We consider a generalization of the discrete-time Self Healing Umbrella Sampling method, which is... more We consider a generalization of the discrete-time Self Healing Umbrella Sampling method, which is an adaptive importance technique useful to sample multimodal target distributions. The importance function is based on the weights (namely the relative probabilities) of disjoint sets which form a partition of the space. These weights are unknown but are learnt on the fly yielding an adaptive algorithm. In the context of computational statistical physics, the logarithm of these weights is, up to a multiplicative constant, the free energy, and the discrete valued function defining the partition is called the collective variable. The algorithm falls into the general class of Wang-Landau type methods, and is a generalization of the original Self Healing Umbrella Sampling method in two ways: (i) the updating strategy leads to a larger penalization strength of already visited sets in order to escape more quickly from metastable states, and (ii) the target distribution is biased using only a fraction of the free energy, in order to increase the effective sample size and reduce the variance of importance sampling estimators. The algorithm can also be seen as a generalization of well-tempered metadynamics. We prove the convergence of the algorithm and analyze numerically its efficiency on a toy example.
2018 IEEE Statistical Signal Processing Workshop (SSP), 2018
Motivated by challenges in Computational Statistics such as Penalized Maximum Likelihood inferenc... more Motivated by challenges in Computational Statistics such as Penalized Maximum Likelihood inference in statistical models with intractable likelihoods, we analyze the convergence of a stochastic perturbation of the Fast Iterative Shrinkage-Thresholding Algorithm (FISTA), when the stochastic approximation relies on a biased Monte Carlo estimation as it happens when the points are drawn from a Markov chain Monte Carlo (MCMC) sampler. We first motivate this general framework and then show a convergence result for the perturbed FISTA algorithm. We discuss the convergence rate of this algorithm and the computational cost of the Monte Carlo approximation to reach a given precision. Finally, through a numerical example, we explore new directions for a better understanding of these Proximal-Gradient based stochastic optimization algorithms.
We study a version of the proximal gradient algorithm for which the gradient is intractable and i... more We study a version of the proximal gradient algorithm for which the gradient is intractable and is approximated by Monte Carlo methods (and in particular Markov Chain Monte Carlo). We derive conditions on the step size and the Monte Carlo batch size under which convergence is guaranteed: both increasing batch size and constant batch size are considered. We also derive non-asymptotic bounds for an averaged version. Our results cover both the cases of biased and unbiased Monte Carlo approximation. To support our findings, we discuss the inference of a sparse generalized linear model with random effect and the problem of learning the edge structure and parameters of sparse undirected graphical models.
We design and analyze an algorithm for estimating the mean of a function of a conditional expecta... more We design and analyze an algorithm for estimating the mean of a function of a conditional expectation when the outer expectation is related to a rare event. The outer expectation is evaluated through the average along the path of an ergodic Markov chain generated by a Markov chain Monte Carlo sampler. The inner conditional expectation is computed as a non-parametric regression, using a least-squares method with a general function basis and a design given by the sampled Markov chain. We establish non-asymptotic bounds for the
This paper is devoted to the convergence analysis of stochastic approximation algorithms of the f... more This paper is devoted to the convergence analysis of stochastic approximation algorithms of the form θ n+1 = θn + γ n+1 H θn (X n+1) where {θn, n ∈ N} is a R d-valued sequence, {γn, n ∈ N} is a deterministic step-size sequence and {Xn, n ∈ N} is a controlled Markov chain. We study the convergence under weak assumptions on smoothness-in-θ of the function θ → H θ (x). It is usually assumed that this function is continuous for any x; in this work, we relax this condition. Our results are illustrated by considering stochastic approximation algorithms for (adaptive) quantile estimation and a penalized version of the vector quantization.
Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, 2016
In this paper, we provide sufficient conditions for the existence of the invariant distribution a... more In this paper, we provide sufficient conditions for the existence of the invariant distribution and for subgeometric rates of convergence in Wasserstein distance for general state-space Markov chains which are (possibly) not irreducible. Compared to (Ann. Appl. Probab. 24 (2) (2014) 526-552), our approach is based on a purely probabilistic coupling construction which allows to retrieve rates of convergence matching those previously reported for convergence in total variation in (Bernoulli 13 (3) (2007) 831-848). Our results are applied to establish the subgeometric ergodicity in Wasserstein distance of non-linear autoregressive models and of the pre-conditioned Crank-Nicolson Markov chain Monte Carlo algorithm in Hilbert space. Résumé. Dans cet article, nous donnons des conditions suffisantes pour l'existence d'une probabilité invariante et qui permettent d'établir des taux de convergence sous-géométriques en distance de Wasserstein, pour des chaînes de Markov définies sur des espaces d'états généraux et non nécessairement irréductibles. Comparée à (Ann. Appl. Probab. 24 (2) (2014) 526-552), notre approche est basée sur une construction par couplage purement probabiliste, ce qui permet de retrouver les taux de convergence obtenus précédement pour la variation total dans (Bernoulli 13 (3) (2007) 831-848). Par application de ces résultats, nous établissons la convergence sous-géométrique en distance de Wasserstein de modèles non linéaires auto-régressifs et l'algorithme de MCMC, l'algorithme de Crank-Nicolson pré-conditionné dans les espaces de Hilbert, pour une certaine classe de mesure cible.
Proceedings of the 1st international conference on Performance evaluation methodolgies and tools - valuetools '06, 2006
Fluid limit techniques have become a central tool to analyze queueing networks over the last deca... more Fluid limit techniques have become a central tool to analyze queueing networks over the last decade, with applications to performance analysis, simulation, and optimization. In this paper some of these techniques are extended to a general class of skip-free Markov chains. As in the case of queueing models, a fluid approximation is obtained by scaling time, space, and the initial condition by a large constant. The resulting fluid limit is the solution of an ODE in "most" of the state space. Stability and finer ergodic properties for the stochastic model then follow from stability of the set of fluid limits. Moreover, similar to the queueing context where fluid models are routinely used to design control policies, the structure of the limiting ODE in this general setting provides an understanding of the dynamics of the Markov chain. These results are illustrated through application to Markov Chain Monte Carlo.
IEEE Journal of Selected Topics in Signal Processing, 2016
This paper introduces a new Markov Chain Monte Carlo method for Bayesian variable selection in hi... more This paper introduces a new Markov Chain Monte Carlo method for Bayesian variable selection in high dimensional settings. The algorithm is a Hastings-Metropolis sampler with a proposal mechanism which combines a Metropolis Adjusted Langevin (MALA) step to propose local moves associated with a shrinkage-thresholding step allowing to propose new models. The geometric ergodicity of this new trans-dimensional Markov Chain Monte Carlo sampler is established. An extensive numerical experiment, on simulated and real data, is presented to illustrate the performance of the proposed algorithm in comparison with some more classical trans-dimensional algorithms.
We provide explicit expressions for the constants involved in the characterisation of ergodicity ... more We provide explicit expressions for the constants involved in the characterisation of ergodicity of subgeometric Markov chains. The constants are determined in terms of those appearing in the assumed drift and one-step minorisation conditions. The results are fundamental for the study of some algorithms where uniform bounds for these constants are needed for a family of Markov kernels. Our results accommodate also some classes of inhomogeneous chains.
In this paper, we consider the random-scan symmetric random walk Metropolis algorithm (RSM) on ℝd... more In this paper, we consider the random-scan symmetric random walk Metropolis algorithm (RSM) on ℝd. This algorithm performs a Metropolis step on just one coordinate at a time (as opposed to the full-dimensional symmetric random walk Metropolis algorithm, which proposes a transition on all coordinates at once). We present various sufficient conditions implying V-uniform ergodicity of the RSM when the target density decreases either subexponentially or exponentially in the tails.
When targeting a distribution that is artificially invariant under some permutations, Markov chai... more When targeting a distribution that is artificially invariant under some permutations, Markov chain Monte Carlo (MCMC) algorithms face the label-switching problem, rendering marginal inference particularly cumbersome. Such a situation arises, for example, in the Bayesian analysis of finite mixture models. Adaptive MCMC algorithms such as adaptive Metropolis (AM), which self-calibrates its proposal distribution using an online estimate of the covariance matrix of the target, are no exception. To address the label-switching issue, relabeling algorithms associate a permutation to each MCMC sample, trying to obtain reasonable marginals. In the case of adaptive Metropolis (Bernoulli 7 (2001) 223-242), an online relabeling strategy is required. This paper is devoted to the AMOR algorithm, a provably consistent variant of AM that can cope with the label-switching problem. The idea is to nest relabeling steps within the MCMC algorithm based on the estimation of a single covariance matrix that is used both for adapting the covariance of the proposal distribution in the Metropolis algorithm step and for online relabeling. We compare the behavior of AMOR to similar relabeling methods. In the case of compactly supported target distributions, we prove a strong law of large numbers for AMOR and its ergodicity. These are the first results on the consistency of an online relabeling algorithm to our knowledge. The proof underlines latent relations between relabeling and vector quantization.
