Probabilistic verification of partially observable dynamical systems

Abstract

The construction and formal verification of dynamical models is important in engineering, biology and other disciplines. We focus on non-linear models containing a set of parameters governing their dynamics. The value of these parameters is often unknown and not directly observable through measurements, which are themselves noisy. When treating parameters as random variables, one can constrain their distribution by conditioning on observations and thereby constructing a posterior probability distribution. We aim to perform model verification with respect to this posterior. The main difficulty in performing verification on a model under the posterior distribution is that in general, it is difficult to obtain \emph{independent} samples from the posterior, especially for non-linear dynamical models. Standard statistical model checking methods require independent realizations of the system and are therefore not applicable in this context. We propose a Markov chain Monte Carlo based statistical model checking framework, which produces a sequence of dependent random realizations of the model dynamics over the parameter posterior. Using this sequence of samples, we use statistical hypothesis tests to verify whether the model satisfies a bounded temporal logic property with a certain probability. We use sample size bounds tailored to the setting of dependent samples for fixed sample size and sequential tests. We apply our method to a case-study from the domain of systems biology, to a model of the JAK-STAT biochemical pathway. The pathway is modeled as a system of non-linear ODEs containing a set of unknown parameters. Noisy, indirect observations of the system state are available from an experiment. The results show that the proposed method enables probabilistic verification with respect to the parameter posterior with specified error bounds.