arXiv subjects

Gareth O. Roberts

Publications and source records attributed to Gareth O. Roberts.

At least 37 records · Page 2Linked to original sources

arXiv2024

Stochastic Gradient Piecewise Deterministic Monte Carlo Samplers

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.

stat.ML↗

arXiv2024

Optimal Scaling Results for Moreau-Yosida Metropolis-adjusted Langevin Algorithms

We consider a recently proposed class of MCMC methods which uses proximity maps instead of gradients to build proposal mechanisms which can be employed for both differentiable and non-differentiable targets. These methods have been shown to be stable for a wide class of targets, making them a valuable alternative to Metropolis-adjusted Langevin algorithms (MALA); and have found wide application in imaging contexts. The wider stability properties are obtained by building the Moreau-Yosida envelope for the target of interest, which depends on a parameter $λ$. In this work, we investigate the optimal scaling problem for this class of algorithms, which encompasses MALA, and provide practical guidelines for the implementation of these methods.

stat.CO↗

arXiv2024

Stereographic Markov Chain Monte Carlo

High-dimensional distributions, especially those with heavy tails, are notoriously difficult for off-the-shelf MCMC samplers: the combination of unbounded state spaces, diminishing gradient information, and local moves results in empirically observed ``stickiness'' and poor theoretical mixing properties -- lack of geometric ergodicity. In this paper, we introduce a new class of MCMC samplers that map the original high-dimensional problem in Euclidean space onto a sphere and remedy these notorious mixing problems. In particular, we develop random-walk Metropolis type algorithms as well as versions of the Bouncy Particle Sampler that are uniformly ergodic for a large class of light and heavy-tailed distributions and also empirically exhibit rapid convergence in high dimensions. In the best scenario, the proposed samplers can enjoy the ``blessings of dimensionality'' that the convergence is faster in higher dimensions.

stat.CO↗

arXiv2023

Flexible Bayesian inference for diffusion processes using splines

We introduce a flexible method to simultaneously infer both the drift and volatility functions of a discretely observed scalar diffusion. We introduce spline bases to represent these functions and develop a Markov chain Monte Carlo algorithm to infer, a posteriori, the coefficients of these functions in the spline basis. A key innovation is that we use spline bases to model transformed versions of the drift and volatility functions rather than the functions themselves. The output of the algorithm is a posterior sample of plausible drift and volatility functions that are not constrained to any particular parametric family. The flexibility of this approach provides practitioners a powerful investigative tool, allowing them to posit a variety of parametric models to better capture the underlying dynamics of their processes of interest. We illustrate the versatility of our method by applying it to challenging datasets from finance, paleoclimatology, and astrophysics. In view of the parametric diffusion models widely employed in the literature for those examples, some of our results are surprising since they call into question some aspects of these models.

stat.ME↗

arXiv2023

Divide-and-Conquer Fusion

Combining several (sample approximations of) distributions, which we term sub-posteriors, into a single distribution proportional to their product, is a common challenge. Occurring, for instance, in distributed 'big data' problems, or when working under multi-party privacy constraints. Many existing approaches resort to approximating the individual sub-posteriors for practical necessity, then find either an analytical approximation or sample approximation of the resulting (product-pooled) posterior. The quality of the posterior approximation for these approaches is poor when the sub-posteriors fall out-with a narrow range of distributional form, such as being approximately Gaussian. Recently, a Fusion approach has been proposed which finds an exact Monte Carlo approximation of the posterior, circumventing the drawbacks of approximate approaches. Unfortunately, existing Fusion approaches have a number of computational limitations, particularly when unifying a large number of sub-posteriors. In this paper, we generalise the theory underpinning existing Fusion approaches, and embed the resulting methodology within a recursive divide-and-conquer sequential Monte Carlo paradigm. This ultimately leads to a competitive Fusion approach, which is robust to increasing numbers of sub-posteriors.

stat.ME↗

arXiv2023

Exact Bayesian inference for diffusion driven Cox processes

In this paper, we present a novel methodology to perform Bayesian inference for Cox processes in which the intensity function is driven by a diffusion process. The novelty lies in the fact that no discretization error is involved, despite the non-tractability of both the likelihood function and the transition density of the diffusion. The methodology is based on an MCMC algorithm and its exactness is built on retrospective sampling techniques. The efficiency of the methodology is investigated in some simulated examples and its applicability is illustrated in some real data analyzes.

stat.ME↗

arXiv2023

Exact Monte Carlo likelihood-based inference for jump-diffusion processes

Statistical inference for discretely observed jump-diffusion processes is a complex problem which motivates new methodological challenges. Thus existing approaches invariably resort to time-discretisations which inevitably lead to approximations in inference. In this paper, we give the first general collection of methodologies for exact (in this context meaning discretisation-free) likelihood-based inference for discretely observed finite activity jump-diffusions. The only sources of error involved are Monte Carlo error and convergence of EM or MCMC algorithms. We shall introduce both frequentist and Bayesian approaches, illustrating the methodology through simulated and real examples.

