The Hamiltonian Monte Carlo (HMC) method has been recognized as a powerful sampling tool in computational statistics. We show that performance of HMC can be significantly improved by incorporating importance sampling and an irreversible part of the dynamics into a chain. This is achieved by replacing Hamiltonians in the Metropolis test with modified Hamiltonians, and a complete momentum update with a partial momentum refreshment. We call the resulting generalized HMC importance sampler---Mix & Match Hamiltonian Monte Carlo (MMHMC). The method is irreversible by construction and further benefits from (i) the efficient algorithms for computation of modified Hamiltonians; (ii) the implicit momentum update procedure and (iii) the multi-stage splitting integrators specially derived for the methods sampling with modified Hamiltonians. MMHMC has been implemented, tested on the popular statistical models and compared in sampling efficiency with HMC, Riemann Manifold Hamiltonian Monte Carlo, Generalized Hybrid Monte Carlo, Generalized Shadow Hybrid Monte Carlo, Metropolis Adjusted Langevin Algorithm and Random Walk Metropolis-Hastings. To make a fair comparison, we propose a metric that accounts for correlations among samples and weights, and can be readily used for all methods which generate such samples. The experiments reveal the superiority of MMHMC over popular sampling techniques, especially in solving high dimensional problems.
We consider the problem of estimating rare event probabilities, focusing on systems whose evolution is governed by differential equations with uncertain input parameters. If the system dynamics is expensive to compute, standard sampling algorithms such as the Monte Carlo method may require infeasible running times to accurately evaluate these probabilities. We propose an importance sampling scheme (which we call BIMC) that relies on solving an auxiliary, fictitious Bayesian inverse problem. The solution of the inverse problem yields a posterior PDF, a local Gaussian approximation to which serv…
Dynamically rescaled Hamiltonian Monte Carlo (DRHMC) is introduced as a computationally fast and easily implemented method for performing full Bayesian analysis in hierarchical statistical models. The method relies on introducing a modified parameterisation so that the re-parameterised target distribution has close to constant scaling properties, and thus is easily sampled using standard (Euclidian metric) Hamiltonian Monte Carlo. Provided that the parameterisations of the conditional distributions specifying the hierarchical model are "constant information parameterisations" (CIP), the relati…
Bayesian inference under a set of priors, called robust Bayesian analysis, allows for estimation of parameters within a model and quantification of epistemic uncertainty in quantities of interest by bounded (or imprecise) probability. Iterative importance sampling can be used to estimate bounds on the quantity of interest by optimizing over the set of priors. A method for iterative importance sampling when the robust Bayesian inference rely on Markov chain Monte Carlo (MCMC) sampling is proposed. To accommodate the MCMC sampling in iterative importance sampling, a new expression for the effect…
Modern imaging methods rely strongly on Bayesian inference techniques to solve challenging imaging problems. Currently, the predominant Bayesian computation approach is convex optimisation, which scales very efficiently to high dimensional image models and delivers accurate point estimation results. However, in order to perform more complex analyses, for example image uncertainty quantification or model selection, it is necessary to use more computationally intensive Bayesian computation techniques such as Markov chain Monte Carlo methods. This paper presents a new and highly efficient Markov …
ABC (approximate Bayesian computation) is a general approach for dealing with models with an intractable likelihood. In this work, we derive ABC algorithms based on QMC (quasi- Monte Carlo) sequences. We show that the resulting ABC estimates have a lower variance than their Monte Carlo counter-parts. We also develop QMC variants of sequential ABC algorithms, which progressively adapt the proposal distribution and the acceptance threshold. We illustrate our QMC approach through several examples taken from the ABC literature. Keywords: Approximate Bayesian computation, Likelihood-free inference,…
Many real-world problems require one to estimate parameters of interest, in a Bayesian framework, from data that are collected sequentially in time. Conventional methods for sampling from posterior distributions, such as {Markov Chain Monte Carlo} can not efficiently address such problems as they do not take advantage of the data's sequential structure. To this end, sequential methods which seek to update the posterior distribution whenever a new collection of data become available are often used to solve these types of problems. Two popular choices of sequential method are the Ensemble Kalman…
While Robert and Rousseau (2010) addressed the foundational aspects of Bayesian analysis, the current chapter details its practical aspects through a review of the computational methods available for approximating Bayesian procedures. Recent innovations like Monte Carlo Markov chain, sequential Monte Carlo methods and more recently Approximate Bayesian Computation techniques have considerably increased the potential for Bayesian applications and they have also opened new avenues for Bayesian inference, first and foremost Bayesian model choice.
