Abstract We propose a flexible nonparametric Bayesian modelling framework for multivariate time series of count data based on tensor factorisations. Our models can be viewed as infinite state space Markov chains of known maximal order with non-linear serial dependence through the introduction of appropriate latent variables. Alternatively, our models can be viewed as Bayesian hierarchical models with conditionally independent Poisson distributed observations. Inference about the important lags and their complex interactions is achieved via MCMC. When the observed counts are large, we deal with the resulting computational complexity of Bayesian inference via a two-step inferential strategy based on an initial analysis of a training set of the data. Our methodology is illustrated using simulation experiments and analysis of real-world data.

We adopt the interpretability offered by a parametric, Hawkes-process-inspired conditional probability mass function for the marks and apply variational inference techniques to derive a general and scalable inferential framework for marked point processes. The framework can handle both exchangeable and non-exchangeable event sequences with minimal tuning and without any pre-training. This contrasts with many parametric and non-parametric state-of-the-art methods that typically require pre-training and/or careful tuning, and can only handle exchangeable event sequences. The framework’s competitive computational and predictive performance against other state-of-the-art methods are illustrated through real data experiments. Its attractiveness for large-scale applications is demonstrated through a case study involving all events occurring in an English Premier League season.

We introduce a novel scheme for Bayesian inference on permanental processes which models the Poisson intensity as the square of a Gaussian process. Combining generalized kernels and a Fourier features-based representation of the Gaussian process with a Laplace approximation to the posterior, we achieve a fast and efficient inference that does not require numerical integration over the input space, allows kernel design and scales linearly with the number of events. Our method builds and improves upon the state-of-theart Laplace Bayesian point process benchmark of Walder and Bishop (2017), demonstrated on both synthetic, real-world temporal and large spatial data sets.

In this paper, we introduce a variant of Bayesian online change point detection with a reducedrank Student-t process (TP) and dependent Student-t noise, as a nonparametric time series model. Our method builds and improves upon the state-of-the-art Gaussian process (GP) change point model benchmark of Saatci et al. (2010). The Student-t process generalizes the concept of a GP and hence yields a more flexible alternative. Additionally, unlike a GP, the predictive variance explicitly depends on the training observations, while the use of an entangled Student-t noise model preserves analytical tractability. Our approach also uses a Hilbert space reduced-rank representation of the TP kernel, derived from an eigenfunction expansion of the Laplace operator (Solin & Sarkka, 2020), to alleviate its computational complexity. Improvements in prediction and training time are demonstrated with real-world data-sets

We provide guarantees for approximate Gaussian Process (GP) regression resulting from two common low-rank kernel approximations: based on random Fourier features, and based on truncating the kernel's Mercer expansion. In particular, we bound the Kullback–Leibler divergence between an exact GP and one resulting from one of the afore-described low-rank approximations to its kernel, as well as between their corresponding predictive densities, and we also bound the error between predictive mean vectors and between predictive covariance matrices computed using the exact versus using the approximate GP. We provide experiments on both simulated data and standard benchmarks to evaluate the effectiveness of our theoretical bounds.

Hamiltonian Monte Carlo (HMC) is a popular Markov Chain Monte Carlo (MCMC) algorithm to sample from an unnormalized probability distribution. A leapfrog integrator is commonly used to implement HMC in practice, but its performance can be sensitive to the choice of mass matrix used therein. We develop a gradient-based algorithm that allows for the adaptation of the mass matrix by encouraging the leapfrog integrator to have high acceptance rates while also exploring all dimensions jointly. In contrast to previous work that adapt the hyperparameters of HMC using some form of expected squared jumping distance, the adaptation strategy suggested here aims to increase sampling efficiency by maximizing an approximation of the proposal entropy. We illustrate that using multiple gradients in the HMC proposal can be beneficial compared to a single gradient-step in Metropolis-adjusted Langevin proposals. Empirical evidence suggests that the adaptation method can outperform different versions of HMC schemes by adjusting the mass matrix to the geometry of the target distribution and by providing some control on the integration time.

