Recent advancements in Markov chain Monte Carlo (MCMC) sampling and surrogate modelling have significantly enhanced the feasibility of Bayesian analysis across engineering fields. However, the selection and integration of surrogate models and cutting-edge MCMC algorithms, often depend on ad-hoc decisions. A systematic assessment of their combined influence on accuracy and efficiency is notably lacking. The present work offers a comprehensive comparative study, employing a scalable case study in computational mechanics focused on the inference of spatially varying material parameters, that sheds light on the impact of methodological choices for surrogate modelling and sampling. We show that a priori training of the surrogate model introduces large errors in the posterior estimation even in low to moderate dimensions. We introduce a simple active learning strategy based on the path of the MCMC algorithm that is superior to all a priori trained models, and determine its training data requirements. We demonstrate that the choice of the MCMC algorithm has only a small influence on the amount of training data but no significant influence on the accuracy of the resulting surrogate model. Further, we show that the accuracy of the posterior estimation largely depends on the surrogate model, but not even a tailored surrogate guarantees convergence of the MCMC. Finally, we identify the forward model as the bottleneck in the inference process, not the MCMC algorithm. While related works focus on employing advanced MCMC algorithms, we demonstrate that the training data requirements render the surrogate modelling approach infeasible before the benefits of these gradient-based MCMC algorithms on cheap models can be reaped.

Piecewise deterministic Markov processes (PDMPs) are a type of continuous-time Markov process that combine deterministic flows with jumps. Recently, PDMPs have garnered attention within the Monte Carlo community as a potential alternative to traditional Markov chain Monte Carlo (MCMC) methods. The Zig-Zag sampler and the Bouncy Particle Sampler are commonly used examples of the PDMP methodology which have also yielded impressive theoretical properties, but little is known about their robustness to extreme dependence or anisotropy of the target density. It turns out that PDMPs may suffer from poor mixing due to anisotropy and this paper investigates this effect in detail in the stylised but important Gaussian case. To this end, we employ a multi-scale analysis framework in this paper. Our results show that when the Gaussian target distribution has two scales, of order 1 and ɛ, the computational cost of the Bouncy Particle Sampler is of order ɛ⁻¹, and the computational cost of the Zig-Zag sampler is ɛ⁻² . In comparison, the cost of the traditional MCMC methods such as RWM is of order ɛ⁻², at least when the dimensionality of the small component is more than 1. Therefore, there is a robustness advantage to using PDMPs in this context.

Strong invariance principles describe the error term of a Brownian approximation to the partial sums of a stochastic process. While these strong approximation results have many applications, results for continuous-time settings have been limited. In this paper, we obtain strong invariance principles for a broad class of ergodic Markov processes. Strong invariance principles provide a unified framework for analysing commonly used estimators of the asymptotic variance in settings with a dependence structure. We demonstrate how this can be used to analyse the batch means method for simulation output of Piecewise Deterministic Monte Carlo samplers. We also derive a fluctuation result for additive functionals of ergodic diffusions using our strong approximation results.

By tuning the steric environment and free pore space in metal-organic frameworks, a large variety of rotor dynamics of the organic linkers can appear. Nitrofunctionalized MIL-53 is a terephthalate-linker-based MOF that shows coupled rotor dynamics between the neighboring linkers along the pore direction. Here, we use classical molecular dynamics up to 6 × 2 × 2 supercells to investigate the range of the correlated linker dynamics. Interestingly, we observe an PNPNPNPN... conformational arrangement (P = nearly planar and N = nonplanar) for the conformations of the linkers in a row along the pore direction in the MOF. We identified correlated linker dynamics emerging among the direct and next nearest neighboring linkers along the pore. Due to 180° rotational flips of the planar linkers along the pore, (1) a change in the width of librations in their direct neighbors (PN) is observed; (2) intriguingly, their next nearest planar neighbors (PP) rotate between 0° and ±180° to reattain aligned (0°, 0°) or (±180°, ±180°) conformations. The presence of correlated dynamics in such linkers over long-length scales occurring at nanoseconds time scales is desirable for applications like ferroelectric switching or diffusion control via geared linker rotation, and this work provides insight into the design for such applications.

In this paper we aim to construct infinite dimensional versions of well established Piecewise Deterministic Monte Carlo methods, such as the Bouncy Particle Sampler, the Zig-Zag Sampler and the Boomerang Sampler. In order to do so we provide an abstract infinite dimensional framework for Piecewise Deterministic Markov Processes (PDMPs) with unbounded event intensities. We further develop exponential convergence to equilibrium of the infinite dimensional Boomerang Sampler, using hypocoercivity techniques. Furthermore we establish how the infinite dimensional Boomerang Sampler admits a finite dimensional approximation, rendering it suitable for computer simulation.

