Maximum Entropy Principles

Events between which we have no epistemic reason to discriminate have equal epistemic probabilities. Bertrand’s chord paradox, however, appears to show this to be false, and thereby poses a general threat to probabilities for continuum sized state spaces. Articulating the nature of such spaces involves some deep mathematics and that is perhaps why the recent literature on Bertrand’s Paradox has been almost entirely from mathematicians and physicists, who have often deployed elegant mathematics of considerable sophistication. At the same time, the (...) philosophy of probability has been left out. In particular, left out entirely are the philosophical ground of the principle of indifference, the nature of the principle itself, the stringent constraint this places on the mathematical representation of the principle needed for its application to continuum sized event spaces, and what these entail for rigour in developing the paradox itself. This book puts the philosophy and its entailments back in and in so doing casts a new light on the paradox, giving original analyses of the paradox, its possible solutions, the source of the paradox, the philosophical errors we make in attempting to solve it and what the paradox proves for the philosophy of probability. The book finishes with the author’s proposed solution—a solution in the spirit of Bertrand’s, indeed—in which an epistemic principle more general than the principle of indifference offers a principled restriction of the domain of the principle of indifference. (shrink)

This insight to art came from chess composition concentrating art in a very dense form. To identify and mathematically assess the uniqueness is the key applicable to other areas eg. computer programming. Maximization of uniqueness is minimization of entropy that coincides as well as goes beyond Information Theory (Shannon, 1948). The reusage of logic as a universal principle to minimize entropy, requires simplified architecture and abstraction. Any structures (e.g. plugins) duplicating or dividing functionality increase entropy and so unreliability (eg. British (...) Airways IT system). The ideas here were verified by my chess compositions, art works and complex information system as an author of each co uk, and were presented at conferences in Santorini, Adelaide, Geneva, Daejon and virtually. (shrink)

An important suggestion of objective Bayesians is that the maximum entropy principle can replace a principle which is known to get into paradoxical difficulties: the principle of indifference. No one has previously determined whether the maximum entropy principle is better able to solve Bertrand’s chord paradox than the principle of indifference. In this paper I show that it is not. Additionally, the course of the analysis brings to light a new paradox, a revenge paradox of the chords, that is unique (...) to the maximum entropy principle. (shrink)

This paper has three main objectives: (a) Discuss the formal analogy between some important symmetry-invariance arguments used in physics, probability and statistics. Specifically, we will focus on Noether’s theorem in physics, the maximum entropy principle in probability theory, and de Finetti-type theorems in Bayesian statistics; (b) Discuss the epistemological and ontological implications of these theorems, as they are interpreted in physics and statistics. Specifically, we will focus on the positivist (in physics) or subjective (in statistics) interpretations vs. objective interpretations that (...) are suggested by symmetry and invariance arguments; (c) Introduce the cognitive constructivism epistemological framework as a solution that overcomes the realism-subjectivism dilemma and its pitfalls. The work of the physicist and philosopher Max Born will be particularly important in our discussion. (shrink)

Abstract -/- The Bekenstein Bound proposed that the maximum information content of a finite region of space is determined by its surface area, not its volume, leading to the holographic principle. This shift reframed information as the fundamental constraint on physical systems rather than matter or energy. It suggested that entropy, rather than being an inherent measure of disorder, is a function of informational constraints at the boundaries of a system. However, existing models that incorporate this bound still rely on (...) probability, treating entropy as a stochastic process governed by statistical mechanics. These interpretations assume that uncertainty is a fundamental property of nature rather than an emergent effect of incomplete structural alignment. -/- Recent refinements propose a toroidal representation of entropy, arguing that physical systems favor structured rotational flows, vortices, and spirals over simple spherical constraints. This development reflects a growing recognition that entropy is not merely an abstract statistical measure but is subject to deeper geometric and topological constraints. The toroidal model acknowledges that energy and information distributions follow organized patterns rather than purely random dispersal. However, despite this refinement, these approaches fail to eliminate the underlying dependence on probabilistic mechanics and continue to frame entropy within the paradigm of uncertainty and statistical randomness. -/- This paper presents a deterministic framework for information and entropy, governed by prime-driven structured resonance. We argue that structured resonance—not stochastic entropy—dictates the fundamental limits of information storage, transfer, and dissipation. By introducing prime-resonant phase-locking as the foundational ordering principle, we demonstrate that information constraints arise from structured interference patterns rather than probabilistic uncertainty. This perspective eliminates the necessity of randomness in entropy models and replaces it with a structured, chirally-constrained mechanism of phase interactions. In doing so, we establish that entropy is not a measure of disorder but a function of resonance alignment, and that probability is an emergent illusion caused by phase misalignment rather than an intrinsic feature of physical law. (shrink)