Functional data registration is a necessary processing step for many applications. The observed data can be inherently noisy, often due to measurement error or natural process uncertainty, which most functional alignment methods cannot handle. A pair of functions can also have multiple optimal alignment solutions, which is not addressed in current literature. In this paper, a flexible Bayesian approach to functional alignment is presented, which appropriately accounts for noise in the data without any pre-smoothing required. Additionally, by running parallel MCMC chains, the method can account…
Riemann manifold Hamiltonian Monte Carlo (RMHMC) has the potential to produce high-quality Markov chain Monte Carlo-output even for very challenging target distributions. To this end, a symmetric positive definite scaling matrix for RMHMC, which derives, via a modified Cholesky factorization, from the potentially indefinite negative Hessian of the target log-density is proposed. The methodology is able to exploit the sparsity of the Hessian, stemming from conditional independence modeling assumptions, and thus admit fast implementation of RMHMC even for high-dimensional target distributions. M…
In the past decade, many Bayesian shrinkage models have been developed for linear regression problems where the number of covariates, $p$, is large. Computing the intractable posterior are often done with three-block Gibbs samplers (3BG), based on representing the shrinkage priors as scale mixtures of Normal distributions. An alternative computing tool is a state of the art Hamiltonian Monte Carlo (HMC) method, which can be easily implemented in the Stan software. However, we found both existing methods to be inefficient and often impractical for large $p$ problems. Following the general idea …
In the following article we consider approximate Bayesian computation (ABC) inference. We introduce a method for numerically approximating ABC posteriors using the multilevel Monte Carlo (MLMC). A sequential Monte Carlo version of the approach is developed and it is shown under some assumptions that for a given level of mean square error, this method for ABC has a lower cost than i.i.d. sampling from the most accurate ABC approximation. Several numerical examples are given.
In the context of Bayesian inversion for scientific and engineering modeling, Markov chain Monte Carlo sampling strategies are the benchmark due to their flexibility and robustness in dealing with arbitrary posterior probability density functions (PDFs). However, these algorithms been shown to be inefficient when sampling from posterior distributions that are high-dimensional or exhibit multi-modality and/or strong parameter correlations. In such contexts, the sequential Monte Carlo technique of transitional Markov chain Monte Carlo (TMCMC) provides a more efficient alternative. Despite the re…
The problem of Bayesian reduced rank regression is considered in this paper. We propose, for the first time, to use Langevin Monte Carlo method in this problem. A spectral scaled Student prior distrbution is used to exploit the underlying low-rank structure of the coefficient matrix. We show that our algorithms are significantly faster than the Gibbs sampler in high-dimensional setting. Simulation results show that our proposed algorithms for Bayesian reduced rank regression are comparable to the state-of-the-art method where the rank is chosen by cross validation.
Approximate Bayesian computation has emerged as a standard computational tool when dealing with the increasingly common scenario of completely intractable likelihood functions in Bayesian inference. We show that many common Markov chain Monte Carlo kernels used to facilitate inference in this setting can fail to be variance bounding, and hence geometrically ergodic, which can have consequences for the reliability of estimates in practice. This phenomenon is typically independent of the choice of tolerance in the approximation. We then prove that a recently introduced Markov kernel in this sett…
The main focus of the analysts who deal with clustered data is usually not on the clustering variables, and hence the group-specific parameters are treated as nuisance. If a fixed effects formulation is preferred and the total number of clusters is large relative to the single-group sizes, classical frequentist techniques relying on the profile likelihood are often misleading. The use of alternative tools, such as modifications to the profile likelihood or integrated likelihoods, for making accurate inference on a parameter of interest can be complicated by the presence of nonstandard modellin…
In this paper we propose to evaluate and compare Markov chain Monte Carlo (MCMC) methods to estimate the parameters in a generalized extreme value model. We employed the Bayesian approach using traditional Metropolis-Hastings methods, Hamiltonian Monte Carlo (HMC) and Riemann manifold HMC (RMHMC) methods to obtain the approximations to the posterior marginal distributions of interest. Applications to real datasets of maxima illustrate illustrate how HMC can be much more efficient computationally than traditional MCMC and simulation studies are conducted to compare the algorithms in terms of ho…