We construct a new class of efficient Monte Carlo methods based on continuous-time piecewise deterministic Markov processes (PDMPs) suitable for inference in high dimensional sparse models, i.e. models for which there is prior knowledge that many coordinates are likely to be exactly 0. This is achieved with the fairly simple idea of endowing existing PDMP samplers with “sticky” coordinate axes, coordinate planes etc. Upon hitting those subspaces, an event is triggered during which the process sticks to the subspace, this way spending some time in a sub-model. This results in non-reversible jumps between different (sub-)models. While we show that PDMP samplers in general can be made sticky, we mainly focus on the Zig-Zag sampler. Compared to the Gibbs sampler for variable selection, we heuristically derive favourable dependence of the Sticky Zig-Zag sampler on dimension and data size. The computational efficiency of the Sticky Zig-Zag sampler is further established through numerical experiments where both the sample size and the dimension of the parameter space are large.

Piecewise deterministic Markov processes are an important new tool in the design of Markov chain Monte Carlo algorithms. Two examples of fundamental importance are the bouncy particle sampler (BPS) and the zig–zag process (ZZ). In this paper scaling limits for both algorithms are determined. Here the dimensionality of the space tends towards infinity and the target distribution is the multivariate standard normal distribution. For several quantities of interest (angular momentum, first coordinate and negative log-density) the scaling limits show qualitatively very different and rich behaviour. Based on these scaling limits the performance of the two algorithms in high dimensions can be compared. Although for angular momentum both processes require only a computational effort of O(d) to obtain approximately independent samples, the computational effort for negative log-density and first coordinate differ: for these BPS requires O(d²) computational effort whereas ZZ requires O(d). Finally we provide a criterion for the choice of the refreshment rate of BPS.

The zigzag process is a variant of the telegraph process with position dependent switching intensities. A characterization of the L2-spectrum for the generator of the one-dimensional zigzag process is obtained in the case where the marginal stationary distribution on R is unimodal and the refreshment intensity is zero. Sufficient conditions are obtained for a spectral mapping theorem, mapping the spectrum of the generator to the spectrum of the corresponding Markov semigroup. Furthermore results are obtained for symmetric stationary distributions and for perturbations of the spectrum, in particular for the case of a non-zero refreshment intensity. In the examples we consider (including a Gaussian target distribution) a slight increase of the refreshment intensity above zero results in a larger L2-spectral gap, corresponding to an improved convergence in L2.

The Bouncy Particle sampler (BPS) and the Zig-Zag sampler (ZZS) are continuous time, non-reversible Monte Carlo methods based on piecewise deterministic Markov processes. Experiments show that the speed of convergence of these samplers can be affected by the shape of the target distribution, as for instance in the case of anisotropic targets. We propose an adaptive scheme that iteratively learns all or part of the covariance matrix of the target and takes advantage of the obtained information to modify the underlying process with the aim of increasing the speed of convergence. Moreover, we define an adaptive scheme that automatically tunes the refreshment rate of the BPS or ZZS. We prove ergodicity and a law of large numbers for all the proposed adaptive algorithms. Finally, we show the benefits of the adaptive samplers with several numerical simulations.

Piecewise deterministic Markov processes (PDMPs) are a class of stochastic processes with applications in several fields of applied mathematics spanning from mathematical modelling of physical phenomena to computational methods. A PDMP is specified by three characteristic quantities: the deterministic motion, the law of the random event times, and the jump kernels. The applicability of PDMPs to real world scenarios is currently limited by the fact that these processes can be simulated only when these three characteristics of the process can be simulated exactly. In order to overcome this problem, we introduce discretisation schemes for PDMPs which make their approximate simulation possible. In particular, we design both first order and higher order schemes that rely on approximations of one or more of the three characteristics. For the proposed approximation schemes we study both pathwise convergence to the continuous PDMP as the step size converges to zero and convergence in law to the invariant measure of the PDMP in the long time limit. Moreover, we apply our theoretical results to several PDMPs that arise from the computational statistics and mathematical biology literature.

We introduce the use of the Zig-Zag sampler to the problem of sampling conditional diffusion processes (diffusion bridges). The Zig-Zag sampler is a rejection-free sampling scheme based on a non-reversible continuous piecewise deterministic Markov process. Similar to the Lévy–Ciesielski construction of a Brownian motion, we expand the diffusion path in a truncated Faber–Schauder basis. The coefficients within the basis are sampled using a Zig-Zag sampler. A key innovation is the use of the fully local algorithm for the Zig-Zag sampler that allows to exploit the sparsity structure implied by the dependency graph of the coefficients and by the subsampling technique to reduce the complexity of the algorithm. We illustrate the performance of the proposed methods in a number of examples.

