Irreversibility in Stochastic Dynamic Models and Efficient Bayesian Inference

This thesis is the summary of an excursion around the topic of reversibility. We start the journal from a classical mechanical view of the “time reversal symmetry”: we look into the details to track the movements of all particles at all times and ask whether the entire system remains the same if both time and momentum flip signs. This description of reversible process is the exact reflection of classical mechanics with a quadratic kinetic energy which generates Boltzmann’s equilibrium thermodynamics. Unfortunately, it heavily depends on the coordinate system the variables reside in and automatically excludes the processes with dissipation or/and fluctuation from being reversible. A related but slightly more relaxed scenario is that the dynamics conserve certain quantities. Fortunately, we are able to generalize thermodynamics to this broader range of systems. For the discussion of reversibility, however, we veer towards a direction that requires much less scrutiny, and provides far more generality. We follow Kolmogorov’s footsteps and only study the statistics of the variables in question. Reversibility in that realm dictates that the probability of observing a path forward equals to that of seeing a path backward. Interestingly though, the aforementioned conservative dynamics are the source of irreversibility in stationarity. We then realize that the general Markov process can be decomposed into reversible and irreversible components, each preserving the entire process’ stationary distribution. This realization lets us continue along the path to develop thermodynamic theory for general stochastic processes and confirm the universal ideal behavior in Orntein …