Stochastic gradient Langevin dynamics combined with Gibbs updates (SGLD--Gibbs) provides a highly scalable approach to approximate Bayesian inference in latent variable models. However, it remains unclear how to tune the algorithm's hyperparameters in a principled manner to ensure the uncertainty estimates are statistically meaningful. In this work, we address this gap in tuning guidance by developing a statistical scaling limit theory for SGLD--Gibbs. We derive a joint asymptotic limit for the global parameters and latent variables under appropriate space-time rescaling. We show that global parameters converge to a diffusion-type limit, while each latent variable converges to a jump process, reflecting the use of intermittent Gibbs updates. This joint jump-diffusion structure reveals how latent-variable randomness contributes to the stationary distribution of the global parameters. We leverage our results to propose explicit guidance on hyperparameter tuning for SGLD--Gibbs that ensures meaningful uncertainty quantification. Numerical experiments show that SGLD--Gibbs with our tuning guidance leads to better parameter estimates, uncertainty quantification, and predictive performance than stochastic variational inference.
Subsampling is an important technique to tackle the computational challenges brought by big data. Many subsampling procedures fall within the framework of importance sampling, which assigns high sampling probabilities to the samples appearing to have big impacts. When the noise level is high, those sampling procedures tend to pick many outliers and thus often do not perform satisfactorily in practice. To tackle this issue, we design a new Markov subsampling strategy based on Huber criterion (HMS) to construct an informative subset from the noisy full data; the constructed subset then serves as…
Markov Chain Monte Carlo (MCMC) requires to evaluate the full data likelihood at different parameter values iteratively and is often computationally infeasible for large data sets. In this paper, we propose to approximate the log-likelihood with subsamples taken according to nonuniform subsampling probabilities, and derive the most likely optimal (MLO) subsampling probabilities for better approximation. Compared with existing subsampled MCMC algorithm with equal subsampling probabilities, our MLO subsampled MCMC has a higher estimation efficiency with the same subsampling ratio. We also derive…
Spectral subsampling MCMC was recently proposed to speed up Markov chain Monte Carlo (MCMC) for long stationary univariate time series by subsampling periodogram observations in the frequency domain. This article extends the approach to multivariate time series using a multivariate generalisation of the Whittle likelihood. To assess the computational gains from spectral subsampling in challenging problems, a multivariate generalisation of the autoregressive tempered fractionally integrated moving average model (ARTFIMA) is introduced and some of its properties derived. Bayesian inference based…
This work demonstrates the ability to produce readily interpretable statistical metrics for model fit, fixed effects covariance coefficients, and prediction confidence. Importantly, this work compares 4 suitable and commonly applied epistemic UQ approaches, BNN, SWAG, MC dropout, and ensemble approaches in their ability to calculate these statistical metrics for the ARMED MEDL models. In our experiment for AD prognosis, not only do the UQ methods provide these benefits, but several UQ methods maintain the high performance of the original ARMED method, some even provide a modest (but not statis…
The rapid development of computing power and efficient Markov Chain Monte Carlo (MCMC) simulation algorithms have revolutionized Bayesian statistics, making it a highly practical inference method in applied work. However, MCMC algorithms tend to be computationally demanding, and are particularly slow for large datasets. Data subsampling has recently been suggested as a way to make MCMC methods scalable on massively large data, utilizing efficient sampling schemes and estimators from the survey sampling literature. These developments tend to be unknown by many survey statisticians who tradition…
Georgia Institute of Technology（佐治亚理工学院）；Google Inc.（谷歌公司）；The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））
Many Markov Chain Monte Carlo (MCMC) methods leverage gradient information of the potential function of target distribution to explore sample space efficiently. However, computing gradients can often be computationally expensive for large scale applications, such as those in contemporary machine learning. Stochastic Gradient (SG-)MCMC methods approximate gradients by stochastic ones, commonly via uniformly subsampled data points, and achieve improved computational efficiency, however at the price of introducing sampling error. We propose a non-uniform subsampling scheme to improve the sampling…
We propose Subsampling MCMC, a Markov Chain Monte Carlo (MCMC) framework where the likelihood function for $n$ observations is estimated from a random subset of $m$ observations. We introduce a highly efficient unbiased estimator of the log-likelihood based on control variates, such that the computing cost is much smaller than that of the full log-likelihood in standard MCMC. The likelihood estimate is bias-corrected and used in two dependent pseudo-marginal algorithms to sample from a perturbed posterior, for which we derive the asymptotic error with respect to $n$ and $m$, respectively. We p…