This paper provides a Central Limit Theorem (CLT) for a process {θn, n ≥ 0} satisfying a stochast... more This paper provides a Central Limit Theorem (CLT) for a process {θn, n ≥ 0} satisfying a stochastic approximation (SA) equation of the form θn+1 = θn + γn+1H(θn, Xn+1); a CLT for the associated average sequence is also established. The originality of this paper is to address the case of controlled Markov chain dynamics {Xn, n ≥ 0} and the case of multiple targets. The framework also accomodates (randomly) truncated SA algorithms. Sufficient conditions for CLT's hold are provided as well as comments on how these conditions extend previous works (such as independent and identically distributed dynamics, the Robbins−Monro dynamic or the single target case). The paper gives a special emphasis on how these conditions hold for SA with controlled Markov chain dynamics and multiple targets; it is proved that this paper improves on existing works.
Computable bounds for subgeometrical and geometrical ergodicity
. This paper discusses general quantitative bounds on the convergence rates of Markov chains,unde... more . This paper discusses general quantitative bounds on the convergence rates of Markov chains,under various ergodicity conditions. This paper extends an earlier work by Roberts and Tweedie (1999),which provides quantitative bounds for the total variation norm under conditions implying geometricergodicity.We rst focus on conditions implying polynomial rate of convergence. Explicit bounds for the totalvariation norm are obtained by evaluating the
Adaptive and interacting Markov Chains Monte Carlo (MCMC) algorithms are a novel class of non-Mar... more Adaptive and interacting Markov Chains Monte Carlo (MCMC) algorithms are a novel class of non-Markovian algorithms aimed at improving the simulation efficiency for complicated target distributions. In this paper, we study a general (non-Markovian) simulation framework covering both the adaptive and interacting MCMC algorithms. We establish a central limit theorem for additive functionals of unbounded functions under a set of verifiable conditions, and identify the asymptotic variance. Our result extends all the results reported so far. An application to the interacting tempering algorithm (a simplified version of the equi-energy sampler) is presented to support our claims.
We consider a form of state-dependent drift condition for a general Markov chain, whereby the cha... more We consider a form of state-dependent drift condition for a general Markov chain, whereby the chain subsampled at some deterministic time satisfies a geometric Foster-Lyapunov condition. We present sufficient criteria for such a drift condition to exist, and use these to partially answer a question posed in [2] concerning the existence of so-called 'tame' Markov chains. Furthermore, we show that our 'subsampled drift condition' implies the existence of finite moments for the return time to a small set.
This paper discusses quantitative bounds on the convergence rates of Markov chains, under conditi... more This paper discusses quantitative bounds on the convergence rates of Markov chains, under conditions implying polynomial convergence rates. This paper extends an earlier work by Roberts and Tweedie (Stochastic Process. Appl. 80(2) (1999) 211), which provides quantitative bounds for the total variation norm under conditions implying geometric ergodicity. Explicit bounds for the total variation norm are obtained by evaluating the moments of an appropriately deÿned coupling time, using a set of drift conditions, adapted from an earlier work by Tuominen and Tweedie (Adv. Appl. Probab. 26(3) (1994) 775). Applications of this result are then presented to study the convergence of random walk Hastings Metropolis algorithm for super-exponential target functions and of general state-space models. Explicit bounds for f-ergodicity are also given, for an appropriately deÿned control function f.
We establish a simple variance inequality for U-statistics whose underlying sequence of random va... more We establish a simple variance inequality for U-statistics whose underlying sequence of random variables is an ergodic Markov Chain. The constants in this inequality are explicit and depend on computable bounds on the mixing rate of the Markov Chain. We apply this result to derive the strong law of large number for U-statistics of a Markov Chain under conditions which are close from being optimal.
