Infinite Shannon entropy

TL;DR

This paper investigates the conditions under which a properly normalizable probability distribution can have infinite Shannon entropy, providing bounds, asymptotic estimates, and clarifying the nature of large entropy distributions.

Contribution

It offers a detailed analysis and necessary conditions for infinite Shannon entropy, emphasizing the dispersion of probability into infinitely many states and simplifying technical computations.

Findings

Infinite entropy occurs when probability is dispersed into infinitely many states.

Large entropies are supported only on exponentially large state spaces.

The paper provides bounds and asymptotic estimates for infinite Shannon entropy.

Abstract

Even if a probability distribution is properly normalizable, its associated Shannon (or von Neumann) entropy can easily be infinite. We carefully analyze conditions under which this phenomenon can occur. Roughly speaking, this happens when arbitrarily small amounts of probability are dispersed into an infinite number of states; we shall quantify this observation and make it precise. We develop several particularly simple, elementary, and useful bounds, and also provide some asymptotic estimates, leading to necessary and sufficient conditions for the occurrence of infinite Shannon entropy. We go to some effort to keep technical computations as simple and conceptually clear as possible. In particular, we shall see that large entropies cannot be localized in state space; large entropies can only be supported on an exponentially large number of states. We are for the time being interested in single-channel Shannon entropy in the information theoretic sense, not entropy in a stochastic field theory or QFT defined over some configuration space, on the grounds that this simple problem is a necessary precursor to understanding infinite entropy in a field theoretic context.