Markov chain Monte Carlo methods are often deemed too computationally intensive to be of any practical use for big data applications, and in particular for inference on datasets containing a large number $n$ of individual data points, also known as tall datasets. In scenarios where data are assumed independent, various approaches to scale up the Metropolis-Hastings algorithm in a Bayesian inference context have been recently proposed in machine learning and computational statistics. These approaches can be grouped into two categories: divide-and-conquer approaches and, subsampling-based algori…
Uncertainty quantification for Markov chain induced martingales with application to temporal difference learning
The Voleon Group（Voleon集团）；Department of Statistics and Data Science, The Wharton School, University of Pennsylvania（统计与数据科学系，沃顿商学院，宾夕法尼亚大学）；Department of Statistics and Data Sciences, University of Texas（统计与数据科学系，德克萨斯大学）
Speeding up Markov Chain Monte Carlo (MCMC) for datasets with many observations by data subsampling has recently received considerable attention. A pseudo-marginal MCMC method is proposed that estimates the likelihood by data subsampling using a block-Poisson estimator. The estimator is a product of Poisson estimators, allowing us to update a single block of subsample indicators in each MCMC iteration so that a desired correlation is achieved between the logs of successive likelihood estimates. This is important since pseudo-marginal MCMC with positively correlated likelihood estimates can use…
Bayesian inference using Markov Chain Monte Carlo (MCMC) on large datasets has developed rapidly in recent years. However, the underlying methods are generally limited to relatively simple settings where the data have specific forms of independence. We propose a novel technique for speeding up MCMC for time series data by efficient data subsampling in the frequency domain. For several challenging time series models, we demonstrate a speedup of up to two orders of magnitude while incurring negligible bias compared to MCMC on the full dataset. We also propose alternative control variates for var…
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…
InstroTek Inc.（InstroTek公司）；North Carolina State University（北卡罗来纳州立大学）
Full Wave Inversion (FWI) imaging scheme has many applications in engineering, geoscience and medical sciences. In this paper, a surrogate deep learning FWI approach is presented to quantify properties of materials using stress waves. Such inverse problems, in general, are ill-posed and nonconvex, especially in cases where the solutions exhibit shocks, heterogeneity, discontinuities, or large gradients. The proposed approach is proven efficient to obtain global minima responses in these cases. This approach is trained based on random sampled set of material properties and sampled trials around…
University of Massachusetts, Amherst（马萨诸塞大学阿默斯特分校）
We present an information-based uncertainty quantification method for general Markov Random Fields. Markov Random Fields (MRF) are structured, probabilistic graphical models over undirected graphs, and provide a fundamental unifying modeling tool for statistical mechanics, probabilistic machine learning, and artificial intelligence. Typically MRFs are complex and high-dimensional with nodes and edges (connections) built in a modular fashion from simpler, low-dimensional probabilistic models and their local connections; in turn, this modularity allows to incorporate available data to MRFs and e…
We propose a generic Markov Chain Monte Carlo (MCMC) algorithm to speed up computations for datasets with many observations. A key feature of our approach is the use of the highly efficient difference estimator from the survey sampling literature to estimate the log-likelihood accurately using only a small fraction of the data. Our algorithm improves on the $O(n)$ complexity of regular MCMC by operating over local data clusters instead of the full sample when computing the likelihood. The likelihood estimate is used in a Pseudo-marginal framework to sample from a perturbed posterior which is w…
Target tracking faces the challenge in coping with large volumes of data which requires efficient methods for real time applications. The complexity considered in this paper is when there is a large number of measurements which are required to be processed at each time step. Sequential Markov chain Monte Carlo (MCMC) has been shown to be a promising approach to target tracking in complex environments, especially when dealing with clutter. However, a large number of measurements usually results in large processing requirements. This paper goes beyond the current state-of-the-art and presents a …
Spectral subsampling MCMC for Lévy-driven continuous-time ARMA models with expensive likelihood contributions
Human Technology Institute, University of Technology Sydney（悉尼科技大学人类技术研究所）；School of Mathematical and Physical Sciences, University of Technology Sydney（悉尼科技大学数理科学学院）；School of Economics, University of New South Wales（新南威尔士大学经济管理学院）；Guanghua School of Management, Peking University（北京大学光华管理学院）
Gibbs sampling is the de facto Markov chain Monte Carlo method used for inference and learning on large scale graphical models. For complicated factor graphs with lots of factors, the performance of Gibbs sampling can be limited by the computational cost of executing a single update step of the Markov chain. This cost is proportional to the degree of the graph, the number of factors adjacent to each variable. In this paper, we show how this cost can be reduced by using minibatching: subsampling the factors to form an estimate of their sum. We introduce several minibatched variants of Gibbs, sh…