stat.ME↗

arXiv2023

Football Group Draw Probabilities and Corrections

This paper considers the challenge of designing football group draw mechanisms which have the uniform distribution over all valid draw assignments, but are also entertaining, practical, and transparent. We explain how to simulate the FIFA Sequential Draw method, to compute the non-uniformity of its draws by comparison to a uniform Rejection Sampler. We then propose two practical methods of achieving the uniform distribution while still using balls and bowls in a way which is suitable for a televised draw. The solutions can also be tried interactively.

stat.AP↗

arXiv2022

Speed Up Zig-Zag

Zig-Zag is Piecewise Deterministic Markov Process, efficiently used for simulation in an MCMC setting. As we show in this article, it fails to be exponentially ergodic on heavy tailed target distributions. We introduce an extension of the Zig-Zag process by allowing the process to move with a non-constant speed function $s$, depending on the current state of the process. We call this process Speed Up Zig-Zag (SUZZ). We provide conditions that guarantee stability properties for the SUZZ process, including non-explosivity, exponential ergodicity in heavy tailed targets and central limit theorem. Interestingly, we find that using speed functions that induce explosive deterministic dynamics may lead to stable algorithms that can even mix faster. We further discuss the choice of an efficient speed function by providing an efficiency criterion for the one-dimensional process and we support our findings with simulation results.

math.PR↗

arXiv2022

Dimension-free Mixing for High-dimensional Bayesian Variable Selection

Yang et al. (2016) proved that the symmetric random walk Metropolis--Hastings algorithm for Bayesian variable selection is rapidly mixing under mild high-dimensional assumptions. We propose a novel MCMC sampler using an informed proposal scheme, which we prove achieves a much faster mixing time that is independent of the number of covariates, under the same assumptions. To the best of our knowledge, this is the first high-dimensional result which rigorously shows that the mixing rate of informed MCMC methods can be fast enough to offset the computational cost of local posterior evaluation. Motivated by the theoretical analysis of our sampler, we further propose a new approach called "two-stage drift condition" to studying convergence rates of Markov chains on general state spaces, which can be useful for obtaining tight complexity bounds in high-dimensional settings. The practical advantages of our algorithm are illustrated by both simulation studies and real data analysis.

stat.ME↗

arXiv2022

The computational cost of blocking for sampling discretely observed diffusions

Many approaches for conducting Bayesian inference on discretely observed diffusions involve imputing diffusion bridges between observations. This can be computationally challenging in settings in which the temporal horizon between subsequent observations is large, due to the poor scaling of algorithms for simulating bridges as observation distance increases. It is common in practical settings to use a blocking scheme, in which the path is split into a (user-specified) number of overlapping segments and a Gibbs sampler is employed to update segments in turn. Substituting the independent simulation of diffusion bridges for one obtained using blocking introduces an inherent trade-off: we are now imputing shorter bridges at the cost of introducing a dependency between subsequent iterations of the bridge sampler. This is further complicated by the fact that there are a number of possible ways to implement the blocking scheme, each of which introduces a different dependency structure between iterations. Although blocking schemes have had considerable empirical success in practice, there has been no analysis of this trade-off nor guidance to practitioners on the particular specifications that should be used to obtain a computationally efficient implementation. In this article we conduct this analysis and demonstrate that the expected computational cost of a blocked path-space rejection sampler applied to Brownian bridges scales asymptotically at a cubic rate with respect to the observation distance and that this rate is linear in the case of the Ornstein-Uhlenbeck process. Numerical experiments suggest applicability both of the results of our paper and of the guidance we provide beyond the class of linear diffusions considered.

stat.CO↗

arXiv2022

Optimal Scaling of MCMC Beyond Metropolis

The problem of optimally scaling the proposal distribution in a Markov chain Monte Carlo algorithm is critical to the quality of the generated samples. Much work has gone into obtaining such results for various Metropolis-Hastings (MH) algorithms. Recently, acceptance probabilities other than MH are being employed in problems with intractable target distributions. There is little resource available on tuning the Gaussian proposal distributions for this situation. We obtain optimal scaling results for a general class of acceptance functions, which includes Barker's and Lazy-MH. In particular, optimal values for the Barker's algorithm are derived and found to be significantly different from that obtained for the MH algorithm. Our theoretical conclusions are supported by numerical simulations indicating that when the optimal proposal variance is unknown, tuning to the optimal acceptance probability remains an effective strategy.

