Multilevel Sequential Monte Carlo Samplers for Normalizing Constants
Center INRIA Bordeaux Sud-Ouest & Institut de Mathematiques de Bordeaux, Universite de Bordeaux I（波尔多南西INRIA中心与波尔多数学研究所，波尔多第一大学）；Department of Statistics & Applied Probability, National University of Singapore（新加坡国立大学统计与应用概率系）；Computer Science and Mathematics Division, Oak Ridge National Laboratory（橡树岭国家实验室计算机科学与数学部）
This article considers the sequential Monte Carlo (SMC) approximation of ratios of normalizing constants associated to posterior distributions which in principle rely on continuum models. Therefore, the Monte Carlo estimation error and the discrete approximation error must be balanced. A multilevel strategy is utilized to substantially reduce the cost to obtain a given error level in the approximation as compared to standard estimators. Two estimators are considered and relative variance bounds are given. The theoretical results are numerically illustrated for the example of identifying a parametrized permeability in an elliptic equation given point-wise observations of the pressure.
Markov Chain Monte Carlo (MCMC) methods are a powerful tool for computation with complex probability distributions. However the performance of such methods is critically dependant on properly tuned parameters, most of which are difficult if not impossible to know a priori for a given target distribution. Adaptive MCMC methods aim to address this by allowing the parameters to be updated during sampling based on previous samples from the chain at the expense of requiring a new theoretical analysis to ensure convergence. In this work we extend the convergence theory of adaptive MCMC methods to a …
In this article we consider the application of multilevel Monte Carlo, for the estimation of normalizing constants. In particular we will make use of the filtering algorithm, the ensemble Kalman-Bucy filter (EnKBF), which is an N-particle representation of the Kalma-Bucy filter (KBF). The EnKBF is of interest as it coincides with the optimal filter in the continuous-linear setting, i.e. the KBF. This motivates our particular setup in the linear setting. The resulting methodology we will use is the multilevel ensemble Kalman-Bucy filter (MLEnKBF). We provide an analysis based on deriving Lq-bou…
We propose a new framework for efficiently sampling from complex probability distributions using a combination of normalizing flows and elliptical slice sampling (Murray et al., 2010). The central idea is to learn a diffeomorphism, through normalizing flows, that maps the non-Gaussian structure of the target distribution to an approximately Gaussian distribution. We then use the elliptical slice sampler, an efficient and tuning-free Markov chain Monte Carlo (MCMC) algorithm, to sample from the transformed distribution. The samples are then pulled back using the inverse normalizing flow, yieldi…
Error Bounds and Normalizing Constants for Sequential Monte Carlo in High Dimensions
在Beskos et al（2011）的最新论文中，在Del Moral et al（2006）、Neal（2001）中引入的序贯蒙特卡罗（SMC）采样器已被证明在状态空间d的维度上是渐近稳定的，其代价仅为d的多项式，当蒙特卡罗样本数N固定时。更准确地说，它已经建立了有效的样本量（ESS）的后续（近似）样本和固定维边际的蒙特卡罗误差将收敛$d$增长，与$\mathcal{O}(Nd^2)$的计算成本。在目前的工作中，进一步的结果SMC方法在高维提供作为$d\to\infty$和$N$固定。我们推导出一个显式的界的蒙特卡罗误差估计使用SMC采样器和精确的渐近相对$\mathbb{L}_2$ -误差的归一化常数的估计。我们还建立了该算法的混沌边缘传播特性。文中还讨论了几种基于SMC的近似滤波方案在高维情况下的精度。
Multilevel Generative Samplers for Investigating Critical Phenomena
BIFOLD；Technische Universität Berlin（技术大学柏林）；Università degli Studi di Torino（都灵大学）；INFN Torino（托斯卡纳国家研究所）；University of Bonn（波恩大学）；Helmholtz Institute for Radiation and Nuclear Physics (HISKP)（辐射和核物理研究所）；Deutsches Elektronen-Synchrotron (DESY)（德国电子同步辐射光源）；RIKEN Center for AIP（理化学研究所AIP中心）
Statisticians often use Monte Carlo methods to approximate probability distributions, primarily with Markov chain Monte Carlo and importance sampling. Sequential Monte Carlo samplers are a class of algorithms that combine both techniques to approximate distributions of interest and their normalizing constants. These samplers originate from particle filtering for state space models and have become general and scalable sampling techniques. This article describes sequential Monte Carlo samplers and their possible implementations, arguing that they remain under-used in statistics, despite their ab…