We present a Bayesian sampling algorithm called adaptive importance sampling or Population Monte ... more We present a Bayesian sampling algorithm called adaptive importance sampling or Population Monte Carlo (PMC), whose computational workload is easily parallelizable and thus has the potential to considerably reduce the wall-clock time required for sampling, along with providing other benefits. To assess the performance of the approach for cosmological problems, we use simulated and actual data consisting of CMB anisotropies, supernovae of type Ia, and weak cosmological lensing, and provide a comparison of results to those obtained using state-of-the-art Markov Chain Monte Carlo (MCMC). For both types of data sets, we find comparable parameter estimates for PMC and MCMC, with the advantage of a significantly lower computational time for PMC. In the case of WMAP5 data, for example, the wall-clock time reduces from several days for MCMC to a few hours using PMC on a cluster of processors. Other benefits of the PMC approach, along with potential difficulties in using the approach, are analysed and discussed.
Monthly Notices of the Royal Astronomical Society, 2010
We use Bayesian model selection techniques to test extensions of the standard flat cold dark matt... more We use Bayesian model selection techniques to test extensions of the standard flat cold dark matter (CDM) paradigm. Dark-energy and curvature scenarios, and primordial perturbation models are considered. To that end, we calculate the Bayesian evidence in favour of each model using Population Monte Carlo (PMC), a new adaptive sampling technique which was recently applied in a cosmological context. In contrast to the case of other sampling-based inference techniques such as Markov chain Monte Carlo (MCMC), the Bayesian evidence is immediately available from the PMC sample used for parameter estimation without further computational effort, and it comes with an associated error evaluation. Also, it provides an unbiased estimator of the evidence after any fixed number of iterations and it is naturally parallelizable, in contrast with MCMC and nested sampling methods. By comparison with analytical predictions for simulated data, we show that our results obtained with PMC are reliable and robust. The variability in the evidence evaluation and the stability for various cases are estimated both from simulations and from data. For the cases we consider, the log-evidence is calculated with a precision of better than 0.08. Using a combined set of recent cosmic microwave background, type Ia supernovae and baryonic acoustic oscillation data, we find inconclusive evidence between flat CDM and simple dark-energy models. A curved universe is moderately to strongly disfavoured with respect to a flat cosmology. Using physically well-motivated priors within the slow-roll approximation of inflation, we find a weak preference for a running spectral index. A Harrison-Zel'dovich spectrum is weakly disfavoured. With the current data, tensor modes are not detected; the large prior volume on the tensor-to-scalar ratio r results in moderate evidence in favour of r = 0.
We consider a generalization of the discrete-time Self Healing Umbrella Sampling method, which is... more We consider a generalization of the discrete-time Self Healing Umbrella Sampling method, which is an adaptive importance technique useful to sample multimodal target distributions. The importance function is based on the weights (namely the relative probabilities) of disjoint sets which form a partition of the space. These weights are unknown but are learnt on the fly yielding an adaptive algorithm. In the context of computational statistical physics, the logarithm of these weights is, up to a multiplicative constant, the free energy, and the discrete valued function defining the partition is called the collective variable. The algorithm falls into the general class of Wang-Landau type methods, and is a generalization of the original Self Healing Umbrella Sampling method in two ways: (i) the updating strategy leads to a larger penalization strength of already visited sets in order to escape more quickly from metastable states, and (ii) the target distribution is biased using only a fraction of the free energy, in order to increase the effective sample size and reduce the variance of importance sampling estimators. The algorithm can also be seen as a generalization of well-tempered metadynamics. We prove the convergence of the algorithm and analyze numerically its efficiency on a toy example.
2018 IEEE Statistical Signal Processing Workshop (SSP), 2018
Motivated by challenges in Computational Statistics such as Penalized Maximum Likelihood inferenc... more Motivated by challenges in Computational Statistics such as Penalized Maximum Likelihood inference in statistical models with intractable likelihoods, we analyze the convergence of a stochastic perturbation of the Fast Iterative Shrinkage-Thresholding Algorithm (FISTA), when the stochastic approximation relies on a biased Monte Carlo estimation as it happens when the points are drawn from a Markov chain Monte Carlo (MCMC) sampler. We first motivate this general framework and then show a convergence result for the perturbed FISTA algorithm. We discuss the convergence rate of this algorithm and the computational cost of the Monte Carlo approximation to reach a given precision. Finally, through a numerical example, we explore new directions for a better understanding of these Proximal-Gradient based stochastic optimization algorithms.
