Abstract
We present a simple and clear foundation for finite inference that unites and significantly extends the approaches of Kolmogorov and Cox. Our approach is based on quantifying lattices of logical statements in a way that satisfies general lattice symmetries. With other applications such as measure theory in mind, our derivations assume minimal symmetries, relying on neither negation nor continuity nor differentiability. Each relevant symmetry corresponds to an axiom of quantification, and these axioms are used to derive a unique set of quantifying rules that form the familiar probability calculus. We also derive a unique quantification of divergence, entropy and information.
Highlights
The quality of an axiom rests on it being both convincing for the application(s) in mind, and compelling in that its denial would be intolerable.We present elementary symmetries as convincing and compelling axioms, initially for measure, subsequently for probability, and for information and entropy
We find that the symmetries of chaining require to be multiplication, yielding the product rule of probability calculus
Whereas the sum rule for measure and probability generalizes to the inclusion/exclusion form for general elements which need not be disjoint, so does the ratio form of probability allow generalization from intervals [3] to generalized intervals, consisting of arbitrary pairs [x, t] which need not be in a chain
Summary
The quality of an axiom rests on it being both convincing for the application(s) in mind, and compelling in that its denial would be intolerable. Cox [1] showed the way by deriving the unique laws of probability from logical systems having a mere three elementary “atomic” propositions By extension, those same laws applied to Boolean systems with arbitrarily many atoms and where appropriate, to well-defined infinite limits. We use arbitrarily many atoms to define the calculus to arbitrarily fine precision Avoiding infinity in this way yields results that cover all practical applications, while avoiding unobservable subtleties. Our approach unites and significantly extends the set-based approach of Kolmogorov [2] and the logic-based approach of Cox [1], to form a foundation for inference that yields not just probability calculus, and the unique quantification of divergence and information
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