Departement of Computer Science, KU Leuven（计算机科学系，鲁汶大学）；Department of Electrical Engineering, ESAT - STADIUS, KU Leuven（电气工程系，ESAT - STADIUS，鲁汶大学）
Bayesian regression remains a simple but effective tool based on Bayesian inference techniques. For large-scale applications, with complicated posterior distributions, Markov Chain Monte Carlo methods are applied. To improve the well-known computational burden of Markov Chain Monte Carlo approach for Bayesian regression, we developed a multilevel Gibbs sampler for Bayesian regression of linear mixed models. The level hierarchy of data matrices is created by clustering the features and/or samples of data matrices. Additionally, the use of correlated samples is investigated for variance reductio…
Multilevel Sequential Importance Sampling for Rare Event Estimation
稀有事件发生概率的估计是可靠性和风险评估中的一项重要任务。我们考虑的故障事件表示的极限状态函数，这取决于一个偏微分方程（PDE）的解决方案。由于偏微分方程的数值计算是昂贵的，估计这样的概率的失败，蒙特卡罗抽样是棘手的。可靠性分析中更有效的抽样方法，如子集模拟，很受欢迎，但如果PDE评估成本很高，则仍然不可行。在这篇文章中，我们开发了一种新的，高效的估计罕见事件的概率。我们的方法是基于序贯重要性采样器使用离散化的PDE为基础的极限状态函数具有不同的精度。一个双重的自适应算法，确保我们得到一个估计的基础上所需的离散化精度。与[Ullmann，Papaioannou 2015; SIAM/ASA J. Uncertain.数量。3（1）：922-953]，我们的估计克服了嵌套问题。在顺序重要性采样算法中，特别重要的是正确选择MCMC内核。而不是流行的自适应条件采样方法，我们提出了一种新的算法，使用独立的建议，从自适应构造的冯Mises-Fischer-Nakagami分布。该算法分别应用于1D和2D空间中的测试问题，并与多级子集模拟估计量和单级版本的序贯重要性抽样和子集模拟进行比较。
Sampling from a complex distribution $π$ and approximating its intractable normalizing constant Z are challenging problems. In this paper, a novel family of importance samplers (IS) and Markov chain Monte Carlo (MCMC) samplers is derived. Given an invertible map T, these schemes combine (with weights) elements from the forward and backward Orbits through points sampled from a proposal distribution $ρ$. The map T does not leave the target $π$ invariant, hence the name NEO, standing for Non-Equilibrium Orbits. NEO-IS provides unbiased estimators of the normalizing constant and self-normalized IS…
Non-reversible, tuning- and rejection-free Markov chain Monte Carlo via iterated random functions
在这项工作中，我们提出了一个不可逆的，调整和拒绝免费的马尔可夫链蒙特卡罗，自然适合在打了就跑的框架。采样器只需要访问对数密度函数的梯度，因此不需要归一化常数。我们证明了所提出的马尔可夫链是不变的目标分布，并说明其适用性，通过广泛的例子。我们表明，本文中介绍的采样器与Peters and de With（2012），Bouchard-Cote et al.（2017）的连续采样器密切相关。特别是，计算是非常相似的，在这个意义上，都是围绕模拟一个非均匀泊松过程。当对数密度的梯度允许将计算有效的方向分解为两个单调函数之和时，计算可以被简化。我们将我们的采样器应用于选择性推理，比以前使用的采样器有了显著的改进（Tian et al. 2016）。
University of Amsterdam（阿姆斯特丹大学）；Qualcomm AI Research, Qualcomm Technologies Netherlands B.V.（高通技术荷兰有限公司高通人工智能研究院）
Deterministic dynamics is an essential part of many MCMC algorithms, e.g. Hybrid Monte Carlo or samplers utilizing normalizing flows. This paper presents a general construction of deterministic measure-preserving dynamics using autonomous ODEs and tools from differential geometry. We show how Hybrid Monte Carlo and other deterministic samplers follow as special cases of our theory. We then demonstrate the utility of our approach by constructing a continuous non-sequential version of Gibbs sampling in terms of an ODE flow and extending it to discrete state spaces. We find that our deterministic…
Estimating the normalizing constant of an unnormalized probability distribution has important applications in computer science, statistical physics, machine learning, and statistics. In this work, we consider the problem of estimating the normalizing constant $Z=\int_{\mathbb{R}^d} e^{-f(x)}\,\mathrm{d}x$ to within a multiplication factor of $1 \pm \varepsilon$ for a $μ$-strongly convex and $L$-smooth function $f$, given query access to $f(x)$ and $\nabla f(x)$. We give both algorithms and lowerbounds for this problem. Using an annealing algorithm combined with a multilevel Monte Carlo method …
This article is a discussion of Zanella and Roberts' paper: Multilevel linear models, gibbs samplers and multigrid decompositions. We consider several extensions in which the multigrid decomposition would bring us interesting insights, including vector hierarchical models, linear mixed effects models and partial centering parametrizations.
