Maximum-Likelihood Estimation of the Discrete Coefficient of Determination in Stochastic Boolean Systems

The discrete Coefficient of Determination (CoD) has become a key component of inference methods for stochastic Boolean models. We develop a parametric maximum-likelihood (ML) method for the inference of the discrete CoD for static Boolean systems and for dynamical Boolean systems in the steady state. Using analytical and numerical approaches, we compare the performance of the parametric ML approach against that of common nonparametric alternatives for CoD estimation, which show that the parametric approach has the least bias, variance, and root mean-square (RMS) error, provided that the system noise level is not too high. Next we consider the application of the proposed estimation approach to the problem of system identification, where only partial knowledge about the system is available. Inference procedures are proposed for both the static and dynamical cases, and their performance in logic gate and wiring identification is assessed through numerical experiments. The results indicate that identification rates converge to 100% as sample size increases, and that the convergence rate is much faster as more prior knowledge is available. For wiring identification, the parametric ML approach is compared to the nonparametric approaches, and it produced superior identification rates, though as the amount of prior knowledge is reduced, its performance approaches that of the nonparametric ML estimator, which was generally the best nonparametric approach in our experiments.