University of Halmstad（哈尔姆斯塔德大学）；Helsinki Institute for Information Technology HIIT（赫尔辛基信息技术研究所 HIIT）；University of Helsinki（赫尔辛基大学）；Aalto University（阿尔托大学）
Recent developments in differentially private (DP) machine learning and DP Bayesian learning have enabled learning under strong privacy guarantees for the training data subjects. In this paper, we further extend the applicability of DP Bayesian learning by presenting the first general DP Markov chain Monte Carlo (MCMC) algorithm whose privacy-guarantees are not subject to unrealistic assumptions on Markov chain convergence and that is applicable to posterior inference in arbitrary models. Our algorithm is based on a decomposition of the Barker acceptance test that allows evaluating the Rényi D…
Statistical machine learning has widespread application in various domains. These methods include probabilistic algorithms, such as Markov Chain Monte-Carlo (MCMC), which rely on generating random numbers from probability distributions. These algorithms are computationally expensive on conventional processors, yet their statistical properties, namely interpretability and uncertainty quantification (UQ) compared to deep learning, make them an attractive alternative approach. Therefore, hardware specialization can be adopted to address the shortcomings of conventional processors in running these…
Calibration of individual based models (IBMs), successful in modeling complex ecological dynamical systems, is often performed only ad-hoc. Bayesian inference can be used for both parameter estimation and uncertainty quantification, but its successful application to realistic scenarios has been hindered by the complex stochastic nature of IBMs. Computationally expensive techniques such as Particle Filter (PF) provide marginal likelihood estimates, where multiple model simulations (particles) are required to get a sample from the state distribution conditional on the observed data. Particle ens…
Stabilised weighted data subsampling for accelerated Bayesian inference in models with recursive likelihoods
University of Technology Sydney（悉尼科技大学）；University of New South Wales（新南威尔士大学）；Human Technology Institute, University of Technology Sydney（悉尼科技大学人类技术研究所）
Markov chain Monte Carlo (MCMC) algorithms are generally regarded as the gold standard technique for Bayesian inference. They are theoretically well-understood and conceptually simple to apply in practice. The drawback of MCMC is that in general performing exact inference requires all of the data to be processed at each iteration of the algorithm. For large data sets, the computational cost of MCMC can be prohibitive, which has led to recent developments in scalable Monte Carlo algorithms that have a significantly lower computational cost than standard MCMC. In this paper, we focus on a partic…
TAM-Chain: Multi-Scale Thyroid Cytology Classification via Absorbing Markov Chains and Shannon Entropy Uncertainty Quantification for False-Negative Suppression and Domain-Shift Adaptation
Faculty of Mathematics, Mechanics and Informatics, VNU University of Science（越南国立大学理学院数学、力学与信息学系）
We consider exchangeable Markov multi-state survival processes -- temporal processes taking values over a state-space$\mathcal{S}$ with at least one absorbing failure state $\flat \in \mathcal{S}$ that satisfy natural invariance properties of exchangeability and consistency under subsampling. The set of processes contains many well-known examples from health and epidemiology -- survival, illness-death, competing risk, and comorbidity processes; an extension leads to recurrent event processes. We characterize exchangeable Markov multi-state survival processes in both discrete and continuous tim…
Technische Universiteit Delft（代尔夫特理工大学）；Osaka University（大阪大学）；University of Warwick（华威大学）
This paper introduces the Boomerang Sampler as a novel class of continuous-time non-reversible Markov chain Monte Carlo algorithms. The methodology begins by representing the target density as a density, $e^{-U}$, with respect to a prescribed (usually) Gaussian measure and constructs a continuous trajectory consisting of a piecewise elliptical path. The method moves from one elliptical orbit to another according to a rate function which can be written in terms of $U$. We demonstrate that the method is easy to implement and demonstrate empirically that it can out-perform existing benchmark piec…
It is common to subsample Markov chain output to reduce the storage burden. Geyer (1992) shows that discarding $k-1$ out of every $k$ observations will not improve statistical efficiency, as quantified through variance in a given computational budget. That observation is often taken to mean that thinning MCMC output cannot improve statistical efficiency. Here we suppose that it costs one unit of time to advance a Markov chain and then $θ>0$ units of time to compute a sampled quantity of interest. For a thinned process, that cost $θ$ is incurred less often, so it can be advanced through more st…
The paper presents a new perspective on the mixture of Dirichlet process model which allows the recovery of full and correct uncertainty quantification associated with the full model, even after having integrated out the random distribution function. The implication is that we can run a simple Markov chain Monte Carlo algorithm and subsequently return the original uncertainty which was removed from the integration. This also has the benefit of avoiding more complicated algorithms which do not perform the integration step. Numerous illustrations are presented.
