The variational principles called maximum entropy (MaxEnt) and maximum caliber (MaxCal) are reviewed. MaxEnt originated in the statistical physics of Boltzmann and Gibbs, as a theoretical tool for predicting the equilibrium states of thermal systems. Later, entropy maximization was also applied to matters of information, signal transmission, and image reconstruction. Recently, since the work of Shore and Johnson, MaxEnt has been regarded as a principle that is broader than either physics or information alone. MaxEnt is a procedure that ensures that inferences drawn from stochastic data satisfy basic self-consistency requirements. The different historical justifications for the entropy $S=-$\sum${}_{i}{p}_{i}\mathrm{log}{}{p}_{i}$ and its corresponding variational principles are reviewed. As an illustration of the broadening purview of maximum entropy principles, maximum caliber, which is path entropy maximization applied to the trajectories of dynamical systems, is also reviewed. Examples are given in which maximum caliber is used to interpret dynamical fluctuations in biology and on the nanoscale, in single-molecule and few-particle systems such as molecular motors, chemical reactions, biological feedback circuits, and diffusion in microfluidics devices.

Significance: Large and complex macromolecular assemblies—like RNA and DNA polymerases, the kinetochore, the ribosome, ATP synthase, and many others—are critical components of the cell. To fully characterize these molecular machines, it is not sufficient to rely solely on traditional structural methods, like cryo-EM and X-ray crystallography that provide great structural detail in vitro; we need experimental and theoretical tools that can describe the organization of these machines in their native cellular environment so as to better understand their function. Here we provide a strategy for extracting precisely this information directly from superresolution imaging data, a state-of-the-art technique for probing biological structures in living cells below the diffraction limit.

We review here Maximum Caliber (Max Cal), a general variational principle for inferring distributions of paths in dynamical processes and networks. Max Cal is to dynamical trajectories what the principle of maximum entropy is to equilibrium states or stationary populations. In Max Cal, you maximize a path entropy over all possible pathways, subject to dynamical constraints, in order to predict relative path weights. Many well-known relationships of non-equilibrium statistical physics—such as the Green-Kubo fluctuation-dissipation relations, Onsager’s reciprocal relations, and Prigogine’s minimum entropy production—are limited to near-equilibrium processes. Max Cal is more general. While it can readily derive these results under those limits, Max Cal is also applicable far from equilibrium. We give examples of Max Cal as a method of inference about trajectory distributions from limited data, finding reaction coordinates in bio-molecular simulations, and modeling the complex dynamics of non-thermal systems such as gene regulatory networks or the collective firing of neurons. We also survey its basis in principle and some limitations.