Entropy vs information

Over at statistical mechanics of statistics we wonder about the connection between statistical mechanics and statistics, which suggests we might consider the connection between entropy and information. Entropy, the physics concept, and information, the computer science concept, looked danged similar and yet they are defined for different things. How do they connect?

I first looked into this 10+ years ago, but I have had my interest piqued again after ignoring it for ages, because de Vries claims to have actioned the idea within the predictive coding theory of mind, which gives the entire idea somewhat more credibility, and inclines me to return to the original MaxEnt work by Caticha, which now has textbooks about it (Caticha 2015, 2008).

Connected somehow: algorithmic statistics, information geometry.

Shalizi and Moore (2003):

We consider the question of whether thermodynamic macrostates are objective consequences of dynamics, or subjective reflections of our ignorance of a physical system. We argue that they are both; more specifically, that the set of macrostates forms the unique maximal partition of phase space which 1) is consistent with our observations (a subjective fact about our ability to observe the system) and 2) obeys a Markov process (an objective fact about the system’s dynamics). We review the ideas of computational mechanics, an information-theoretic method for finding optimal causal models of stochastic processes, and argue that macrostates coincide with the “causal states” of computational mechanics. Defining a set of macrostates thus consists of an inductive process where we start with a given set of observables, and then refine our partition of phase space until we reach a set of states which predict their own future, i.e. which are Markovian. Macrostates arrived at in this way are provably optimal statistical predictors of the future values of our observables.

• Statistical Physics of Inference and Bayesian Estimation Informational entropy versus thermodynamics entropy.

• John Baez’s A Characterisation of Entropy etc. See also There is now a book (John Carlos Baez 2024)! the post What is Entropy? introduces with the pitch including

  I have largely avoided the second law of thermodynamics, which says that entropy always increases. While fascinating, this is so problematic that a good explanation would require another book! I have also avoided the role of entropy in biology, black hole physics, etc. Thus, the aspects of entropy most beloved by physics popularizers will not be found here. I also never say that entropy is ‘disorder’.

• Wolpert (2006)

• Daniel Ellerman’s Logical Entropy stuff and which he has now written up as Ellerman (2017).

• Information loss and entropy

• Feldman, A Brief Introduction to: Information Theory, Excess Entropy and Computational Mechanics

• It Took Me 10 Years to Understand Entropy, Here is What I Learned. | by Aurelien Pelissier

• Decisional states

  “This article introduces both a new algorithm for reconstructing epsilon-machines from data, as well as the decisional states. These are defined as the internal states of a system that lead to the same decision, based on a user-provided utility or pay-off function.”

• CRS’s CSSR