Approximations of Piecewise Deterministic Markov Processes and their convergence properties

Bertazzi, Andrea; Bierkens, Joris; Dobson, Paul

doi:10.1016/j.spa.2022.09.004

Cited by 7 publications

(11 citation statements)

References 29 publications

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“…Comparison to related works. The work in this paper can be seen as a continuation of the work that two of the authors started with their coauthors in [3], in which a general framework to approximate PDMPs is introduced and studied. In this previous work, the focus is not a specific scheme and thus the results are mostly general and not tailored for particular processes or schemes, though the ZZS and BPS are considered as recurrent examples.…”

Section: Introductionmentioning

confidence: 98%

“…Similarly to rejection sampling which requires a good reference measure, an efficient Poisson thinning algorithm requires the knowledge of good bounds for the jump rate λ along the trajectory of the ODE. In this work, we 1 Delft Institute of Applied Mathematics, TU Delft 2 University of Edinburgh 3 Sorbonne Université E-mail addresses: a.bertazzi@tudelft.nl, pdobson@ed.ac.uk, pierre.monmarche@sorbonne-universite.fr. Date: January 9, 2023. focus on the case in which such bounds are not available, or are so crude that thinning would not be numerically efficient.…”

Section: Introductionmentioning

confidence: 99%

“…In that case, the random event times have to be approximated even if the ODE can be solved exactly. This question has recently been addressed in [3,38,14] with three different schemes. Here, rather than designing an ad hoc numerical schemes, we work in the general framework of splitting schemes, which are widely used for e.g.…”

Section: Introductionmentioning

confidence: 99%

“…In this previous work, the focus is not a specific scheme and thus the results are mostly general and not tailored for particular processes or schemes, though the ZZS and BPS are considered as recurrent examples. In particular, the schemes introduced in [3] leave considerable freedom to the user in the choice of some crucial components of the algorithm, namely an approximation of the switching rates or a numerical integrator in place of the exact flow map. On the other hand, in this paper we follow the philosophy of splitting schemes to describe a simple recipe to approximate PDMPs.…”

Section: Introductionmentioning

confidence: 99%

“…On the other hand, in this paper we follow the philosophy of splitting schemes to describe a simple recipe to approximate PDMPs. Note that splitting schemes are not considered in [3]. The main advantage of splitting schemes is the second order of accuracy with one gradient evaluation per iteration, whereas second order algorithms considered in [3] relied on approximations of second order of the switching rates, which can be usually obtained with the expensive computation of the Hessian of the negative log-target.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Splitting schemes for second order approximations of piecewise-deterministic Markov processes

Bertazzi¹,

Dobson²,

Monmarché³

2023

Preprint

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Numerical approximations of piecewise-deterministic Markov processes based on splitting schemes are introduced, together with their Metropolis-adjusted versions. The unadjusted schemes are shown to have a weak error of order two in the step size in a general framework. Focusing then on unadjusted schemes based on the Bouncy Particle and Zig-Zag samplers, we provide conditions ensuring ergodicity and consider the expansion of the invariant measure in terms of the step size. The dependency of the leading term in this expansion in terms of the refreshment rate, depending of the splitting order, is analyzed. Finally, we present numerical experiments on Gaussian targets, a Bayesian imaging inverse problem and a system of interacting particles.

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Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Splitting schemes for second order approximations of piecewise-deterministic Markov processes

Bertazzi¹,

Dobson²,

Monmarché³

2023

Preprint

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Automatic Zig-Zag sampling in practice

2022

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Novel Monte Carlo methods to generate samples from a target distribution, such as a posterior from a Bayesian analysis, have rapidly expanded in the past decade. Algorithms based on Piecewise Deterministic Markov Processes (PDMPs), non-reversible continuous-time processes, are developing into their own research branch, thanks their important properties (e.g., super-efficiency). Nevertheless, practice has not caught up with the theory in this field, and the use of PDMPs to solve applied problems is not widespread. This might be due, firstly, to several implementational challenges that PDMP-based samplers present with and, secondly, to the lack of papers that showcase the methods and implementations in applied settings. Here, we address both these issues using one of the most promising PDMPs, the Zig-Zag sampler, as an archetypal example. After an explanation of the key elements of the Zig-Zag sampler, its implementation challenges are exposed and addressed. Specifically, the formulation of an algorithm that draws samples from a target distribution of interest is provided. Notably, the only requirement of the algorithm is a closed-form differentiable function to evaluate the log-target density of interest, and, unlike previous implementations, no further information on the target is needed. The performance of the algorithm is evaluated against canonical Hamiltonian Monte Carlo, and it is proven to be competitive, in simulation and real-data settings. Lastly, we demonstrate that the super-efficiency property, i.e. the ability to draw one independent sample at a lesser cost than evaluating the likelihood of all the data, can be obtained in practice.

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NuZZ: Numerical Zig-Zag for general models

Pagani,

Chevallier,

Power

et al. 2024

Stat Comput

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Markov chain Monte Carlo (MCMC) is a key algorithm in computational statistics, and as datasets grow larger and models grow more complex, many popular MCMC algorithms become too computationally expensive to be practical. Recent progress has been made on this problem through development of MCMC algorithms based on Piecewise Deterministic Markov Processes (PDMPs), irreversible processes which can be engineered to converge at a rate which is independent of the size of the dataset. While there has understandably been a surge of theoretical studies following these results, PDMPs have so far only been implemented for models where certain gradients can be bounded in closed form, which is not possible in many relevant statistical problems. Furthermore, there has been substantionally less focus on practical implementation, or the efficiency of PDMP dynamics in exploring challenging densities. Focusing on the Zig-Zag process, we present the Numerical Zig-Zag (NuZZ) algorithm, which is applicable to general statistical models without the need for bounds on the gradient of the log posterior. This allows us to perform numerical experiments on: (i) how the Zig-Zag dynamics behaves on some test problems with common challenging features; and (ii) how the error between the target and sampled distributions evolves as a function of computational effort for different MCMC algorithms including NuZZ. Moreover, due to the specifics of the NuZZ algorithms, we are able to give an explicit bound on the Wasserstein distance between the exact posterior and its numerically perturbed counterpart in terms of the user-specified numerical tolerances of NuZZ.

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Approximations of Piecewise Deterministic Markov Processes and their convergence properties

Cited by 7 publications

References 29 publications

Splitting schemes for second order approximations of piecewise-deterministic Markov processes

Splitting schemes for second order approximations of piecewise-deterministic Markov processes

Automatic Zig-Zag sampling in practice

NuZZ: Numerical Zig-Zag for general models

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