Approximate Bayesian computation allows for inference of complicated probabilistic models with intractable likelihoods using model simulations. The Markov chain Monte Carlo implementation of approximate Bayesian computation is often sensitive to the tolerance parameter: low tolerance leads to poor mixing and large tolerance entails excess bias. We consider an approach using a relatively large tolerance for the Markov chain Monte Carlo sampler to ensure its sufficient mixing, and post-processing the output leading to estimators for a range of finer tolerances. We introduce an approximate confid…
Multifidelity approximate Bayesian computation (MF-ABC) is a likelihood-free technique for parameter inference that exploits model approximations to significantly increase the speed of ABC algorithms (Prescott and Baker, 2020). Previous work has considered MF-ABC only in the context of rejection sampling, which does not explore parameter space particularly efficiently. In this work, we integrate the multifidelity approach with the ABC sequential Monte Carlo (ABC-SMC) algorithm into a new MF-ABC-SMC algorithm. We show that the improvements generated by each of ABC-SMC and MF-ABC to the efficien…
We study probit regression from a Bayesian perspective and give an alternative form for the posterior distribution when the prior distribution for the regression parameters is the uniform distribution. This new form allows simple Monte Carlo simulation of the posterior as opposed to MCMC simulation studied in much of the literature and may therefore be more efficient computationally. We also provide alternative explicit expression for the first and second moments. Additionally we provide analogous results for Gaussian priors.
Riemannian manifold Hamiltonian (RMHMC) and Lagrangian Monte Carlo (LMC) have emerged as powerful methods of Bayesian inference. Unlike Euclidean Hamiltonian Monte Carlo (EHMC) and the Metropolis-adjusted Langevin algorithm (MALA), the geometric ergodicity of these Riemannian algorithms has not been extensively studied. On the other hand, the manifold Metropolis-adjusted Langevin algorithm (MMALA) has recently been shown to exhibit geometric ergodicity under certain conditions. This work investigates the mixture of the LMC and RMHMC transition kernels with MMALA in order to equip the resulting…
We describe an "embarrassingly parallel" method for Bayesian phylogenetic inference, annealed Sequential Monte Carlo, based on recent advances in the Sequential Monte Carlo literature such as adaptive determination of annealing parameters. The algorithm provides an approximate posterior distribution over trees and evolutionary parameters as well as an unbiased estimator for the marginal likelihood. This unbiasedness property can be used for the purpose of testing the correctness of posterior simulation software. We evaluate the performance of phylogenetic annealed Sequential Monte Carlo by rev…
In this paper we address the problem of Monte Carlo approximation of posterior probability distributions in stochastic kinetic models (SKMs). SKMs are multivariate Markov jump processes that model the interactions among species in biochemical systems according to a set of uncertain parameters. Markov chain Monte Carlo (MCMC) methods have been typically preferred for this Bayesian inference problem. Specifically, the particle MCMC (pMCMC) method has been recently shown to be an effective, while computationally demanding, method applicable to this problem. Within the pMCMC framework, importance …
Bayesian filtering aims at tracking sequentially a hidden process from an observed one. In particular, sequential Monte Carlo (SMC) techniques propagate in time weighted trajectories which represent the posterior probability density function (pdf) of the hidden process given the available observations. On the other hand, Conditional Monte Carlo (CMC) is a variance reduction technique which replaces the estimator of a moment of interest by its conditional expectation given another variable. In this paper we show that up to some adaptations, one can make use of the time recursive nature of SMC a…
Markov Chain Monte Carlo based Bayesian data analysis has now become the method of choice for analyzing and interpreting data in almost all disciplines of science. In astronomy, over the last decade, we have also seen a steady increase in the number of papers that employ Monte Carlo based Bayesian analysis. New, efficient Monte Carlo based methods are continuously being developed and explored. In this review, we first explain the basics of Bayesian theory and discuss how to set up data analysis problems within this framework. Next, we provide an overview of various Monte Carlo based methods fo…
We propose a Monte Carlo algorithm to sample from high dimensional probability distributions that combines Markov chain Monte Carlo and importance sampling. We provide a careful theoretical analysis, including guarantees on robustness to high dimensionality, explicit comparison with standard Markov chain Monte Carlo methods and illustrations of the potential improvements in efficiency. Simple and concrete intuition is provided for when the novel scheme is expected to outperform standard schemes. When applied to Bayesian variable-selection problems, the novel algorithm is orders of magnitude mo…
The University of Manchester（曼彻斯特大学）；Oak Ridge National Laboratory（橡树岭国家实验室）