Bayesian max-margin models have shown superiority in various practical applications, such as text categorization, collaborative prediction, social network link prediction and crowdsourcing, and they conjoin the flexibility of Bayesian modeling and predictive strengths of max-margin learning. However, Monte Carlo sampling for these models still remains challenging, especially for applications that involve large-scale datasets. In this paper, we present the stochastic subgradient Hamiltonian Monte Carlo (HMC) methods, which are easy to implement and computationally efficient. We show the approxi…
Lehigh University（理海大学）；Sandia National Laboratories（桑迪亚国家实验室）
We consider the problem of modeling heterogeneous materials where micro-scale dynamics and interactions affect global behavior. In the presence of heterogeneities in material microstructure it is often impractical, if not impossible, to provide quantitative characterization of material response. The goal of this work is to develop a Bayesian framework for uncertainty quantification (UQ) in material response prediction when using nonlocal models. Our approach combines the nonlocal operator regression (NOR) technique and Bayesian inference. Specifically, we use a Markov chain Monte Carlo (MCMC) …
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 …
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…
In many real-world engineering systems, the performance or reliability of the system is characterised by a scalar parameter. The distribution of this performance parameter is important in many uncertainty quantification problems, ranging from risk management to utility optimisation. In practice, this distribution usually cannot be derived analytically and has to be obtained numerically by simulations. To this end, standard Monte Carlo simulations are often used, however, they cannot efficiently reconstruct the tail of the distribution which is essential in many applications. One possible remed…
Many problems in science and engineering require uncertainty quantification that accounts for observed data. For example, in computational neuroscience, Neural Population Models (NPMs) are mechanistic models that describe brain physiology in a range of different states. Within computational neuroscience there is growing interest in the inverse problem of inferring NPM parameters from recordings such as the EEG (Electroencephalogram). Uncertainty quantification is essential in this application area in order to infer the mechanistic effect of interventions such as anaesthesia. This paper present…
The problem of estimating certain distributions over $\{0,1\}^d$ is considered here. The distribution represents a quantum system of $d$ qubits, where there are non-trivial dependencies between the qubits. A maximum entropy approach is adopted to reconstruct the distribution from exact moments or observed empirical moments. The Robbins Monro algorithm is used to solve the intractable maximum entropy problem, by constructing an unbiased estimator of the un-normalized target with a sequential Monte Carlo sampler at each iteration. In the case of empirical moments, this coincides with a maximum l…
Uncertainty quantification is a critical missing component in radio interferometric imaging that will only become increasingly important as the big-data era of radio interferometry emerges. Since radio interferometric imaging requires solving a high-dimensional, ill-posed inverse problem, uncertainty quantification is difficult but also critical to the accurate scientific interpretation of radio observations. Statistical sampling approaches to perform Bayesian inference, like Markov Chain Monte Carlo (MCMC) sampling, can in principle recover the full posterior distribution of the image, from w…
In inverse problems, uncertainty quantification (UQ) deals with a probabilistic description of the solution nonuniqueness and data noise sensitivity. Setting seismic imaging into a Bayesian framework allows for a principled way of studying uncertainty by solving for the model posterior distribution. Imaging, however, typically constitutes only the first stage of a sequential workflow, and UQ becomes even more relevant when applied to subsequent tasks that are highly sensitive to the inversion outcome. In this paper, we focus on how UQ trickles down to horizon tracking for the determination of …
KLATASDS - MOE School of Statistics East China Normal University（KLATASDS - 教育部统计学系华东师范大学）
Reliable uncertainty quantification is crucial for reinforcement learning (RL) in high-stakes settings. We propose a unified conformal prediction framework for infinite-horizon policy evaluation that constructs distribution-free prediction intervals {for returns} in both on-policy and off-policy settings. Our method integrates distributional RL with conformal calibration, addressing challenges such as unobserved returns, temporal dependencies, and distributional shifts. We propose a modular pseudo-return construction based on truncated rollouts and a time-aware calibration strategy using exper…
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…
Monte Carlo sampling for Bayesian posterior inference is a common approach used in machine learning. The Markov Chain Monte Carlo procedures that are used are often discrete-time analogues of associated stochastic differential equations (SDEs). These SDEs are guaranteed to leave invariant the required posterior distribution. An area of current research addresses the computational benefits of stochastic gradient methods in this setting. Existing techniques rely on estimating the variance or covariance of the subsampling error, and typically assume constant variance. In this article, we propose …
