Paul Fearnhead

Jie Li, Paul Fearnhead, Piotr Fryzlewicz et al.

Detecting change-points in data is challenging because of the range of possible types of change and types of behaviour of data when there is no change. Statistically efficient methods for detecting a change will depend on both of these features, and it can be difficult for a practitioner to develop an appropriate detection method for their application of interest. We show how to automatically generate new offline detection methods based on training a neural network. Our approach is motivated by many existing tests for the presence of a change-point being representable by a simple neural network, and thus a neural network trained with sufficient data should have performance at least as good as these methods. We present theory that quantifies the error rate for such an approach, and how it depends on the amount of training data. Empirical results show that, even with limited training data, its performance is competitive with the standard CUSUM-based classifier for detecting a change in mean when the noise is independent and Gaussian, and can substantially outperform it in the presence of auto-correlated or heavy-tailed noise. Our method also shows strong results in detecting and localising changes in activity based on accelerometer data.

Paul Fearnhead, Christopher Nemeth, Chris J. Oates et al.

This book aims to provide a graduate-level introduction to advanced topics in Markov chain Monte Carlo (MCMC) algorithms, as applied broadly in the Bayesian computational context. Most, if not all of these topics (stochastic gradient MCMC, non-reversible MCMC, continuous time MCMC, and new techniques for convergence assessment) have emerged as recently as the last decade, and have driven substantial recent practical and theoretical advances in the field. A particular focus is on methods that are scalable with respect to either the amount of data, or the data dimension, motivated by the emerging high-priority application areas in machine learning and AI.

Srshti Putcha, Christopher Nemeth, Paul Fearnhead

Stochastic gradient MCMC (SGMCMC) offers a scalable alternative to traditional MCMC, by constructing an unbiased estimate of the gradient of the log-posterior with a small, uniformly-weighted subsample of the data. While efficient to compute, the resulting gradient estimator may exhibit a high variance and impact sampler performance. The problem of variance control has been traditionally addressed by constructing a better stochastic gradient estimator, often using control variates. We propose to use a discrete, non-uniform probability distribution to preferentially subsample data points that have a greater impact on the stochastic gradient. In addition, we present a method of adaptively adjusting the subsample size at each iteration of the algorithm, so that we increase the subsample size in areas of the sample space where the gradient is harder to estimate. We demonstrate that such an approach can maintain the same level of accuracy while substantially reducing the average subsample size that is used.

Christopher Nemeth, Paul Fearnhead

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 particular class of scalable Monte Carlo algorithms, stochastic gradient Markov chain Monte Carlo (SGMCMC) which utilises data subsampling techniques to reduce the per-iteration cost of MCMC. We provide an introduction to some popular SGMCMC algorithms and review the supporting theoretical results, as well as comparing the efficiency of SGMCMC algorithms against MCMC on benchmark examples. The supporting R code is available online.

Toby Dylan Hocking, Guillem Rigaill, Paul Fearnhead et al.

Changepoint detection is a central problem in time series and genomic data. For some applications, it is natural to impose constraints on the directions of changes. One example is ChIP-seq data, for which adding an up-down constraint improves peak detection accuracy, but makes the optimization problem more complicated. We show how a recently proposed functional pruning technique can be adapted to solve such constrained changepoint detection problems. This leads to a new algorithm which can solve problems with arbitrary affine constraints on adjacent segment means, and which has empirical time complexity that is log-linear in the amount of data. This algorithm achieves state-of-the-art accuracy in a benchmark of several genomic data sets, and is orders of magnitude faster than existing algorithms that have similar accuracy. Our implementation is available as the PeakSegPDPA function in the coseg R package, https://github.com/tdhock/coseg

Paul Fearnhead, Sebastiano Grazzi, Chris Nemeth et al.

