Inverse sensitivity analysis of mathematical models avoiding the curse of dimensionality

1AbstractBiological systems have evolved a degree of robustness with respect to perturbations in their environment and this capability is essential for their survival. In applications ranging from therapeutics to conservation, it is important to understand not only the sensitivity of biological systems to changes in their environments, but which features of these systems are necessary to achieve a given outcome. Mathematical models are increasingly employed to understand these mechanisms. Sensitivity analyses of such mathematical models provide insight into the responsiveness of the system when experimental manipulation is difficult. One common approach is to seek the probability distribution of the outputs of the system corresponding to a known distribution of inputs. By contrast, inverse sensitivity analysis determines the probability distribution of model inputs which produces a known distribution of outputs. The computational complexity of the methods used to conduct inverse sensitivity analyses for deterministic systems has limited their application to models with relatively few parameters. Here we describe a novel Markov Chain Monte Carlo method we call “Contour Monte Carlo”, which can be used to invert systems with a large number of parameters. We demonstrate the utility of this method by inverting a range of frequently-used deterministic models of biological systems, including the logistic growth equation, the Michaelis-Menten equation, and an SIR model of disease transmission with nine input parameters. We argue that the simplicity of our approach means it is amenable to a large class of problems of practical significance and, more generally, provides a probabilistic framework for understanding the inversion of deterministic models.2Author summaryMathematical models of complex systems are constructed to provide insight into their underlying functioning. Statistical inversion can probe the often unobserved processes underlying biological systems, by proceeding from a given distribution of a model’s outputs (the aggregate “effects”) to a distribution over input parameters (the constituent “causes”). The process of inversion is well-defined for systems involving randomness and can be described by Bayesian inference. The inversion of a deterministic system, however, cannot be performed by the standard Bayesian approach. We develop a conceptual framework that describes the inversion of deterministic systems with fewer outputs than input parameters. Like Bayesian inference, our approach uses probability distributions to describe the uncertainty over inputs and outputs, and requires a prior input distribution to ensure a unique “posterior” probability distribution over inputs. We describe a computational Monte Carlo method that allows efficient sampling from the posterior distribution even as the dimension of the input parameter space grows. This is a two-step process where we first estimate a “contour volume density” associated with each output value which is then used to define a sampling algorithm that yields the requisite input distribution asymptotically. Our approach is simple, broadly applicable and could be widely adopted.