Markov chain Monte Carlo (MCMC) methods have not been broadly adopted in Bayesian neural networks (BNNs). This paper initially reviews the main challenges in sampling from the parameter posterior of a neural network via MCMC. Such challenges culminate to lack of convergence to the parameter posterior. Nevertheless, this paper shows that a non-converged Markov chain, generated via MCMC sampling from the parameter space of a neural network, can yield via Bayesian marginalization a valuable posterior predictive distribution of the output of the neural network. Classification examples based on mul…
Bayesian methods are critical for quantifying the behaviors of systems. They capture our uncertainty about a system's behavior using probability distributions and update this understanding as new information becomes available. Probabilistic predictions that incorporate this uncertainty can then be made to evaluate system performance and make decisions. While Bayesian methods are very useful, they are often computationally intensive. This necessitates the development of more efficient algorithms. Here, we discuss a group of population Markov Chain Monte Carlo (MCMC) methods for Bayesian updatin…
Stochastic approximation methods play a central role in maximum likelihood estimation problems involving intractable likelihood functions, such as marginal likelihoods arising in problems with missing or incomplete data, and in parametric empirical Bayesian estimation. Combined with Markov chain Monte Carlo algorithms, these stochastic optimisation methods have been successfully applied to a wide range of problems in science and industry. However, this strategy scales poorly to large problems because of methodological and theoretical difficulties related to using high-dimensional Markov chain …
Proximal Markov Chain Monte Carlo is a novel construct that lies at the intersection of Bayesian computation and convex optimization, which helped popularize the use of nondifferentiable priors in Bayesian statistics. Existing formulations of proximal MCMC, however, require hyperparameters and regularization parameters to be prespecified. In this work, we extend the paradigm of proximal MCMC through introducing a novel new class of nondifferentiable priors called epigraph priors. As a proof of concept, we place trend filtering, which was originally a nonparametric regression problem, in a para…
We present a sequential Monte Carlo sampler variant of the partial rejection control algorithm, and show that this variant can be considered as a sequential Monte Carlo sampler with a modified mutation kernel. We prove that the new sampler can reduce the variance of the incremental importance weights when compared with standard sequential Monte Carlo samplers. We provide a study of theoretical properties of the new algorithm, and make connections with some existing algorithms. Finally, the sampler is adapted for application under the challenging "likelihood free," approximate Bayesian computat…
Gaussian latent variable models are a key class of Bayesian hierarchical models with applications in many fields. Performing Bayesian inference on such models can be challenging as Markov chain Monte Carlo algorithms struggle with the geometry of the resulting posterior distribution and can be prohibitively slow. An alternative is to use a Laplace approximation to marginalize out the latent Gaussian variables and then integrate out the remaining hyperparameters using dynamic Hamiltonian Monte Carlo, a gradient-based Markov chain Monte Carlo sampler. To implement this scheme efficiently, we der…
We propose an importance sampling (IS)-based transport map Hamiltonian Monte Carlo procedure for performing full Bayesian analysis in general nonlinear high-dimensional hierarchical models. Using IS techniques to construct a transport map, the proposed method transforms the typically highly challenging target distribution of a hierarchical model into a target which is easily sampled using standard Hamiltonian Monte Carlo. Conventional applications of high-dimensional IS, where infinite variance of IS weights can be a serious problem, require computationally costly high-fidelity IS distribution…
The availability of data sets with large numbers of variables is rapidly increasing. The effective application of Bayesian variable selection methods for regression with these data sets has proved difficult since available Markov chain Monte Carlo methods do not perform well in typical problem sizes of interest. The current paper proposes new adaptive Markov chain Monte Carlo algorithms to address this shortcoming. The adaptive design of these algorithms exploits the observation that in large $p$ small $n$ settings, the majority of the $p$ variables will be approximately uncorrelated a posteri…
We extend the application of Hamiltonian Monte Carlo to allow for sampling from probability distributions defined over symmetric or Hermitian positive definite matrices. To do so, we exploit the Riemannian structure induced by Cartan's century-old canonical metric. The geodesics that correspond to this metric are available in closed-form and---within the context of Lagrangian Monte Carlo---provide a principled way to travel around the space of positive definite matrices. Our method improves Bayesian inference on such matrices by allowing for a broad range of priors, so we are not limited to co…