Recent work has suggested using Monte Carlo methods based on piecewise deterministic Markov processes (PDMPs) to sample from target distributions of interest. PDMPs are non-reversible continuous-time processes endowed with momentum, and hence can mix better than standard reversible MCMC samplers. Furthermore, they can incorporate exact sub-sampling schemes which only require access to a single (randomly selected) data point at each iteration, yet without introducing bias to the algorithm's stationary distribution. However, the range of models for which PDMPs can be used, particularly with sub-sampling, is limited. We propose approximate simulation of PDMPs with sub-sampling for scalable sampling from posterior distributions. The approximation takes the form of an Euler approximation to the true PDMP dynamics, and involves using an estimate of the gradient of the log-posterior based on a data sub-sample. We thus call this class of algorithms stochastic-gradient PDMPs. Importantly, the trajectories of stochastic-gradient PDMPs are continuous and can leverage recent ideas for sampling from measures with continuous and atomic components. We show these methods are easy to implement, present results on their approximation error and demonstrate numerically that this class of algorithms has similar efficiency to, but is more robust than, stochastic gradient Langevin dynamics.

C. Trojan, P. Fearnhead, C. Nemeth

Divide-and-conquer MCMC is a strategy for parallelising Markov Chain Monte Carlo sampling by running independent samplers on disjoint subsets of a dataset and merging their output. An ongoing challenge in the literature is to efficiently perform this merging without imposing distributional assumptions on the posteriors. We propose using diffusion generative modelling to fit density approximations to the subposterior distributions. This approach outperforms existing methods on challenging merging problems, while its computational cost scales more efficiently to high dimensional problems than existing density estimation approaches.

Augustin Chevallier, Frédéric Cazals, Paul Fearnhead

Computing the volume of a polytope in high dimensions is computationally challenging but has wide applications. Current state-of-the-art algorithms to compute such volumes rely on efficient sampling of a Gaussian distribution restricted to the polytope, using e.g. Hamiltonian Monte Carlo. We present a new sampling strategy that uses a Piecewise Deterministic Markov Process. Like Hamiltonian Monte Carlo, this new method involves simulating trajectories of a non-reversible process and inherits similar good mixing properties. However, importantly, the process can be simulated more easily due to its piecewise linear trajectories - and this leads to a reduction of the computational cost by a factor of the dimension of the space. Our experiments indicate that our method is numerically robust and is one order of magnitude faster (or better) than existing methods using Hamiltonian Monte Carlo. On a single core processor, we report computational time of a few minutes up to dimension 500.

Gaetano Romano, Idris Eckley, Paul Fearnhead et al.

Many modern applications of online changepoint detection require the ability to process high-frequency observations, sometimes with limited available computational resources. Online algorithms for detecting a change in mean often involve using a moving window, or specifying the expected size of change. Such choices affect which changes the algorithms have most power to detect. We introduce an algorithm, Functional Online CuSUM (FOCuS), which is equivalent to running these earlier methods simultaneously for all sizes of window, or all possible values for the size of change. Our theoretical results give tight bounds on the expected computational cost per iteration of FOCuS, with this being logarithmic in the number of observations. We show how FOCuS can be applied to a number of different change in mean scenarios, and demonstrate its practical utility through its state-of-the art performance at detecting anomalous behaviour in computer server data.

Augustin Chevallier, Paul Fearnhead, Matthew Sutton

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 discrete space of models and the continuous space of parameters. Our framework is general: it takes any existing PDMP sampler, and adds two types of trans-dimensional moves that allow for the addition or removal of a variable from the model. We show how the rates of these trans-dimensional moves can be calculated so that the sampler has the correct invariant distribution. Simulations show that the new samplers can mix better than standard MCMC algorithms. Our empirical results show they are also more efficient than gradient-based samplers that avoid model choice through use of continuous spike-and-slab priors which replace a point mass at zero for each parameter with a density concentrated around zero.

Alexander T. M. Fisch, Idris A. Eckley, P. Fearnhead

In this paper, we propose CE-BASS, a particle mixture Kalman filter which is robust to both innovative and additive outliers, and able to fully capture multi-modality in the distribution of the hidden state. Furthermore, the particle sampling approach re-samples past states, which enables CE-BASS to handle innovative outliers which are not immediately visible in the observations, such as trend changes. The filter is computationally efficient as we derive new, accurate approximations to the optimal proposal distributions for the particles. The proposed algorithm is shown to compare well with existing approaches and is applied to both machine temperature and server data.

