Statistical Model Checking : An Overview

Quantitative properties of stochastic systems are usually specified in logics that allow one to compare the measure of executions satisfying certain temporal properties with thresholds. The model checking problem for stochastic systems with respect to such logics is typically solved by a numerical approach that iteratively computes (or approximates) the exact measure of paths satisfying relevant subformulas; the algorithms themselves depend on the class of systems being analyzed as well as the logic used for specifying the properties. Another approach to solve the model checking problem is to \emph{simulate} the system for finitely many runs, and use \emph{hypothesis testing} to infer whether the samples provide a \emph{statistical} evidence for the satisfaction or violation of the specification. In this short paper, we survey the statistical approach, and outline its main advantages in terms of efficiency, uniformity, and simplicity.Comment: non

Techniques for automated parameter estimation in computational models of probabilistic systems

The main contribution of this dissertation is the design of two new algorithms for automatically synthesizing values of numerical parameters of computational models of complex stochastic systems such that the resultant model meets user-specified behavioral specifications. These algorithms are designed to operate on probabilistic systems – systems that, in general, behave differently under identical conditions. The algorithms work using an approach that combines formal verification and mathematical optimization to explore a model\u27s parameter space. The problem of determining whether a model instantiated with a given set of parameter values satisfies the desired specification is first defined using formal verification terminology, and then reformulated in terms of statistical hypothesis testing. Parameter space exploration involves determining the outcome of the hypothesis testing query for each parameter point and is guided using simulated annealing. The first algorithm uses the sequential probability ratio test (SPRT) to solve the hypothesis testing problems, whereas the second algorithm uses an approach based on Bayesian statistical model checking (BSMC). The SPRT-based parameter synthesis algorithm was used to validate that a given model of glucose-insulin metabolism has the capability of representing diabetic behavior by synthesizing values of three parameters that ensure that the glucose-insulin subsystem spends at least 20 minutes in a diabetic scenario. The BSMC-based algorithm was used to discover the values of parameters in a physiological model of the acute inflammatory response that guarantee a set of desired clinical outcomes. These two applications demonstrate how our algorithms use formal verification, statistical hypothesis testing and mathematical optimization to automatically synthesize parameters of complex probabilistic models in order to meet user-specified behavioral propertie

Bayesian Statistical Model-Checking of Continuous Stochastic Logic

Master of ScienceDepartment of StatisticsChristopher VahlAutonomous systems are transforming the society by enabling sophisticated technologies such as robotic surgery and driverless cars. On one hand, increased automation through removal of the human-in-the-loop promises enhanced efficiency, while, on the other hand, the highly uncertain and safety critical environments, such as, varying weather and road conditions, and the presence of pedestrians on the road, pose challenge to the design of reliable autonomous systems. Hence, there is an immediate need for a robust framework for certifying the correctness of autonomous systems. In this report, we explore verifying the correctness of uncertain autonomous systems modeled as discrete-time Markov chains (DTMCs) against correctness criteria provided as continuous stochastic logic (CSL) formulae. Statistical model-checking (SMC) is a paradigm for verification based on formulating the verification problem as a hypothesis testing prob- lem. We propose a novel statistical model-checking algorithm based on Bayesian hypothesis testing. While Bayesian approaches for simpler logics without nested probabilistic operators and Frequentist approaches for nested logic have been previously explored, the Bayesian ap- proach for CSL that has nested probabilistic operators has not been addressed. The challenge in the nested case arises from the fact that unlike in probabilistic model-checking (PMC), where we obtain a definitive answer for the model-checking problem for the sub-formulae, we only obtain a correct answer with a certain confidence, which needs to be factored into the recursive SMC algorithm. We have implemented our algorithm in a Python Toolbox, and present our evaluation on some benchmark examples. We observe that while both the Bayesian and frequentist SMC perform well in terms of inference, Bayesian SMC is more efficient in terms of the number of samples. On several examples, it even beats the state-of- the-art probabilistic model-checker PRISM

