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Most natural and engineered information-processing systems transmitinformation via signals that vary in time. Computing the informationtransmission rate or the information encoded in the temporal characteristics ofthese signals, requires the mutual information between the input and outputsignals as a function of time, i.e. between the input and output trajectories.Yet, this is notoriously difficult because of the high-dimensional nature ofthe trajectory space, and all existing techniques require approximations. Wepresent an exact Monte Carlo technique called Path Weight Sampling (PWS) that,for the first time, makes it possible to compute the mutual information betweeninput and output trajectories for any stochastic system that is described by amaster equation. The principal idea is to use the master equation to evaluatethe exact conditional probability of an individual output trajectory for agiven input trajectory, and average this via Monte Carlo sampling in trajectoryspace to obtain the mutual information. We present three variants of PWS, whichall generate the trajectories using the standard stochastic simulationalgorithm. While Direct PWS is a brute-force method, Rosenbluth-Rosenbluth PWSexploits the analogy between signal trajectory sampling and polymer sampling,and Thermodynamic Integration PWS is based on a reversible work calculation intrajectory space. PWS also makes it possible to compute the mutual informationbetween input and output trajectories for systems with hidden internal statesas well as systems with feedback from output to input. Applying PWS to thebacterial chemotaxis system, consisting of 182 coupled chemical reactions,demonstrates not only that the scheme is highly efficient, but also that thenumber of receptor clusters is much smaller than hitherto believed, while theirsize is much larger.

Path Weight Sampling: Exact Monte Carlo Computation of the Mutual Information between Stochastic Trajectories