines several randomized decision trees and aggregates their predictions by averaging, has shown excellent performance in settings where the number of variables is much larger than the number of observations. Moreover, it is versatile enough to be applied to large-scale problems, is easily adapted to various ad-hoc learning tasks, and returns measures of variable importance. The present article reviews the most recent theoretical and methodological developments for random forests. Emphasis is placed on the mathematical forces driving the algorithm, with special attention given to the selection of parameters, the resampling mechanism, and variable importance measures. This review is intended to provide non-experts easy access to the main ideas.

A Random Forest Guided Tour

Random forests are a scheme proposed by Leo Breiman in the 2000's for building a predictor ensemble with a set of decision trees that grow in randomly selected subspaces of data. Despite growing interest and practical use, there has been little exploration of the statistical properties of random forests, and little is known about the mathematical forces driving the algorithm. In this paper, we offer an in-depth analysis of a random forests model suggested by Breiman (2004), which is very close to the original algorithm. We show in particular that the procedure is consistent and adapts to sparsity, in the sense that its rate of convergence depends only on the number of strong features and not on how many noise variables are present.

/pdf/analysis-of-a-random-forests-model-3uegmd8tez.pdf

Analysis of a random forests model

A method to generate reactive trajectories, namely equilibrium trajectories leaving a metastable state and ending in another one is proposed. The algorithm is based on simulating in parallel many copies of the system, and selecting the replicas which have reached the highest values along a chosen one-dimensional reaction coordinate. This reaction coordinate does not need to precisely describe all the metastabilities of the system for the method to give reliable results. An extension of the algorithm to compute transition times from one metastable state to another one is also presented. We demonstrate the interest of the method on two simple cases: a one-dimensional two-well potential and a two-dimensional potential exhibiting two channels to pass from one metastable state to another one.

A multiple replica approach to simulate reactive trajectories

This article introduces a new nonparametric method for estimating a univariate regression function of bounded variation. The method exploits the Jordan decomposition which states that a function of bounded variation can be decomposed as the sum of a non-decreasing function and a non-increasing function. This suggests combining the backfitting algorithm for estimating additive functions with isotonic regression for estimating monotone functions. The resulting iterative algorithm is called Iterative Isotonic Regression (I.I.R.). The main technical result in this paper is the consistency of the proposed estimator when the number of iterations $k_n$ grows appropriately with the sample size $n$. The proof requires two auxiliary results that are of interest in and by themselves: firstly, we generalize the well-known consistency property of isotonic regression to the framework of a non-monotone regression function, and secondly, we relate the backfitting algorithm to Von Neumann's algorithm in convex analysis.

Iterative Isotonic Regression

The invention relates to a computer checking tool that can repeatedly process a plurality of data sets including data distributed according to a statistical rule. The tool includes an estimator (8) that can establish, for a data set, a value characterising the reproduction of a criterion concerning the data contained therein, a driver (6) designed to call the estimator (8) with a plurality of data sets in order to determine a plurality of values and establish a new plurality of sets from the plurality of values, and repeat the estimator (8) call with a new previously established plurality of sets until a condition is verified that involves an extremum of said plurality of values and/or number of repetitions. The tool also includes a mixer (10) that can establish a new set of data on the basis of an existing data set while maintaining the distribution according to the statistical rule, and the driver (6) is designed to call the mixer (10) with at least some of the data sets according to a rule based on the plurality of values associated with said sets.

Computer checking tool

. In this paper, we apply the median-of-means principle to derive robust versions of local averaging rules in non-parametric regression. For various estimates, including nearest neighbors and kernel procedures, we obtain non-asymptotic exponential inequali-ties, with only a second moment assumption on the noise. We then show that these bounds cannot be signiﬁcantly improved by establishing a corresponding lower bound on the tail probabilities.

Robust non-parametric regression via median-of-means

Let g : Ω = [0, 1] d → R denote a Lipschitz function that can be evaluated at each point, but at the price of a heavy computational time. Let X stand for a random variable with values in Ω such that one is able to simulate, at least approximately, according to the restriction of the law of X to any subset of Ω. For example, thanks to Markov chain Monte Carlo techniques, this is always possible when X admits a density that is known up to a normalizing constant. In this context, given a deterministic threshold T such that the failure probability p := P(g(X) > T) may be very low, our goal is to estimate the latter with a minimal number of calls to g. In this aim, building on Cohen et al. [9], we propose a recursive and optimal algorithm that selects on the fly areas of interest and estimate their respective probabilities.

/pdf/recursive-estimation-of-a-failure-probability-for-a-47f0cvoz6d.pdf

Arnaud Guyader

Papers

A multiple replica approach to simulate reactive trajectories

Iterative Isotonic Regression

Computer checking tool

Robust non-parametric regression via median-of-means

Recursive Estimation of a Failure Probability for a Lipschitz Function