Nonlinear differential equations (DEs) are used in a wide range of scientific problems to model complex dynamic systems. The differential equations often contain unknown parameters that are of scientific interest, which have to be estimated from noisy measurements of the dynamic system. Generally, there is no closed-form solution for nonlinear DEs, and the likelihood surface for the parameter of interest is multi-modal and very sensitive to different parameter values. We propose a Bayesian framework for nonlinear DE systems. A flexible nonparametric function is used to represent the dynamic pr…
Interest is in evaluating, by Markov chain Monte Carlo (MCMC) simulation, the expected value of a function with respect to a, possibly unnormalized, probability distribution. A general purpose variance reduction technique for the MCMC estimator, based on the zero-variance principle introduced in the physics literature, is proposed. Conditions for asymptotic unbiasedness of the zero-variance estimator are derived. A central limit theorem is also proved under regularity conditions. The potential of the idea is illustrated with real applications to probit, logit and GARCH Bayesian models. For all…
In cohort studies binary outcomes are very often analyzed by logistic regression. However, it is well-known that when the goal is to estimate a risk ratio, the logistic regression is inappropriate if the outcome is common. In these cases, a log-binomial regression model is preferable. On the other hand, the estimation of the regression coefficients of the log-binomial model is difficult due to the constraints that must be imposed on these coefficients. Bayesian methods allow a straightforward approach for log-binomial regression models, produce smaller mean squared errors and the posterior inf…
In this article we consider a Bayesian inverse problem associated to elliptic partial differential equations (PDEs) in two and three dimensions. This class of inverse problems is important in applications such as hydrology, but the complexity of the link function between unknown field and measurements can make it difficult to draw inference from the associated posterior. We prove that for this inverse problem a basic SMC method has a Monte Carlo rate of convergence with constants which are independent of the dimension of the discretization of the problem; indeed convergence of the SMC method i…
Likelihood-free methods, such as approximate Bayesian computation, are powerful tools for practical inference problems with intractable likelihood functions. Markov chain Monte Carlo and sequential Monte Carlo variants of approximate Bayesian computation can be effective techniques for sampling posterior distributions in an approximate Bayesian computation setting. However, without careful consideration of convergence criteria and selection of proposal kernels, such methods can lead to very biased inference or computationally inefficient sampling. In contrast, rejection sampling for approximat…
Sampling from hierarchical Bayesian models is often difficult for MCMC methods, because of the strong correlations between the model parameters and the hyperparameters. Recent Riemannian manifold Hamiltonian Monte Carlo (RMHMC) methods have significant potential advantages in this setting, but are computationally expensive. We introduce a new RMHMC method, which we call semi-separable Hamiltonian Monte Carlo, which uses a specially designed mass matrix that allows the joint Hamiltonian over model parameters and hyperparameters to decompose into two simpler Hamiltonians. This structure is explo…
Stochastic reaction network models are often used to explain and predict the dynamics of gene regulation in single cells. These models usually involve several parameters, such as the kinetic rates of chemical reactions, that are not directly measurable and must be inferred from experimental data. Bayesian inference provides a rigorous probabilistic framework for identifying these parameters by finding a posterior parameter distribution that captures their uncertainty. Traditional computational methods for solving inference problems such as Markov Chain Monte Carlo methods based on classical Me…
Advances in digital sensors, digital data storage and communications have resulted in systems being capable of accumulating large collections of data. In the light of dealing with the challenges that massive data present, this work proposes solutions to inference and filtering problems within the Bayesian framework. Two novel Bayesian inference algorithms are developed for non-linear and non-Gaussian state space models, able to deal with large volumes of data (or observations). These are sequential Markov chain Monte Carlo (MCMC) approaches relying on two key ideas: 1) subsample the massive da…
We show how to speed up Sequential Monte Carlo (SMC) for Bayesian inference in large data problems by data subsampling. SMC sequentially updates a cloud of particles through a sequence of distributions, beginning with a distribution that is easy to sample from such as the prior and ending with the posterior distribution. Each update of the particle cloud consists of three steps: reweighting, resampling, and moving. In the move step, each particle is moved using a Markov kernel; this is typically the most computationally expensive part, particularly when the dataset is large. It is crucial to h…
