arXiv subjects

Randal Douc

Publications and source records attributed to Randal Douc.

At least 19 recordsLinked to original sources

arXiv2025

Solving the Poisson equation using coupled Markov chains

This article shows how coupled Markov chains that meet exactly after a random number of iterations can be used to generate unbiased estimators of the solutions of the Poisson equation. Through this connection, we re-derive known unbiased estimators of expectations with respect to the stationary distribution of a Markov chain and provide conditions for the finiteness of their moments. We further construct unbiased estimators of the asymptotic variance of Markov chain ergodic averages, and provide conditions for the finiteness of the estimators' moments of any order. If their second moment is finite, the average of independent copies of such estimators converges to the asymptotic variance at the Monte Carlo rate, comparing favorably to known rates for batch means and spectral variance estimators. The results are illustrated with numerical experiments.

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arXiv2025

Self-Organizing State-Space Models with Artificial Dynamics

We consider the problem of performing parameter and state inference in a state-space model (SSM) parametrized by a static parameter $θ$. A popular idea to address this problem consists of incorporating $θ$ in the state of the system and allowing its time evolution, modelled as a Markov chain $(θ_t)_{t\geq 1}$. This proxy model defines a so-called self-organizing SSM (SO-SSM) to which one may apply standard particle filters. However, the practical implementation of this idea in a theoretically justified manner has remained an open problem until now. In this paper we fill this gap and in particular show that theoretically consistent SO-SSMs can be defined such that $\|\mathrm{Var}(θ_{t+1}|θ_{t})\|\rightarrow 0$ slowly as $t\rightarrow\infty$. This, in turn, leads to particle filter algorithms for online inference in SSMs which we find to be robust in simulation. We also develop constructions of $(θ_t)_{t\geq 1}$ and associated theoretical guarantees tailored to the application of SO-SSMs to maximum likelihood estimation in SSMs, leading to novel iterated filtering algorithms. The algorithms developed in this work have the advantage of being simple to implement and to require minimal tuning to perform well.

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arXiv2025

On the Asymptotics of Importance Weighted Variational Inference

For complex latent variable models, the likelihood function is not available in closed form. In this context, a popular method to perform parameter estimation is Importance Weighted Variational Inference. It essentially maximizes the expectation of the logarithm of an importance sampling estimate of the likelihood with respect to both the latent variable model parameters and the importance distribution parameters, the expectation being itself with respect to the importance samples. Despite its great empirical success in machine learning, a theoretical analysis of the limit properties of the resulting estimates is still lacking. We fill this gap by establishing consistency when both the Monte Carlo and the observed data sample sizes go to infinity simultaneously. We also establish asymptotic normality and efficiency under additional conditions relating the rate of growth between the Monte Carlo and the observed data samples sizes. We distinguish several regimes related to the smoothness of the importance ratio.

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arXiv2024

Variational Diffusion Posterior Sampling with Midpoint Guidance

Diffusion models have recently shown considerable potential in solving Bayesian inverse problems when used as priors. However, sampling from the resulting denoising posterior distributions remains a challenge as it involves intractable terms. To tackle this issue, state-of-the-art approaches formulate the problem as that of sampling from a surrogate diffusion model targeting the posterior and decompose its scores into two terms: the prior score and an intractable guidance term. While the former is replaced by the pre-trained score of the considered diffusion model, the guidance term has to be estimated. In this paper, we propose a novel approach that utilises a decomposition of the transitions which, in contrast to previous methods, allows a trade-off between the complexity of the intractable guidance term and that of the prior transitions. We validate the proposed approach through extensive experiments on linear and nonlinear inverse problems, including challenging cases with latent diffusion models as priors. We then demonstrate its applicability to various modalities and its promising impact on public health by tackling cardiovascular disease diagnosis through the reconstruction of incomplete electrocardiograms. The code is publicly available at \url{https://github.com/yazidjanati/mgps}.

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arXiv2024

The Importance Markov Chain

The Importance Markov chain is a novel algorithm bridging the gap between rejection sampling and importance sampling, moving from one to the other through a tuning parameter. Based on a modified sample of an instrumental Markov chain targeting an instrumental distribution (typically via a MCMC kernel), the Importance Markov chain produces an extended Markov chain where the marginal distribution of the first component converges to the target distribution. For example, when targeting a multimodal distribution, the instrumental distribution can be chosen as a tempered version of the target which allows the algorithm to explore its modes more efficiently. We obtain a Law of Large Numbers and a Central Limit Theorem as well as geometric ergodicity for this extended kernel under mild assumptions on the instrumental kernel. Computationally, the algorithm is easy to implement and preexisting libraries can be used to sample from the instrumental distribution.

