Construction of Lyapunov functions for piecewise-deterministic Markov processes

Mesquita, Alexandre R.; Hespanha, João P.

doi:10.1109/cdc.2010.5717973

Cited by 4 publications

(3 citation statements)

References 23 publications

(33 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In particular, non-reversible MCMC algorithms based on piecewise deterministic Markov processes [9,10] have recently emerged in applied probability [4,11,13,26], automatic control [23,24], physics [20,22,29] and statistics [3,5,32,33,34]. These algorithms perform well empirically so they have already found many applications; see, e.g., [11,20,15,28].…”

Section: Introductionmentioning

confidence: 99%

“…These algorithms perform well empirically so they have already found many applications; see, e.g., [11,20,15,28]. However, to the best of our knowledge, quantitative convergence rates for this class of MCMC algorithms have only been established under stringent assumptions: [23] establishes geometric ergodicity of such a scheme but only for targets with exponentially decaying tails, [26] obtains sharp results but requires the state-space to be compact, while [2,4,13] consider targets on the real line. Similar restrictions apply to limit theorems for ergodic averages, where for example in [2], a Central Limit Theorem (CLT) has been obtained but this result is restricted to targets on the real line.…”

Section: Introductionmentioning

confidence: 99%

“…In[23], the Lyapunov function e U (x)/2 λ(x, v) 1/2 is used to establish the geometric ergodicity of a different piecewise deterministic MCMC scheme for targets with exponential tails but we found this function did not apply to BPS.…”

mentioning

confidence: 99%

See 2 more Smart Citations

Exponential ergodicity of the bouncy particle sampler

Deligiannidis¹,

Bouchard‐Côté²,

Doucet³

2019

Ann. Statist.

View full text Add to dashboard Cite

Non-reversible Markov chain Monte Carlo schemes based on piecewise deterministic Markov processes have been recently introduced in applied probability, automatic control, physics and statistics. Although these algorithms demonstrate experimentally good performance and are accordingly increasingly used in a wide range of applications, geometric ergodicity results for such schemes have only been established so far under very restrictive assumptions. We give here verifiable conditions on the target distribution under which the Bouncy Particle Sampler algorithm introduced in [29] is geometrically ergodic. This holds whenever the target satisfies a curvature condition and has tails decaying at least as fast as an exponential and at most as fast as a Gaussian distribution. This allows us to provide a central limit theorem for the associated ergodic averages. When the target has tails thinner than a Gaussian distribution, we propose an original modification of this scheme that is geometrically ergodic. For thick-tailed target distributions, such as t-distributions, we extend the idea pioneered in [19] in a random walk Metropolis context. We apply a change of variable to obtain a transformed target satisfying the tail conditions for geometric ergodicity. By sampling the transformed target using the Bouncy Particle Sampler and mapping back the Markov process to the original parameterization, we obtain a geometrically ergodic algorithm.

show abstract

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Exponential ergodicity of the bouncy particle sampler

Deligiannidis¹,

Bouchard‐Côté²,

Doucet³

2019

Ann. Statist.

View full text Add to dashboard Cite

show abstract

Stability analysis for stochastic hybrid systems: A survey

2014

View full text Add to dashboard Cite