Alexander T M Fisch, Idris A Eckley, Paul Fearnhead

In recent years, there has been a growing interest in identifying anomalous structure within multivariate data streams. We consider the problem of detecting collective anomalies, corresponding to intervals where one or more of the data streams behaves anomalously. We first develop a test for a single collective anomaly that has power to simultaneously detect anomalies that are either rare, that is affecting few data streams, or common. We then show how to detect multiple anomalies in a way that is computationally efficient but avoids the approximations inherent in binary segmentation-like approaches. This approach, which we call MVCAPA, is shown to consistently estimate the number and location of the collective anomalies, a property that has not previously been shown for competing methods. MVCAPA can be made robust to point anomalies and can allow for the anomalies to be imperfectly aligned. We show the practical usefulness of allowing for imperfect alignments through a resulting increase in power to detect regions of copy number variation.

Christopher Aicher, Srshti Putcha, Christopher Nemeth et al.

State space models (SSMs) provide a flexible framework for modeling complex time series via a latent stochastic process. Inference for nonlinear, non-Gaussian SSMs is often tackled with particle methods that do not scale well to long time series. The challenge is two-fold: not only do computations scale linearly with time, as in the linear case, but particle filters additionally suffer from increasing particle degeneracy with longer series. Stochastic gradient MCMC methods have been developed to scale Bayesian inference for finite-state hidden Markov models and linear SSMs using buffered stochastic gradient estimates to account for temporal dependencies. We extend these stochastic gradient estimators to nonlinear SSMs using particle methods. We present error bounds that account for both buffering error and particle error in the case of nonlinear SSMs that are log-concave in the latent process. We evaluate our proposed particle buffered stochastic gradient using stochastic gradient MCMC for inference on both long sequential synthetic and minute-resolution financial returns data, demonstrating the importance of this class of methods.

Jack Baker, Paul Fearnhead, Emily B Fox et al.

Stochastic gradient Markov chain Monte Carlo (SGMCMC) has become a popular method for scalable Bayesian inference. These methods are based on sampling a discrete-time approximation to a continuous time process, such as the Langevin diffusion. When applied to distributions defined on a constrained space the time-discretization error can dominate when we are near the boundary of the space. We demonstrate that because of this, current SGMCMC methods for the simplex struggle with sparse simplex spaces; when many of the components are close to zero. Unfortunately, many popular large-scale Bayesian models, such as network or topic models, require inference on sparse simplex spaces. To avoid the biases caused by this discretization error, we propose the stochastic Cox-Ingersoll-Ross process (SCIR), which removes all discretization error and we prove that samples from the SCIR process are asymptotically unbiased. We discuss how this idea can be extended to target other constrained spaces. Use of the SCIR process within a SGMCMC algorithm is shown to give substantially better performance for a topic model and a Dirichlet process mixture model than existing SGMCMC approaches.

Alexander T. M. Fisch, Idris A. Eckley, Paul Fearnhead

The challenge of efficiently identifying anomalies in data sequences is an important statistical problem that now arises in many applications. Whilst there has been substantial work aimed at making statistical analyses robust to outliers, or point anomalies, there has been much less work on detecting anomalous segments, or collective anomalies, particularly in those settings where point anomalies might also occur. In this article, we introduce Collective And Point Anomalies (CAPA), a computationally efficient approach that is suitable when collective anomalies are characterised by either a change in mean, variance, or both, and distinguishes them from point anomalies. Theoretical results establish the consistency of CAPA at detecting collective anomalies and, as a by-product, the consistency of a popular penalised cost based change in mean and variance detection method. Empirical results show that CAPA has close to linear computational cost as well as being more accurate at detecting and locating collective anomalies than other approaches. We demonstrate the utility of CAPA through its ability to detect exoplanets from light curve data from the Kepler telescope.