We study a version of the proximal gradient algorithm for which the gradient is intractable and i... more We study a version of the proximal gradient algorithm for which the gradient is intractable and is approximated by Monte Carlo methods (and in particular Markov Chain Monte Carlo). We derive conditions on the step size and the Monte Carlo batch size under which convergence is guaranteed: both increasing batch size and constant batch size are considered. We also derive non-asymptotic bounds for an averaged version. Our results cover both the cases of biased and unbiased Monte Carlo approximation. To support our findings, we discuss the inference of a sparse generalized linear model with random effect and the problem of learning the edge structure and parameters of sparse undirected graphical models.
We design and analyze an algorithm for estimating the mean of a function of a conditional expecta... more We design and analyze an algorithm for estimating the mean of a function of a conditional expectation when the outer expectation is related to a rare event. The outer expectation is evaluated through the average along the path of an ergodic Markov chain generated by a Markov chain Monte Carlo sampler. The inner conditional expectation is computed as a non-parametric regression, using a least-squares method with a general function basis and a design given by the sampled Markov chain. We establish non-asymptotic bounds for the
This paper is devoted to the convergence analysis of stochastic approximation algorithms of the f... more This paper is devoted to the convergence analysis of stochastic approximation algorithms of the form θ n+1 = θn + γ n+1 H θn (X n+1) where {θn, n ∈ N} is a R d-valued sequence, {γn, n ∈ N} is a deterministic step-size sequence and {Xn, n ∈ N} is a controlled Markov chain. We study the convergence under weak assumptions on smoothness-in-θ of the function θ → H θ (x). It is usually assumed that this function is continuous for any x; in this work, we relax this condition. Our results are illustrated by considering stochastic approximation algorithms for (adaptive) quantile estimation and a penalized version of the vector quantization.
Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, 2016
In this paper, we provide sufficient conditions for the existence of the invariant distribution a... more In this paper, we provide sufficient conditions for the existence of the invariant distribution and for subgeometric rates of convergence in Wasserstein distance for general state-space Markov chains which are (possibly) not irreducible. Compared to (Ann. Appl. Probab. 24 (2) (2014) 526-552), our approach is based on a purely probabilistic coupling construction which allows to retrieve rates of convergence matching those previously reported for convergence in total variation in (Bernoulli 13 (3) (2007) 831-848). Our results are applied to establish the subgeometric ergodicity in Wasserstein distance of non-linear autoregressive models and of the pre-conditioned Crank-Nicolson Markov chain Monte Carlo algorithm in Hilbert space. Résumé. Dans cet article, nous donnons des conditions suffisantes pour l'existence d'une probabilité invariante et qui permettent d'établir des taux de convergence sous-géométriques en distance de Wasserstein, pour des chaînes de Markov définies sur des espaces d'états généraux et non nécessairement irréductibles. Comparée à (Ann. Appl. Probab. 24 (2) (2014) 526-552), notre approche est basée sur une construction par couplage purement probabiliste, ce qui permet de retrouver les taux de convergence obtenus précédement pour la variation total dans (Bernoulli 13 (3) (2007) 831-848). Par application de ces résultats, nous établissons la convergence sous-géométrique en distance de Wasserstein de modèles non linéaires auto-régressifs et l'algorithme de MCMC, l'algorithme de Crank-Nicolson pré-conditionné dans les espaces de Hilbert, pour une certaine classe de mesure cible.
Proceedings of the 1st international conference on Performance evaluation methodolgies and tools - valuetools '06, 2006
Fluid limit techniques have become a central tool to analyze queueing networks over the last deca... more Fluid limit techniques have become a central tool to analyze queueing networks over the last decade, with applications to performance analysis, simulation, and optimization. In this paper some of these techniques are extended to a general class of skip-free Markov chains. As in the case of queueing models, a fluid approximation is obtained by scaling time, space, and the initial condition by a large constant. The resulting fluid limit is the solution of an ODE in "most" of the state space. Stability and finer ergodic properties for the stochastic model then follow from stability of the set of fluid limits. Moreover, similar to the queueing context where fluid models are routinely used to design control policies, the structure of the limiting ODE in this general setting provides an understanding of the dynamics of the Markov chain. These results are illustrated through application to Markov Chain Monte Carlo.
IEEE Journal of Selected Topics in Signal Processing, 2016
This paper introduces a new Markov Chain Monte Carlo method for Bayesian variable selection in hi... more This paper introduces a new Markov Chain Monte Carlo method for Bayesian variable selection in high dimensional settings. The algorithm is a Hastings-Metropolis sampler with a proposal mechanism which combines a Metropolis Adjusted Langevin (MALA) step to propose local moves associated with a shrinkage-thresholding step allowing to propose new models. The geometric ergodicity of this new trans-dimensional Markov Chain Monte Carlo sampler is established. An extensive numerical experiment, on simulated and real data, is presented to illustrate the performance of the proposed algorithm in comparison with some more classical trans-dimensional algorithms.
