  Here we define and study the properties of retrodictive inference. We derive
equations relating retrodiction entropy and thermodynamic entropy, and as a
special case, show that under equilibrium conditions, the two are identical. We
demonstrate relations involving the KL-divergence and retrodiction probability,
and bound the time rate of change of retrodiction entropy. As a specific case,
we invert various Langevin processes, inferring the initial condition of \(N\)
particles given their final positions at some later time. We evaluate the
retrodiction entropy for Langevin dynamics exactly for special cases, and find
that one's ability to infer the initial state of a system can exhibit two
possible qualitative behaviors depending on the potential energy landscape,
either decreasing indefinitely, or asymptotically approaching a fixed value. We
also study how well we can retrodict points that evolve based on the logistic
map. We find singular changes in the retrodictivity near bifurcations.
Counterintuitively, the transition to chaos is accompanied by maximal
retrodictability.