Jack Baker, Paul Fearnhead, Emily B. Fox et al.

This paper introduces the R package sgmcmc; which can be used for Bayesian inference on problems with large datasets using stochastic gradient Markov chain Monte Carlo (SGMCMC). Traditional Markov chain Monte Carlo (MCMC) methods, such as Metropolis-Hastings, are known to run prohibitively slowly as the dataset size increases. SGMCMC solves this issue by only using a subset of data at each iteration. SGMCMC requires calculating gradients of the log likelihood and log priors, which can be time consuming and error prone to perform by hand. The sgmcmc package calculates these gradients itself using automatic differentiation, making the implementation of these methods much easier. To do this, the package uses the software library TensorFlow, which has a variety of statistical distributions and mathematical operations as standard, meaning a wide class of models can be built using this framework. SGMCMC has become widely adopted in the machine learning literature, but less so in the statistics community. We believe this may be partly due to lack of software; this package aims to bridge this gap.

Jack Baker, Paul Fearnhead, Emily B. Fox et al.

It is well known that Markov chain Monte Carlo (MCMC) methods scale poorly with dataset size. A popular class of methods for solving this issue is stochastic gradient MCMC. These methods use a noisy estimate of the gradient of the log posterior, which reduces the per iteration computational cost of the algorithm. Despite this, there are a number of results suggesting that stochastic gradient Langevin dynamics (SGLD), probably the most popular of these methods, still has computational cost proportional to the dataset size. We suggest an alternative log posterior gradient estimate for stochastic gradient MCMC, which uses control variates to reduce the variance. We analyse SGLD using this gradient estimate, and show that, under log-concavity assumptions on the target distribution, the computational cost required for a given level of accuracy is independent of the dataset size. Next we show that a different control variate technique, known as zero variance control variates can be applied to SGMCMC algorithms for free. This post-processing step improves the inference of the algorithm by reducing the variance of the MCMC output. Zero variance control variates rely on the gradient of the log posterior; we explore how the variance reduction is affected by replacing this with the noisy gradient estimate calculated by SGMCMC.

Robert Maidstone, Paul Fearnhead, Adam Letchford

Whilst there are many approaches to detecting changes in mean for a univariate time-series, the problem of detecting multiple changes in slope has comparatively been ignored. Part of the reason for this is that detecting changes in slope is much more challenging. For example, simple binary segmentation procedures do not work for this problem, whilst efficient dynamic programming methods that work well for the change in mean problem cannot be directly used for detecting changes in slope. We present a novel dynamic programming approach, CPOP, for finding the "best" continuous piecewise-linear fit to data. We define best based on a criterion that measures fit to data using the residual sum of squares, but penalises complexity based on an $L_0$ penalty on changes in slope. We show that using such a criterion is more reliable at estimating changepoint locations than approaches that penalise complexity using an $L_1$ penalty. Empirically CPOP has good computational properties, and can analyse a time-series with over 10,000 observations and over 100 changes in a few minutes. Our method is used to analyse data on the motion of bacteria, and provides fits to the data that both have substantially smaller residual sum of squares and are more parsimonious than two competing approaches.

Paul Fearnhead, Guillem Rigaill

Many traditional methods for identifying changepoints can struggle in the presence of outliers, or when the noise is heavy-tailed. Often they will infer additional changepoints in order to fit the outliers. To overcome this problem, data often needs to be pre-processed to remove outliers, though this is difficult for applications where the data needs to be analysed online. We present an approach to changepoint detection that is robust to the presence of outliers. The idea is to adapt existing penalised cost approaches for detecting changes so that they use loss functions that are less sensitive to outliers. We argue that loss functions that are bounded, such as the classical biweight loss, are particularly suitable -- as we show that only bounded loss functions are robust to arbitrarily extreme outliers. We present an efficient dynamic programming algorithm that can find the optimal segmentation under our penalised cost criteria. Importantly, this algorithm can be used in settings where the data needs to be analysed online. We show that we can consistently estimate the number of changepoints, and accurately estimate their locations, using the biweight loss function. We demonstrate the usefulness of our approach for applications such as analysing well-log data, detecting copy number variation, and detecting tampering of wireless devices.