We provide explicit expressions for the constants involved in the characterisation of ergodicity ... more We provide explicit expressions for the constants involved in the characterisation of ergodicity of subgeometric Markov chains. The constants are determined in terms of those appearing in the assumed drift and one-step minorisation conditions. The results are fundamental for the study of some algorithms where uniform bounds for these constants are needed for a family of Markov kernels. Our results accommodate also some classes of inhomogeneous chains.
In this paper, we consider the random-scan symmetric random walk Metropolis algorithm (RSM) on ℝd... more In this paper, we consider the random-scan symmetric random walk Metropolis algorithm (RSM) on ℝd. This algorithm performs a Metropolis step on just one coordinate at a time (as opposed to the full-dimensional symmetric random walk Metropolis algorithm, which proposes a transition on all coordinates at once). We present various sufficient conditions implying V-uniform ergodicity of the RSM when the target density decreases either subexponentially or exponentially in the tails.
When targeting a distribution that is artificially invariant under some permutations, Markov chai... more When targeting a distribution that is artificially invariant under some permutations, Markov chain Monte Carlo (MCMC) algorithms face the label-switching problem, rendering marginal inference particularly cumbersome. Such a situation arises, for example, in the Bayesian analysis of finite mixture models. Adaptive MCMC algorithms such as adaptive Metropolis (AM), which self-calibrates its proposal distribution using an online estimate of the covariance matrix of the target, are no exception. To address the label-switching issue, relabeling algorithms associate a permutation to each MCMC sample, trying to obtain reasonable marginals. In the case of adaptive Metropolis (Bernoulli 7 (2001) 223-242), an online relabeling strategy is required. This paper is devoted to the AMOR algorithm, a provably consistent variant of AM that can cope with the label-switching problem. The idea is to nest relabeling steps within the MCMC algorithm based on the estimation of a single covariance matrix that is used both for adapting the covariance of the proposal distribution in the Metropolis algorithm step and for online relabeling. We compare the behavior of AMOR to similar relabeling methods. In the case of compactly supported target distributions, we prove a strong law of large numbers for AMOR and its ergodicity. These are the first results on the consistency of an online relabeling algorithm to our knowledge. The proof underlines latent relations between relabeling and vector quantization.
This paper provides a Central Limit Theorem (CLT) for a process {θn, n ≥ 0} satisfying a stochast... more This paper provides a Central Limit Theorem (CLT) for a process {θn, n ≥ 0} satisfying a stochastic approximation (SA) equation of the form θn+1 = θn + γn+1H(θn, Xn+1); a CLT for the associated average sequence is also established. The originality of this paper is to address the case of controlled Markov chain dynamics {Xn, n ≥ 0} and the case of multiple targets. The framework also accomodates (randomly) truncated SA algorithms. Sufficient conditions for CLT's hold are provided as well as comments on how these conditions extend previous works (such as independent and identically distributed dynamics, the Robbins−Monro dynamic or the single target case). The paper gives a special emphasis on how these conditions hold for SA with controlled Markov chain dynamics and multiple targets; it is proved that this paper improves on existing works.
Computable bounds for subgeometrical and geometrical ergodicity
. This paper discusses general quantitative bounds on the convergence rates of Markov chains,unde... more . This paper discusses general quantitative bounds on the convergence rates of Markov chains,under various ergodicity conditions. This paper extends an earlier work by Roberts and Tweedie (1999),which provides quantitative bounds for the total variation norm under conditions implying geometricergodicity.We rst focus on conditions implying polynomial rate of convergence. Explicit bounds for the totalvariation norm are obtained by evaluating the
Adaptive and interacting Markov Chains Monte Carlo (MCMC) algorithms are a novel class of non-Mar... more Adaptive and interacting Markov Chains Monte Carlo (MCMC) algorithms are a novel class of non-Markovian algorithms aimed at improving the simulation efficiency for complicated target distributions. In this paper, we study a general (non-Markovian) simulation framework covering both the adaptive and interacting MCMC algorithms. We establish a central limit theorem for additive functionals of unbounded functions under a set of verifiable conditions, and identify the asymptotic variance. Our result extends all the results reported so far. An application to the interacting tempering algorithm (a simplified version of the equi-energy sampler) is presented to support our claims.