The sampling of probability distributions specified up to a normalization constant is an important problem in both machine learning and statistical mechanics. While classical stochastic sampling methods such as Markov Chain Monte Carlo (MCMC) or Langevin Dynamics (LD) can suffer from slow mixing times there is a growing interest in using normalizing flows in order to learn the transformation of a simple prior distribution to the given target distribution. Here we propose a generalized and combined approach to sample target densities: Stochastic Normalizing Flows (SNF) -- an arbitrary sequence …
Non-Vacuous Certification of Transport MCMC via Oscillation-Controlled Normalizing Flows
China Sanya Science and Education Innovation Park, Wuhan University of Technology, Sanya 572025, China（中国三亚科技教育创新园，武汉理工大学，三亚572025，中国）；School of Civil Engineering and Architecture, Wuhan University of Technology, Wuhan 430070, China（武汉理工大学土木工程与建筑学院，武汉430070，中国）
Faster Hamiltonian Monte Carlo by Learning Leapfrog Scale: an offline randomized solution
CEREMADE, Université Paris-Dauphine, Université PSL, CNRS（CEREMADE，巴黎第九大学，巴黎文理研究大学，法国国家科学研究中心）；Aix Marseille Univ, CNRS, I2M（艾克斯-马赛大学，法国国家科学研究中心，数学研究所）；Department of Statistics, University of Warwick（华威大学统计系）；Université Paris-Saclay, INRAE, AgroParisTech, UMR MIA Paris-Saclay（巴黎萨克雷大学，法国国家农业食品与环境研究院，巴黎农学院，MIA巴黎萨克雷联合研究单位）
Models of stochastic processes are widely used in almost all fields of science. Theory validation, parameter estimation, and prediction all require model calibration and statistical inference using data. However, data are almost always incomplete observations of reality. This leads to a great challenge for statistical inference because the likelihood function will be intractable for almost all partially observed stochastic processes. This renders many statistical methods, especially within a Bayesian framework, impossible to implement. Therefore, computationally expensive likelihood-free appro…
Bayesian computation with generative diffusion models by Multilevel Monte Carlo
School of Mathematical and Computer Sciences, Heriot-Watt University（赫瑞斯泰学院数学与计算机科学系）；Departament de Matemàtiques and IMAC, Universitat Jaume I（数学系和IMAC， Jaime I大学）；School of Mathematics, University of Edinburgh（爱丁堡大学数学系）；Maxwell Institute for Mathematical Sciences, Edinburgh（爱丁堡数学科学研究所）
针对扩散模型用于贝叶斯逆问题 Monte Carlo 计算时神经评估成本高的问题，提出 Multilevel Monte Carlo 多层耦合策略，在不损失精度的情况下于三个成像基准上降低 4 至 8 倍成本。
We consider the problem of estimating expectations with respect to a target distribution with an unknown normalizing constant, and where even the unnormalized target needs to be approximated at finite resolution. This setting is ubiquitous across science and engineering applications, for example in the context of Bayesian inference where a physics-based model governed by an intractable partial differential equation (PDE) appears in the likelihood. A multi-index Sequential Monte Carlo (MISMC) method is used to construct ratio estimators which provably enjoy the complexity improvements of multi-…
