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Dynamic Bayesian networks (DBNs) are a general model for stochastic processes with partially observed states. Belief filtering in DBNs is the task of inferring the belief state (i.e. the probability distribution over process states) based on incomplete and noisy observations. This can be a hard problem in complex processes with large state spaces. In this article, we explore the idea of accelerating the filtering task by automatically exploiting causality in the process. We consider a specific type of causal relation, called passivity, which pertains to how state variables cause changes in other variables. We present the Passivity-based Selective Belief Filtering (PSBF) method, which maintains a factored belief representation and exploits passivity to perform selective updates over the belief factors. PSBF produces exact belief states under certain assumptions and approximate belief states otherwise, where the approximation error is bounded by the degree of uncertainty in the process. We show empirically, in synthetic processes with varying sizes and degrees of passivity, that PSBF is faster than several alternative methods while achieving competitive accuracy. Furthermore, we demonstrate how passivity occurs naturally in a complex system such as a multi-robot warehouse, and how PSBF can exploit this to accelerate the filtering task.