stat.CO↗

arXiv2021

Simulating bridges using confluent diffusions

Diffusions are a fundamental class of models in many fields, including finance, engineering, and biology. Simulating diffusions is challenging as their sample paths are infinite-dimensional and their transition functions are typically intractable. In statistical settings such as parameter inference for discretely observed diffusions, we require simulation techniques for diffusions conditioned on hitting a given endpoint, which introduces further complication. In this paper we introduce a Markov chain Monte Carlo algorithm for simulating bridges of ergodic diffusions which (i) is exact in the sense that there is no discretisation error, (ii) has computational cost that is linear in the duration of the bridges, and (iii) provides bounds on local maxima and minima of the simulated trajectory. Our approach works directly on diffusion path space, by constructing a proposal (which we term a confluence) that is then corrected with an accept/reject step in a pseudo-marginal algorithm. Our method requires only the simulation of unconditioned diffusion sample paths. We apply our approach to the simulation of Langevin diffusion bridges, a practical problem arising naturally in many situations, such as statistical inference in distributed settings.

stat.ME↗

arXiv2021

Skew Brownian Motion and Complexity of the ALPS Algorithm

Simulated tempering is a popular method of allowing MCMC algorithms to move between modes of a multimodal target density π. The paper [24] introduced the Annealed Leap-Point Sampler (ALPS) to allow for rapid movement between modes. In this paper, we prove that, under appropriate assumptions, a suitably scaled version of the ALPS algorithm converges weakly to skew Brownian motion. Our results show that under appropriate assumptions, the ALPS algorithm mixes in time O(d[log(d)]^2 ) or O(d), depending on which version is used.

math.PR↗

arXiv2021

Efficient Bernoulli factory MCMC for intractable posteriors

Accept-reject based Markov chain Monte Carlo (MCMC) algorithms have traditionally utilised acceptance probabilities that can be explicitly written as a function of the ratio of the target density at the two contested points. This feature is rendered almost useless in Bayesian posteriors with unknown functional forms. We introduce a new family of MCMC acceptance probabilities that has the distinguishing feature of not being a function of the ratio of the target density at the two points. We present two stable Bernoulli factories that generate events within this class of acceptance probabilities. The efficiency of our methods rely on obtaining reasonable local upper or lower bounds on the target density and we present two classes of problems where such bounds are viable: Bayesian inference for diffusions and MCMC on constrained spaces. The resulting portkey Barker's algorithms are exact and computationally more efficient that the current state-of-the-art.

stat.CO↗

arXiv2021

Rao-Blackwellization in the MCMC era

Rao-Blackwellization is a notion often occurring in the MCMC literature, with possibly different meanings and connections with the original Rao--Blackwell theorem (Rao, 1945 and Blackwell,1947), including a reduction of the variance of the resulting Monte Carlo approximations. This survey reviews some of the meanings of the term.

stat.CO↗

arXiv2020

An epidemic model for an evolving pathogen with strain-dependent immunity

Between pandemics, the influenza virus exhibits periods of incremental evolution via a process known as antigenic drift. This process gives rise to a sequence of strains of the pathogen that are continuously replaced by newer strains, preventing a build up of immunity in the host population. In this paper, a parsimonious epidemic model is defined that attempts to capture the dynamics of evolving strains within a host population. The `evolving strains' epidemic model has many properties that lie in-between the Susceptible-Infected-Susceptible and the Susceptible-Infected-Removed epidemic models, due to the fact that individuals can only be infected by each strain once, but remain susceptible to reinfection by newly emerged strains. Coupling results are used to identify key properties, such as the time to extinction. A range of reproduction numbers are explored to characterize the model, including a novel quasi-stationary reproduction number that can be used to describe the re-emergence of the pathogen into a population with `average' levels of strain immunity, analogous to the beginning of the winter peak in influenza. Finally the quasi-stationary distribution of the evolving strains model is explored via simulation.

q-bio.PE↗

arXiv2020

Regeneration-enriched Markov processes with application to Monte Carlo

We study a class of Markov processes that combine local dynamics, arising from a fixed Markov process, with regenerations arising at a state-dependent rate. We give conditions under which such processes possess a given target distribution as their invariant measures, thus making them amenable for use within Monte Carlo methodologies. Since the regeneration mechanism can compensate the choice of local dynamics, while retaining the same invariant distribution, great flexibility can be achieved in selecting local dynamics, and the mathematical analysis is simplified. We give straightforward conditions for the process to possess a central limit theorem, and additional conditions for uniform ergodicity and for a coupling from the past construction to hold, enabling exact sampling from the invariant distribution. We further consider and analyse a natural approximation of the process which may arise in the practical simulation of some classes of continuous-time dynamics.

math.PR↗