The complexity of the Metropolis-Hastings (MH) algorithm arises from the requirement of a likelihood evaluation for the full data set in each iteration. Payne and Mallick (2015) propose to speed up the algorithm by a delayed acceptance approach where the acceptance decision proceeds in two stages. In the first stage, an estimate of the likelihood based on a random subsample determines if it is likely that the draw will be accepted and, if so, the second stage uses the full data likelihood to decide upon final acceptance. Evaluating the full data likelihood is thus avoided for draws that are unlikely to be accepted. We propose a more precise likelihood estimator which incorporates auxiliary information about the full data likelihood while only operating on a sparse set of the data. We prove that the resulting delayed acceptance MH is more efficient compared to that of Payne and Mallick (2015). The caveat of this approach is that the full data set needs to be evaluated in the second stage. We therefore propose to substitute this evaluation by an estimate and construct a state-dependent approximation thereof to use in the first stage. This results in an algorithm that (i) can use a smaller subsample m by leveraging on recent advances in Pseudo-Marginal MH (PMMH) and (ii) is provably within $O(m^{-2})$ of the true posterior.
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…
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…
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…
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…
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）
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…
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…
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 …
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）
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…
Markov chain Monte Carlo (MCMC) is widely used for Bayesian inference in models of complex systems. Performance, however, is often unsatisfactory in models with many latent variables due to so-called poor mixing, necessitating development of application specific implementations. This paper introduces "posterior-based proposals" (PBPs), a new type of MCMC update applicable to a huge class of statistical models (whose conditional dependence structures are represented by directed acyclic graphs). PBPs generates large joint updates in parameter and latent variable space, whilst retaining good acce…
This paper introduces a framework for speeding up Bayesian inference conducted in presence of large datasets. We design a Markov chain whose transition kernel uses an (unknown) fraction of (fixed size) of the available data that is randomly refreshed throughout the algorithm. Inspired by the Approximate Bayesian Computation (ABC) literature, the subsampling process is guided by the fidelity to the observed data, as measured by summary statistics. The resulting algorithm, Informed Sub-Sampling MCMC (ISS-MCMC), is a generic and flexible approach which, contrary to existing scalable methodologies…
In big data context, traditional MCMC methods, such as Metropolis-Hastings algorithms and hybrid Monte Carlo, scale poorly because of their need to evaluate the likelihood over the whole data set at each iteration. In order to resurrect MCMC methods, numerous approaches belonging to two categories: divide-and-conquer and subsampling, are proposed. In this article, we study the parallel MCMC and propose a new combination method in the divide-and-conquer framework. Compared with some parallel MCMC methods, such as consensus Monte Carlo, Weierstrass Sampler, instead of sampling from subposteriors…
Performing exact Bayesian inference for complex models is computationally intractable. Markov chain Monte Carlo (MCMC) algorithms can provide reliable approximations of the posterior distribution but are expensive for large datasets and high-dimensional models. A standard approach to mitigate this complexity consists in using subsampling techniques or distributing the data across a cluster. However, these approaches are typically unreliable in high-dimensional scenarios. We focus here on a recent alternative class of MCMC schemes exploiting a splitting strategy akin to the one used by the cele…
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…
Particle MCMC involves using a particle filter within an MCMC algorithm. For inference of a model which involves an unobserved stochastic process, the standard implementation uses the particle filter to propose new values for the stochastic process, and MCMC moves to propose new values for the parameters. We show how particle MCMC can be generalised beyond this. Our key idea is to introduce new latent variables. We then use the MCMC moves to update the latent variables, and the particle filter to propose new values for the parameters and stochastic process given the latent variables. A generic…
UC Berkeley Computer Science（加州大学伯克利分校计算机科学系）；Technical University of Denmark（丹麦技术大学）
We propose a new class of learning algorithms that combines variational approximation and Markov chain Monte Carlo (MCMC) simulation. Naive algorithms that use the variational approximation as proposal distribution can perform poorly because this approximation tends to underestimate the true variance and other features of the data. We solve this problem by introducing more sophisticated MCMC algorithms. One of these algorithms is a mixture of two MCMC kernels: a random walk Metropolis kernel and a blockMetropolis-Hastings (MH) kernel with a variational approximation as proposaldistribution. Th…
Probabilistic programming languages can simplify the development of machine learning techniques, but only if inference is sufficiently scalable. Unfortunately, Bayesian parameter estimation for highly coupled models such as regressions and state-space models still scales poorly; each MCMC transition takes linear time in the number of observations. This paper describes a sublinear-time algorithm for making Metropolis-Hastings (MH) updates to latent variables in probabilistic programs. The approach generalizes recently introduced approximate MH techniques: instead of subsampling data items assum…
Variational inference is a powerful paradigm for approximate Bayesian inference with a number of appealing properties, including support for model learning and data subsampling. By contrast MCMC methods like Hamiltonian Monte Carlo do not share these properties but remain attractive since, contrary to parametric methods, MCMC is asymptotically unbiased. For these reasons researchers have sought to combine the strengths of both classes of algorithms, with recent approaches coming closer to realizing this vision in practice. However, supporting data subsampling in these hybrid methods can be a c…
Non-linear state space models are a widely-used class of models for biological, economic, and physical processes. Fitting these models to observed data is a difficult inference problem that has no straightforward solution. We take a Bayesian approach to the inference of unknown parameters of a non-linear state model; this, in turn, requires the availability of efficient Markov Chain Monte Carlo (MCMC) sampling methods for the latent (hidden) variables and model parameters. Using the ensemble technique of Neal (2010) and the embedded HMM technique of Neal (2003), we introduce a new Markov Chain…
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…
In an era where big and high-dimensional data is readily available, data scientists are inevitably faced with the challenge of reducing this data for expensive downstream computation or analysis. To this end, we present here a new method for reducing high-dimensional big data into a representative point set, called projected support points (PSPs). A key ingredient in our method is the so-called sparsity-inducing (SpIn) kernel, which encourages the preservation of low-dimensional features when reducing high-dimensional data. We begin by introducing a unifying theoretical framework for data redu…
It is common practice to use Laplace approximations to compute marginal likelihoods in Bayesian versions of generalised linear models (GLM). Marginal likelihoods combined with model priors are then used in different search algorithms to compute the posterior marginal probabilities of models and individual covariates. This allows performing Bayesian model selection and model averaging. For large sample sizes, even the Laplace approximation becomes computationally challenging because the optimisation routine involved needs to evaluate the likelihood on the full set of data in multiple iterations…
A new class of Markov chain Monte Carlo (MCMC) algorithms, based on simulating piecewise deterministic Markov processes (PDMPs), have recently shown great promise: they are non-reversible, can mix better than standard MCMC algorithms, and can use subsampling ideas to speed up computation in big data scenarios. However, current PDMP samplers can only sample from posterior densities that are differentiable almost everywhere, which precludes their use for model choice. Motivated by variable selection problems, we show how to develop reversible jump PDMP samplers that can jointly explore the discr…