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arXiv2023

Boost your favorite Markov Chain Monte Carlo sampler using Kac's theorem: the Kick-Kac teleportation algorithm

The present paper focuses on the problem of sampling from a given target distribution $π$ defined on some general state space. To this end, we introduce a novel class of non-reversible Markov chains, each chain being defined on an extended state space and having an invariant probability measure admitting $π$ as a marginal distribution. The proposed methodology is inspired by a new formulation of Kac's theorem and allows global and local dynamics to be smoothly combined. Under mild conditions, the corresponding Markov transition kernel can be shown to be irreducible and Harris recurrent. In addition, we establish that geometric ergodicity holds under appropriate conditions on the global and local dynamics. Finally, we illustrate numerically the use of the proposed method and its potential benefits in comparison to existing Markov chain Monte Carlo (MCMC) algorithms.

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arXiv2023

Monotonic Alpha-divergence Minimisation for Variational Inference

In this paper, we introduce a novel family of iterative algorithms which carry out $α$-divergence minimisation in a Variational Inference context. They do so by ensuring a systematic decrease at each step in the $α$-divergence between the variational and the posterior distributions. In its most general form, the variational distribution is a mixture model and our framework allows us to simultaneously optimise the weights and components parameters of this mixture model. Our approach permits us to build on various methods previously proposed for $α$-divergence minimisation such as Gradient or Power Descent schemes and we also shed a new light on an integrated Expectation Maximization algorithm. Lastly, we provide empirical evidence that our methodology yields improved results on several multimodal target distributions and on a real data example.

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arXiv2023

Asymptotic convergence of iterative optimization algorithms

This paper introduces a general framework for iterative optimization algorithms and establishes under general assumptions that their convergence is asymptotically geometric. We also prove that under appropriate assumptions, the rate of convergence can be lower bounded. The convergence is then only geometric, and we provide the exact asymptotic convergence rate. This framework allows to deal with constrained optimization and encompasses the Expectation Maximization algorithm and the mirror descent algorithm, as well as some variants such as the alpha-Expectation Maximization or the Mirror Prox algorithm.Furthermore, we establish sufficient conditions for the convergence of the Mirror Prox algorithm, under which the method converges systematically to the unique minimizer of a convex function on a convex compact set.

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arXiv2022

A Global Stochastic Optimization Particle Filter Algorithm

We introduce a new online algorithm for expected log-likelihood maximization in situations where the objective function is multi-modal and/or has saddle points, that we term G-PFSO. The key element underpinning G-PFSO is a probability distribution which (a) is shown to concentrate on the target parameter value as the sample size increases and (b) can be efficiently estimated by means of a standard particle filter algorithm. This distribution depends on a learning rate, where the faster the learning rate the quicker it concentrates on the desired element of the search space, but the less likely G-PFSO is to escape from a local optimum of the objective function. In order to achieve a fast convergence rate with a slow learning rate, G-PFSO exploits the acceleration property of averaging, well-known in the stochastic gradient literature. Considering several challenging estimation problems, the numerical experiments show that, with high probability, G-PFSO successfully finds the highest mode of the objective function and converges to its global maximizer at the optimal rate. While the focus of this work is expected log-likelihood maximization, the proposed methodology and its theory apply more generally for optimizing a function defined through an expectation.

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arXiv2021

Mixture weights optimisation for Alpha-Divergence Variational Inference

This paper focuses on $α$-divergence minimisation methods for Variational Inference. More precisely, we are interested in algorithms optimising the mixture weights of any given mixture model, without any information on the underlying distribution of its mixture components parameters. The Power Descent, defined for all $α\neq 1$, is one such algorithm and we establish in our work the full proof of its convergence towards the optimal mixture weights when $α<1$. Since the $α$-divergence recovers the widely-used forward Kullback-Leibler when $α\to 1$, we then extend the Power Descent to the case $α= 1$ and show that we obtain an Entropic Mirror Descent. This leads us to investigate the link between Power Descent and Entropic Mirror Descent: first-order approximations allow us to introduce the Renyi Descent, a novel algorithm for which we prove an $O(1/N)$ convergence rate. Lastly, we compare numerically the behavior of the unbiased Power Descent and of the biased Renyi Descent and we discuss the potential advantages of one algorithm over the other.

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arXiv2021

General-order observation-driven models: ergodicity and consistency of the maximum likelihood estimator

The class of observation-driven models (ODMs) includes many models of non-linear time series which, in a fashion similar to, yet different from, hidden Markov models (HMMs), involve hidden variables. Interestingly, in contrast to most HMMs, ODMs enjoy likelihoods that can be computed exactly with computational complexity of the same order as the number of observations, making maximum likelihood estimation the privileged approach for statistical inference for these models. A celebrated example of general order ODMs is the GARCH$(p,q)$ model, for which ergodicity and inference has been studied extensively. However little is known on more general models, in particular integer-valued ones, such as the log-linear Poisson GARCH or the NBIN-GARCH of order $(p,q)$ about which most of the existing results seem restricted to the case $p=q=1$. Here we fill this gap and derive ergodicity conditions for general ODMs. The consistency and the asymptotic normality of the maximum likelihood estimator (MLE) can then be derived using the method already developed for first order ODMs.

