(384c) Information Thermodynamics and Bayesian Optimization: Applications to Single Molecule Experiments

We present a fundamental analysis of the information content
for dynamics trajectories that is applicable to all systems governed by over-damped
Langevin dynamics.  The Fisher information metric for this
class of time propagators is presented and the trajectory entropy functional is
derived.  This subsequently permits
evaluation of the magnitude of descriptive features in the dynamic model parameterized
by the potential of mean force and diffusion coefficient along a characteristic
system coordinate(s).  

Armed with a measure of "entropy" for a systems equilibrium
trajectories, we are able to define a new manifold of information
thermodynamics to balance the inputs of "internal energy" or log-likelihood statistical
information from simulation and experiment with the desire to generate the
simplest models of system behavior that contain only statistically significant
features.  This formulation permits
the derivation of the least informative dynamic models subject to kinetic
constraints such as mean-first passage times.  Finally, we provide specific examples
from performing Bayesian optimization of model parameters from single-molecule
FRET experiments.  Exploring what is
the equivalent of a phase diagram for the operation of inverse mapping of data
to models, allows for an optimal choice of regularizing parameters for model
selection.

Figure 1.  Functional approximation of the
log-likelihood surface for a multi-dimensional parameter set θ which is a
composite of the potential of mean force V(x) and the
diffusion constant D.  The dotted lines are hyper-planes in the
functional space generated by projecting derivatives of the
log-likelihood.  Red lines are the
convex hull created by the set of these upper bounds, and blue line is the true
experimental/simulation likelihood as a parameterized by θ.