Efficient Low-Order Approximation of First-Passage Time Distributions

Schnoerr, David; Cseke, Botond; Grima, Ramon; Sanguinetti, Guido

doi:10.1103/physrevlett.119.210601

Cited by 19 publications

(24 citation statements)

References 45 publications

(78 reference statements)

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“…Building upon [16,17] Schnoerr et al [33] have recently formulated timebounded reachability as a Bayesian inference problem. Using this formulation, they proposed a method where the entire distribution of first-passage times can be approximated by taking advantage of some well-established methodologies in the Bayesian inference and statistical physics literature.…”

Section: Time-bounded Reachability As Bayesian Inferencementioning

confidence: 99%

See 1 more Smart Citation

Probabilistic Model Checking for Continuous-Time Markov Chains via Sequential Bayesian Inference

Milios

Sanguinetti

Schnoerr

2018

Quantitative Evaluation of Systems

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Probabilistic model checking for systems with large or unbounded state space is a challenging computational problem in formal modelling and its applications. Numerical algorithms require an explicit representation of the state space, while statistical approaches require a large number of samples to estimate the desired properties with high confidence. Here, we show how model checking of time-bounded path properties can be recast exactly as a Bayesian inference problem. In this novel formulation the problem can be efficiently approximated using techniques from machine learning. Our approach is inspired by a recent result in statistical physics which derived closed form differential equations for the first-passage time distribution of stochastic processes. We show on a number of non-trivial case studies that our method achieves both high accuracy and significant computational gains compared to statistical model checking.

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Section: Time-bounded Reachability As Bayesian Inferencementioning

confidence: 99%

“…In this paper, we propose an alternative approach to solving the probabilistic model checking problem which draws on a recently proposed technique from statistical physics [33]. We show that the model checking problem is equivalent to a sequential Bayesian computation of the marginal likelihood of an auxiliary observation process.…”

Section: Introductionmentioning

confidence: 99%

Probabilistic Model Checking for Continuous-Time Markov Chains via Sequential Bayesian Inference

Milios

Sanguinetti

Schnoerr

2018

Quantitative Evaluation of Systems

Self Cite

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“…The only approach dealing with more general subsets, [19], imposes restrictions on the behaviour of the mean-field approximation, whose trajectory has to enter the reachability region in a finite time. Another interesting approach has been developed in [47,42], where model checking of time-bounded properties for CTMCs is expressed as a Bayesian inference problem, and approximated model checking algorithms are derived. However, no guarantees on the convergence of the resulting algorithms is given.…”

Section: Introductionmentioning

confidence: 99%

Central Limit Model Checking

Bortolussi

Cardelli

Kwiatkowska

et al. 2019

ACM Trans. Comput. Logic

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We consider probabilistic model checking for continuoustime Markov chains (CTMCs) induced from Stochastic Reaction Networks (SRNs) against a fragment of Continuous Stochastic Logic (CSL) extended with reward operators. Classical numerical algorithms for CSL model checking based on uniformisation are limited to finite CTMCs and suffer from exponential growth of the state space with respect to the number of species. On the other hand, approximate techniques such as mean-field approximations and simulations combined with statistical inference are more scalable, but can be time consuming and do not support the full expressiveness of CSL. In this paper we employ a continuousspace approximation of the CTMC in terms of a Gaussian process based on the Central Limit Approximation (CLA), also known as the Linear Noise Approximation (LNA), whose solution requires solving a number of differential equations that is quadratic in the number of species and independent of the population size. We then develop efficient and scalable approximate model checking algorithms on the resulting Gaussian process, where we restrict the target regions for probabilistic reachability to convex polytopes. This allows us to derive an abstraction in terms of a time-inhomogeneous discrete-time Markov chain (DTMC), whose dimension is independent of the number of species, on which model checking is performed. Using results from probability theory, we prove the convergence in distribution of our algorithms to the corresponding measures on the original CTMC. We implement the techniques and, on a set of examples, demonstrate that they allow us to overcome the state space explosion problem, while still correctly characterizing the stochastic behaviour of the system. Our methods can be used for formal analysis of a wide range of distributed stochastic systems, including biochemical systems, sensor networks and population protocols.

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“…There exists a rich literature on exit times, especially in physics and biomathematics [31,41,49]. Recently, there has been renewed interest in exit times of continuous-time Markov chains with discrete state space [2,23,55,51], such as those we study in this paper. While the exit problem from a small finite domain is tractable [25,26], the exit problem from an infinite or large domain can only be solved in special cases [17,23,47].…”

mentioning

confidence: 99%

“…While the exit problem from a small finite domain is tractable [25,26], the exit problem from an infinite or large domain can only be solved in special cases [17,23,47]. Various approximation schemes have been developed to address this issue [3,12,29,51]. However, most of them do not provide bounds or error estimates on their accuracy.…”

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confidence: 99%

The Exit Time Finite State Projection Scheme: Bounding Exit Distributions and Occupation Measures of Continuous-Time Markov Chains

Kuntz¹,

Thomas²,

Stan³

et al. 2019

SIAM J. Sci. Comput.

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We introduce the exit time finite state projection (ETFSP) scheme, a truncationbased method that yields approximations to the exit distribution and occupation measure associated with the time of exit from a domain (i.e., the time of first passage to the complement of the domain) of time-homogeneous continuous-time Markov chains. We prove that: (i) the computed approximations bound the measures from below; (ii) the total variation distances between the approximations and the measures decrease monotonically as states are added to the truncation; and (iii) the scheme converges, in the sense that, as the truncation tends to the entire state space, the total variation distances tend to zero. Furthermore, we give a computable bound on the total variation distance between the exit distribution and its approximation, and we delineate the cases in which the bound is sharp. We also revisit the related finite state projection scheme and give a comprehensive account of its theoretical properties. We demonstrate the use of the ETFSP scheme by applying it to two biological examples: the computation of the first passage time associated with the expression of a gene, and the fixation times of competing species subject to demographic noise.

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Efficient Low-Order Approximation of First-Passage Time Distributions

Cited by 19 publications

References 45 publications

Probabilistic Model Checking for Continuous-Time Markov Chains via Sequential Bayesian Inference

Probabilistic Model Checking for Continuous-Time Markov Chains via Sequential Bayesian Inference

Central Limit Model Checking

The Exit Time Finite State Projection Scheme: Bounding Exit Distributions and Occupation Measures of Continuous-Time Markov Chains

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