Abstract We introduce a Gaussian process latent factor model for multi-label classification that can capture correlations among class labels by using a small set of latent Gaussian process functions. To address computational challenges, when the number of training instances is very large, we introduce several techniques based on variational sparse Gaussian process approximations and stochastic optimization. Specifically, we apply doubly stochastic variational inference that sub-samples data instances and classes which allows us to cope with Big Data. Furthermore, we show it is possible and beneficial to optimize over inducing points, using gradient-based methods, even in very high dimensional input spaces involving up to hundreds of thousands of dimensions. We demonstrate the usefulness of our approach on several real-world large-scale multi-label learning problems.

We present a scalable approach to performing approximate fully Bayesian inference in generic state space models. The proposed method is an alternative to particle MCMC that provides fully Bayesian inference of both the dynamic latent states and the static pa- rameters of the model. We build up on recent advances in computational statistics that combine variational methods with sequential Monte Carlo sampling and we demonstrate the advantages of performing full Bayesian inference over the static parameters rather than just performing variational EM approxima- tions. We illustrate how our approach enables scalable inference in multivariate stochastic volatility models and self-exciting point pro- cess models that allow for flexible dynamics in the latent intensity function.

This paper considers a new family of variational distributions motivated by Sklar's theorem. This family is based on new copula-like densities on the hypercube with non-uniform marginals which can be sampled efficiently, i.e. with a complexity linear in the dimension d of the state space. Then, the proposed variational densities that we suggest can be seen as arising from these copula-like densities used as base distributions on the hypercube with Gaussian quantile functions and sparse rotation matrices as normalizing flows. The latter correspond to a rotation of the marginals with complexity O(d log d). We provide some empirical evidence that such a variational family can also approximate non-Gaussian posteriors and can be beneficial compared to Gaussian approximations. Our method performs largely comparably to state-of-the-art variational approximations on standard regression and classification benchmarks for Bayesian Neural Networks.

We introduce a gradient-based learning method to automatically adapt Markov chain Monte Carlo (MCMC) proposal distributions to intractable targets. We define a maximum entropy regularised objective function, referred to as generalised speed measure, which can be robustly optimised over the parameters of the proposal distribution by applying stochastic gradient optimisation. An advantage of our method compared to traditional adaptive MCMC methods is that the adaptation occurs even when candidate state values are rejected. This is a highly desirable property of any adaptation strategy because the adaptation starts in early iterations even if the initial proposal distribution is far from optimum. We apply the framework for learning multivariate random walk Metropolis and Metropolis-adjusted Langevin proposals with full covariance matrices, and provide empirical evidence that our method can outperform other MCMC algorithms, including Hamiltonian Monte Carlo schemes.

MOTIVATION: As the number of studies looking at differences between DNA methylation increases, there is a growing demand to develop and benchmark statistical methods to analyse these data. To date no objective approach for the comparison of these methods has been developed and as such it remains difficult to assess which analysis tool is most appropriate for a given experiment. As a result, there is an unmet need for a DNA methylation data simulator that can accurately reproduce a wide range of experimental setups, and can be routinely used to compare the performance of different statistical models. RESULTS: We have developed WGBSSuite, a flexible stochastic simulation tool that generates single-base resolution DNA methylation data genome-wide. Several simulator parameters can be derived directly from real datasets provided by the user in order to mimic real case scenarios. Thus, it is possible to choose the most appropriate statistical analysis tool for a given simulated design. To show the usefulness of our simulator, we also report a benchmark of commonly used methods for differential methylation analysis. AVAILABILITY AND IMPLEMENTATION: WGBS code and documentation are available under GNU licence at http://www.wgbssuite.org.uk/ CONTACT: : [email protected] or [email protected] SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.