[srean]

Bethe ansatz is one. It took a toure de force by Yedidia to recognize that loopy belief propagation is computing the stationary point of Bethe's approximation to Free Energy.

Many statistical thermodynamics ideas were reinvented in ML.

Same is true for mirror descent. It was independently discovered by Warmuth and his students as Bregman divergence proximal minimization, or as a special case would have it, exponential gradient algorithms.

One can keep going.

[ogogmad]

The connections of deep learning to stat-mech and thermodynamics are really cool.

It's led me to wonder about the origin of the probability distributions in stat-mech. Physical randomness is mostly a fiction (outside maybe quantum mechanics) so probability theory must be a convenient fiction. But objectively speaking, where then do the probabilities in stat-mech come from? So far, I've noticed that the (generalised) Boltzmann distribution serves as the bridge between probability theory and thermodynamics: It lets us take non-probabilistic physics and invent probabilities in a useful way.

[srean]

In Boltzmann's formulation of stat-mech it comes from the assumption that when a system is in "equilibrium", then all the micro-states that are consistent with the macro-state are equally occupied. That's the basis of the theory. A prime mover is thermal agitation.

It can be circular if one defines equilibrium to be that situation when all the micro-states are equally occupied. One way out is to define equilibrium in temporal terms - when the macro-states are not changing with time.

[mitthrowaway2]

The Bayesian reframing of that would be that when all you have measured is the macrostate, and you have no further information by which to assign a higher probability to any compatible microstate than any other, you follow the principle of indifference and assign a uniform distribution.

[srean]

Yes indeed, thanks for pointing this out. There are strong relationships between max-ent and Bayesian formulations.

For example one can use a non-uniform prior over the micro-states. If that prior happens to be in the Darmois-Koopman family that implicitly means that there are some non explicitly stated constraints that bind the micro-state statistics.