Christopher Nemeth, Chris Sherlock, Paul Fearnhead

This paper proposes a new sampling scheme based on Langevin dynamics that is applicable within pseudo-marginal and particle Markov chain Monte Carlo algorithms. We investigate this algorithm's theoretical properties under standard asymptotics, which correspond to an increasing dimension of the parameters, $n$. Our results show that the behaviour of the algorithm depends crucially on how accurately one can estimate the gradient of the log target density. If the error in the estimate of the gradient is not sufficiently controlled as dimension increases, then asymptotically there will be no advantage over the simpler random-walk algorithm. However, if the error is sufficiently well-behaved, then the optimal scaling of this algorithm will be $O(n^{-1/6})$ compared to $O(n^{-1/2})$ for the random walk. Our theory also gives guidelines on how to tune the number of Monte Carlo samples in the likelihood estimate and the proposal step-size.

Kaylea Haynes, Idris A. Eckley, Paul Fearnhead

In the multiple changepoint setting, various search methods have been proposed which involve optimising either a constrained or penalised cost function over possible numbers and locations of changepoints using dynamic programming. Such methods are typically computationally intensive. Recent work in the penalised optimisation setting has focussed on developing a pruning-based approach which gives an improved computational cost that, under certain conditions, is linear in the number of data points. Such an approach naturally requires the specification of a penalty to avoid under/over-fitting. Work has been undertaken to identify the appropriate penalty choice for data generating processes with known distributional form, but in many applications the model assumed for the data is not correct and these penalty choices are not always appropriate. Consequently it is desirable to have an approach that enables us to compare segmentations for different choices of penalty. To this end we present a method to obtain optimal changepoint segmentations of data sequences for all penalty values across a continuous range. This permits an evaluation of the various segmentations to identify a suitably parsimonious penalty choice. The computational complexity of this approach can be linear in the number of data points and linear in the difference between the number of changepoints in the optimal segmentations for the smallest and largest penalty values. This can be orders of magnitude faster than alternative approaches that find optimal segmentations for a range of the number of changepoints.

Chris Nemeth, Paul Fearnhead

Particle MCMC is a class of algorithms that can be used to analyse state-space models. They use MCMC moves to update the parameters of the models, and particle filters to propose values for the path of the state-space model. Currently the default is to use random walk Metropolis to update the parameter values. We show that it is possible to use information from the output of the particle filter to obtain better proposal distributions for the parameters. In particular it is possible to obtain estimates of the gradient of the log posterior from each run of the particle filter, and use these estimates within a Langevin-type proposal. We propose using the recent computationally efficient approach of Nemeth et al. (2013) for obtaining such estimates. We show empirically that for a variety of state-space models this proposal is more efficient than the standard random walk Metropolis proposal in terms of: reducing autocorrelation of the posterior samples, reducing the burn-in time of the MCMC sampler and increasing the squared jump distance between posterior samples.

Christopher Nemeth, Paul Fearnhead, Lyudmila Mihaylova

Poyiadjis et al. (2011) show how particle methods can be used to estimate both the score and the observed information matrix for state space models. These methods either suffer from a computational cost that is quadratic in the number of particles, or produce estimates whose variance increases quadratically with the amount of data. This paper introduces an alternative approach for estimating these terms at a computational cost that is linear in the number of particles. The method is derived using a combination of kernel density estimation, to avoid the particle degeneracy that causes the quadratically increasing variance, and Rao-Blackwellisation. Crucially, we show the method is robust to the choice of bandwidth within the kernel density estimation, as it has good asymptotic properties regardless of this choice. Our estimates of the score and observed information matrix can be used within both online and batch procedures for estimating parameters for state space models. Empirical results show improved parameter estimates compared to existing methods at a significantly reduced computational cost. Supplementary materials including code are available.