The zig-zag process is a piecewise deterministic Markov process in position and velocity space. The process can be designed to have an arbitrary Gibbs type marginal probability density for its position coordinate, which makes it suitable for Monte Carlo simulation of continuous probability distributions. An important question in assessing the efficiency of this method is how fast the empirical measure converges to the stationary distribution of the process. In this paper we provide a partial answer to this question by characterizing the large deviations of the empirical measure from the stationary distribution. Based on the Feng–Kurtz approach, we develop an abstract framework aimed at encompassing piecewise deterministic Markov processes in position-velocity space. We derive explicit conditions for the zig-zag process to allow the Donsker–Varadhan variational formulation of the rate function, both for a compact setting (the torus) and one-dimensional Euclidean space. Finally we derive an explicit expression for the Donsker–Varadhan functional for the case of a compact state space and use this form of the rate function to address a key question concerning the optimal choice of the switching rate of the zig-zag process.

Suppose X is a multidimensional diffusion process. Assume that at time zero the state of X is fully observed, but at time 0$ ]]> only linear combinations of its components are observed. That is, one only observes the vector for a given matrix L. In this paper we show how samples from the conditioned process can be generated. The main contribution of this paper is to prove that guided proposals, introduced in [35], can be used in a unified way for both uniformly elliptic and hypo-elliptic diffusions, even when L is not the identity matrix. This is illustrated by excellent performance in two challenging cases: a partially observed twice-integrated diffusion with multiple wells and the partially observed FitzHugh-Nagumo model.

Piecewise Deterministic Monte Carlo algorithms enable simulation from a posterior distribution, whilst only needing to access a sub-sample of data at each iteration. We show how they can be implemented in settings where the parameters live on a restricted domain.

Recently, there have been conceptually new developments in Monte Carlo methods through the introduction of new MCMC and sequential Monte Carlo (SMC) algorithms which are based on continuous-time, rather than discrete-time, Markov processes. This has led to some fundamentally new Monte Carlo algorithms which can be used to sample from, say, a posterior distribution. Interestingly, continuous-time algorithms seem particularly well suited to Bayesian analysis in big-data settings as they need only access a small sub-set of data points at each iteration, and yet are still guaranteed to target the true posterior distribution. Whilst continuous-time MCMC and SMC methods have been developed independently we show here that they are related by the fact that both involve simulating a piecewise deterministic Markov process. Furthermore, we show that the methods developed to date are just specific cases of a potentially much wider class of continuous-time Monte Carlo algorithms.We give an informal introduction to piecewise deterministic Markov processes, covering the aspects relevant to these new Monte Carlo algorithms, with a view to making the development of new continuoustime Monte Carlo more accessible. We focus on how and why sub-sampling ideas can be used with these algorithms, and aim to give insight into how these new algorithms can be implemented, and what are some of the issues that affect their efficiency.

In Turitsyn, Chertkov and Vucelja [Phys. D 240 (2011) 410-414] a nonreversible Markov Chain Monte Carlo (MCMC) method on an augmented state space was introduced, here referred to as Lifted Metropolis-Hastings (LMH). A scaling limit of the magnetization process in the Curie-Weiss model is derived for LMH, as well as for Metropolis-Hastings (MH). The required jump rate in the high (supercritical) temperature regime equals n^1/2 for LMH, which should be compared to n for MH. At the critical temperature, the required jump rate equals n^3/4 for LMH and n^3/2 for MH, in agreement with experimental results of Turitsyn, Chertkov and Vucelja (2011). The scaling limit of LMH turns out to be a nonreversible piecewise deterministic exponentially ergodic "zig-zag" Markov process.

Markov chain Monte Carlo (MCMC) methods provide an essential tool in statistics for sampling from complex probability distributions. While the standard approach to MCMC involves constructing discrete-time reversible Markov chains whose transition kernel is obtained via the Metropolis-Hastings algorithm, there has been recent interest in alternative schemes based on piecewise deterministic Markov processes (PDMPs). One such approach is based on the zig-zag process, introduced in Bierkens and Roberts (2016), which proved to provide a highly scalable sampling scheme for sampling in the big data regime; see Bierkens et al. (2016). In this paper we study the performance of the zig-zag sampler, focusing on the one-dimensional case. In particular, we identify conditions under which a central limit theorem holds and characterise the asymptotic variance. Moreover, we study the influence of the switching rate on the diffusivity of the zig-zag process by identifying a diffusion limit as the switching rate tends to. Based on our results we compare the performance of the zig-zag sampler to existing Monte Carlo methods, both analytically and through simulations.