We consider a form of state-dependent drift condition for a general Markov chain, whereby the cha... more We consider a form of state-dependent drift condition for a general Markov chain, whereby the chain subsampled at some deterministic time satisfies a geometric Foster-Lyapunov condition. We present sufficient criteria for such a drift condition to exist, and use these to partially answer a question posed in [2] concerning the existence of so-called 'tame' Markov chains. Furthermore, we show that our 'subsampled drift condition' implies the existence of finite moments for the return time to a small set.
This paper discusses quantitative bounds on the convergence rates of Markov chains, under conditi... more This paper discusses quantitative bounds on the convergence rates of Markov chains, under conditions implying polynomial convergence rates. This paper extends an earlier work by Roberts and Tweedie (Stochastic Process. Appl. 80(2) (1999) 211), which provides quantitative bounds for the total variation norm under conditions implying geometric ergodicity. Explicit bounds for the total variation norm are obtained by evaluating the moments of an appropriately deÿned coupling time, using a set of drift conditions, adapted from an earlier work by Tuominen and Tweedie (Adv. Appl. Probab. 26(3) (1994) 775). Applications of this result are then presented to study the convergence of random walk Hastings Metropolis algorithm for super-exponential target functions and of general state-space models. Explicit bounds for f-ergodicity are also given, for an appropriately deÿned control function f.
We establish a simple variance inequality for U-statistics whose underlying sequence of random va... more We establish a simple variance inequality for U-statistics whose underlying sequence of random variables is an ergodic Markov Chain. The constants in this inequality are explicit and depend on computable bounds on the mixing rate of the Markov Chain. We apply this result to derive the strong law of large number for U-statistics of a Markov Chain under conditions which are close from being optimal.
We present a Bayesian sampling algorithm called adaptive importance sampling or Population Monte ... more We present a Bayesian sampling algorithm called adaptive importance sampling or Population Monte Carlo (PMC), whose computational workload is easily parallelizable and thus has the potential to considerably reduce the wall-clock time required for sampling, along with providing other benefits. To assess the performance of the approach for cosmological problems, we use simulated and actual data consisting of CMB anisotropies, supernovae of type Ia, and weak cosmological lensing, and provide a comparison of results to those obtained using state-of-the-art Markov Chain Monte Carlo (MCMC). For both types of data sets, we find comparable parameter estimates for PMC and MCMC, with the advantage of a significantly lower computational time for PMC. In the case of WMAP5 data, for example, the wall-clock time reduces from several days for MCMC to a few hours using PMC on a cluster of processors. Other benefits of the PMC approach, along with potential difficulties in using the approach, are analysed and discussed.
Monthly Notices of the Royal Astronomical Society, 2010
We use Bayesian model selection techniques to test extensions of the standard flat cold dark matt... more We use Bayesian model selection techniques to test extensions of the standard flat cold dark matter (CDM) paradigm. Dark-energy and curvature scenarios, and primordial perturbation models are considered. To that end, we calculate the Bayesian evidence in favour of each model using Population Monte Carlo (PMC), a new adaptive sampling technique which was recently applied in a cosmological context. In contrast to the case of other sampling-based inference techniques such as Markov chain Monte Carlo (MCMC), the Bayesian evidence is immediately available from the PMC sample used for parameter estimation without further computational effort, and it comes with an associated error evaluation. Also, it provides an unbiased estimator of the evidence after any fixed number of iterations and it is naturally parallelizable, in contrast with MCMC and nested sampling methods. By comparison with analytical predictions for simulated data, we show that our results obtained with PMC are reliable and robust. The variability in the evidence evaluation and the stability for various cases are estimated both from simulations and from data. For the cases we consider, the log-evidence is calculated with a precision of better than 0.08. Using a combined set of recent cosmic microwave background, type Ia supernovae and baryonic acoustic oscillation data, we find inconclusive evidence between flat CDM and simple dark-energy models. A curved universe is moderately to strongly disfavoured with respect to a flat cosmology. Using physically well-motivated priors within the slow-roll approximation of inflation, we find a weak preference for a running spectral index. A Harrison-Zel'dovich spectrum is weakly disfavoured. With the current data, tensor modes are not detected; the large prior volume on the tensor-to-scalar ratio r results in moderate evidence in favour of r = 0.
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