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arXiv2021

The $f$-Divergence Expectation Iteration Scheme

This paper introduces the $f$-EI$(ϕ)$ algorithm, a novel iterative algorithm which operates on measures and performs $f$-divergence minimisation in a Bayesian framework. We prove that for a rich family of values of $(f,ϕ)$ this algorithm leads at each step to a systematic decrease in the $f$-divergence and show that we achieve an optimum. In the particular case where we consider a weighted sum of Dirac measures and the $α$-divergence, we obtain that the calculations involved in the $f$-EI$(ϕ)$ algorithm simplify to gradient-based computations. Empirical results support the claim that the $f$-EI$(ϕ)$ algorithm serves as a powerful tool to assist Variational methods.

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arXiv2020

Infinite-dimensional gradient-based descent for alpha-divergence minimisation

This paper introduces the $(α, Γ)$-descent, an iterative algorithm which operates on measures and performs $α$-divergence minimisation in a Bayesian framework. This gradient-based procedure extends the commonly-used variational approximation by adding a prior on the variational parameters in the form of a measure. We prove that for a rich family of functions $Γ$, this algorithm leads at each step to a systematic decrease in the $α$-divergence and derive convergence results. Our framework recovers the Entropic Mirror Descent algorithm and provides an alternative algorithm that we call the Power Descent. Moreover, in its stochastic formulation, the $(α, Γ)$-descent allows to optimise the mixture weights of any given mixture model without any information on the underlying distribution of the variational parameters. This renders our method compatible with many choices of parameters updates and applicable to a wide range of Machine Learning tasks. We demonstrate empirically on both toy and real-world examples the benefit of using the Power descent and going beyond the Entropic Mirror Descent framework, which fails as the dimension grows.

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arXiv2020

Necessary and sufficient conditions for the identifiability of observation-driven models

In this contribution we are interested in proving that a given observation-driven model is identifiable. In the case of a GARCH(p, q) model, a simple sufficient condition has been established in [1] for showing the consistency of the quasi-maximum likelihood estimator. It turns out that this condition applies for a much larger class of observation-driven models, that we call the class of linearly observation-driven models. This class includes standard integer valued observation-driven time series, such as the log-linear Poisson GARCH or the NBIN-GARCH models.

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arXiv2017

Numerically stable online estimation of variance in particle filters

This paper discusses variance estimation in sequential Monte Carlo methods, alternatively termed particle filters. The variance estimator that we propose is a natural modification of that suggested by H. P. Chan and T. L. Lai [A general theory of particle filters in hidden Markov models and some applications. Ann. Statist., 41(6):2877-2904, 2013], which allows the variance to be estimated in a single run of the particle filter by tracing the genealogical history of the particles. However, due particle lineage degeneracy, the estimator of the mentioned work becomes numerically unstable as the number of sequential particle updates increases. Thus, by tracing only a part of the particles' genealogy rather than the full one, our estimator gains long-term numerical stability at the cost of a bias. The scope of the genealogical tracing is regulated by a lag, and under mild, easily checked model assumptions, we prove that the bias tends to zero geometrically fast as the lag increases. As confirmed by our numerical results, this allows the bias to be tightly controlled also for moderate particle sample sizes.

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arXiv2016

Posterior consistency for partially observed Markov models

In this work we establish the posterior consistency for a parametrized family of partially observed, fully dominated Markov models. As a main assumption, we suppose that the prior distribution assigns positive probability to all neighborhoods of the true parameter, for a distance induced by the expected Kullback-Leibler divergence between the family members' Markov transition densities. This assumption is easily checked in general. In addition, under some additional, mild assumptions we show that the posterior consistency is implied by the consistency of the maximum likelihood estimator. The latter has recently been established also for models with non-compact state space. The result is then extended to possibly non-compact parameter spaces and non-stationary observations. Finally, we check our assumptions on examples including the partially observed Gaussian linear model with correlated noise and a widely used stochastic volatility model.

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arXiv2015

The maximizing set of the asymptotic normalized log-likelihood for partially observed Markov chains

This paper deals with a parametrized family of partially observed bivariate Markov chains. We establish that, under very mild assumptions, the limit of the normalized log-likelihood function is maximized when the parameters belong to the equivalence class of the true parameter, which is a key feature for obtaining the consistency of the maximum likelihood estimators (MLEs) in well-specified models. This result is obtained in the general framework of partially dominated models. We examine two specific cases of interest, namely, hidden Markov models (HMMs) and observation-driven time series models. In contrast with previous approaches, the identifiability is addressed by relying on the uniqueness of the invariant distribution of the Markov chain associated to the complete data, regardless its rate of convergence to the equilibrium.

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arXiv2015

Handy sufficient conditions for the convergence of the maximum likelihood estimator in observation-driven models

This paper generalizes asymptotic properties obtained in the observation-driven times series models considered by \cite{dou:kou:mou:2013} in the sense that the conditional law of each observation is also permitted to depend on the parameter. The existence of ergodic solutions and the consistency of the Maximum Likelihood Estimator (MLE) are derived under easy-to-check conditions. The obtained conditions appear to apply for a wide class of models. We illustrate our results with specific observation-driven times series, including the recently introduced NBIN-GARCH and NM-GARCH models, demonstrating the consistency of the MLE for these two models.

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