This paper introduces a set of algorithms for Monte-Carlo Bayesian reinforcement learning. Firstly, Monte-Carlo estimation of upper bounds on the Bayes-optimal value function is employed to construct an optimistic policy. Secondly, gradient-based algorithms for approximate upper and lower bounds are introduced. Finally, we introduce a new class of gradient algorithms for Bayesian Bellman error minimisation. We theoretically show that the gradient methods are sound. Experimentally, we demonstrate the superiority of the upper bound method in terms of reward obtained. However, we also show that t…
Bayesian phylogenetic inference is often conducted via local or sequential search over topologies and branch lengths using algorithms such as random-walk Markov chain Monte Carlo (MCMC) or Combinatorial Sequential Monte Carlo (CSMC). However, when MCMC is used for evolutionary parameter learning, convergence requires long runs with inefficient exploration of the state space. We introduce Variational Combinatorial Sequential Monte Carlo (VCSMC), a powerful framework that establishes variational sequential search to learn distributions over intricate combinatorial structures. We then develop nes…
For modeling multivariate financial time series we propose a single factor copula model together with stochastic volatility margins. This model generalizes single factor models relying on the multivariate normal distribution and allows for symmetric and asymmetric tail dependence. We develop joint Bayesian inference using Hamiltonian Monte Carlo (HMC) within Gibbs sampling. Thus we avoid information loss caused by the two-step approach for margins and dependence in copula models as followed by Schamberger et al(2017). Further, the Bayesian approach allows for high dimensional parameter spaces …
In this article, we describe a {\tt R} package for sampling from an empirical likelihood-based posterior using a Hamiltonian Monte Carlo method. Empirical likelihood-based methodologies have been used in Bayesian modeling of many problems of interest in recent times. This semiparametric procedure can easily combine the flexibility of a non-parametric distribution estimator together with the interpretability of a parametric model. The model is specified by estimating equations-based constraints. Drawing an inference from a Bayesian empirical likelihood (BayesEL) posterior is challenging. The li…
The R package walker extends standard Bayesian general linear models to the case where the effects of the explanatory variables can vary in time. This allows, for example, to model the effects of interventions such as changes in tax policy which gradually increases their effect over time. The Markov chain Monte Carlo algorithms powering the Bayesian inference are based on Hamiltonian Monte Carlo provided by Stan software, using a state space representation of the model to marginalise over the regression coefficients for efficient low-dimensional sampling.
Detection of induced polarisation (IP) effects in airborne electromagnetic (AEM) measurements does not yet have an established methodology. This contribution develops a Bayesian approach to the IP-detectability problem using decoupled transdimensional layered models, and applies an approach novel to geophysics whereby transdimensional proposals are used within the embarrassingly parallelisable and robust static Sequential Monte Carlo (SMC) class of algorithms for the simultaneous inference of parameters and models. Henceforth referring to this algorithm as Reversible Jump Sequential Monte Carl…
Gradient-based Monte Carlo sampling algorithms, like Langevin dynamics and Hamiltonian Monte Carlo, are important methods for Bayesian inference. In large-scale settings, full-gradients are not affordable and thus stochastic gradients evaluated on mini-batches are used as a replacement. In order to reduce the high variance of noisy stochastic gradients, Dubey et al. [2016] applied the standard variance reduction technique on stochastic gradient Langevin dynamics and obtained both theoretical and experimental improvements. In this paper, we apply the variance reduction tricks on Hamiltonian Mon…
Tsinghua University（清华大学）；University of California（加利福尼亚大学）；Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）
We propose a general purpose Bayesian inference algorithm for expensive likelihoods, replacing the stochastic term in the Langevin equation with a deterministic density gradient term. The particle density is evaluated from the current particle positions using a Normalizing Flow (NF), which is differentiable and has good generalization properties in high dimensions. We take advantage of NF preconditioning and NF based Metropolis-Hastings updates for a faster convergence. We show on various examples that the method is competitive against state of the art sampling methods.
We introduce several novel change of measure inequalities for two families of divergences: $f$-divergences and $α$-divergences. We show how the variational representation for $f$-divergences leads to novel change of measure inequalities. We also present a multiplicative change of measure inequality for $α$-divergences and a generalized version of Hammersley-Chapman-Robbins inequality. Finally, we present several applications of our change of measure inequalities, including PAC-Bayesian bounds for various classes of losses and non-asymptotic intervals for Monte Carlo estimates.