Алгоритм перевірки на коректність моделі сплайнової регресії

Побудовано алгоритм перевірки на коректність моделі двофазної лінійної регресії з невідомою точкою перемикання у випадку, коли треба зробити вибір між такою моделлю та лінійною. Алгоритм заснований на загальних принципах перевірки статистичних гіпотез у регресійному аналізі.The algorithm of checking for correctness of two-phase regression model with unknown switch point is constructed in the case when it is necessary to do a choice between such model and linear. The algorithm is based on the general principles of statistical hypothesis testing in regression analysis

Posterior predictive checking for gravitational-wave detection with pulsar timing arrays: I. The optimal statistic

A gravitational-wave background can be detected in pulsar-timing-array data as Hellings--Downs correlations among the timing residuals measured for different pulsars. The optimal statistic implements this concept as a classical null-hypothesis statistical test: a null model with no correlations can be rejected if the observed value of the statistic is very unlikely under that model. To address the dependence of the statistic on the uncertain pulsar noise parameters, the pulsar-timing-array community has adopted a hybrid classical--Bayesian scheme (Vigeland et al. 2018) in which the posterior distribution of the noise parameters induces a posterior distribution for the statistic. In this article we propose a rigorous interpretation of the hybrid scheme as an instance of posterior predictive checking, and we introduce a new summary statistic (the Bayesian signal-to-noise ratio) that should be used to accurately quantify the statistical significance of an observation instead of the mean posterior signal-to-noise ratio, which does not support such a direct interpretation. In addition to falsifying the no-correlation hypothesis, the Bayesian signal-to-noise ratio can also provide evidence supporting the presence of Hellings--Downs correlations. We demonstrate our proposal with simulated datasets based on NANOGrav's 12.5-yr data release. We also establish a relation between the posterior distribution of the statistic and the Bayes factor in favor of correlations, thus calibrating the Bayes factor in terms of hypothesis-testing significance.Comment: 12 pages, 8 figure

Exploring behaviors of stochastic differential equation models of biological systems using change of measures

Stochastic Differential Equations (SDE) are often used to model the stochastic dynamics of biological systems. Unfortunately, rare but biologically interesting behaviors (e. g., oncogenesis) can be difficult to observe in stochastic models. Consequently, the analysis of behaviors of SDE models using numerical simulations can be challenging. We introduce a method for solving the following problem: given a SDE model and a high-level behavioral specification about the dynamics of the model, algorithmically decide whether the model satisfies the specification. While there are a number of techniques for addressing this problem for discrete-state stochastic models, the analysis of SDE and other continuous-state models has received less attention. Our proposed solution uses a combination of Bayesian sequential hypothesis testing, non-identically distributed samples, and Girsanov\u27s theorem for change of measures to examine rare behaviors. We use our algorithm to analyze two SDE models of tumor dynamics. Our use of non-identically distributed samples sampling contributes to the state of the art in statistical verification and model checking of stochastic models by providing an effective means for exposing rare events in SDEs, while retaining the ability to compute bounds on the probability that those events occur

Exploring behaviors of stochastic differential equation models of biological systems using change of measures

Stochastic Differential Equations (SDE) are often used to model the stochastic dynamics of biological systems. Unfortunately, rare but biologically interesting behaviors (e.g., oncogenesis) can be difficult to observe in stochastic models. Consequently, the analysis of behaviors of SDE models using numerical simulations can be challenging. We introduce a method for solving the following problem: given a SDE model and a high-level behavioral specification about the dynamics of the model, algorithmically decide whether the model satisfies the specification. While there are a number of techniques for addressing this problem for discrete-state stochastic models, the analysis of SDE and other continuous-state models has received less attention. Our proposed solution uses a combination of Bayesian sequential hypothesis testing, non-identically distributed samples, and Girsanov's theorem for change of measures to examine rare behaviors. We use our algorithm to analyze two SDE models of tumor dynamics. Our use of non-identically distributed samples sampling contributes to the state of the art in statistical verification and model checking of stochastic models by providing an effective means for exposing rare events in SDEs, while retaining the ability to compute bounds on the probability that those events occur