  Schr\"odinger bridges have emerged as an enabling framework for unveiling the
stochastic dynamics of systems based on marginal observations at different
points in time. The terminology "bridge'' refers to a probability law that
suitably interpolates such marginals. The theory plays a pivotal role in a
variety of contemporary developments in machine learning, stochastic control,
thermodynamics, and biology, to name a few, impacting disciplines such as
single-cell genomics, meteorology, and robotics. In this work, we extend
Schr\"odinger's paradigm of bridges to account for integral constraints along
paths, in a way akin to Maximum Caliber - a Maximum Entropy principle applied
in a dynamic context. The Maximum Caliber principle has proven useful to infer
the dynamics of complex systems e.g., that model gene circuits and protein
folding. We unify these two problems via a maximum likelihood formulation to
reconcile stochastic dynamics with ensemble-path data. A variety of data types
can be encompassed, ranging from distribution moments to average currents along
paths. The framework enables inference of time-varying potential landscapes
that drive the process. The resulting forces can be interpreted as the optimal
control that drives the system in a way that abides by specified integral
constraints. Analogous results are presented in a discrete-time, discrete-space
setting and specialized to steady-state dynamics. We finish by illustrating the
practical applicability of the framework through paradigmatic examples, such as
that of bit erasure or protein folding. In doing so, we highlight the strengths
of the proposed framework, namely, the generality of the theory, the ease of
computation, and the ability to interpret results in terms of system dynamics.
This is in contrast to Maximum-Caliber problems where the focus is typically on
updating a probability law on paths.
