Foundations of Inference, Kevin H. Knuth, John Skilling, arXiv:1008.4831v1 [math.PR]
We present a foundation for inference that unites and significantly extends the approaches of Kolmogorov and Cox. Our approach is based on quantifying finite lattices of logical statements in a way that satisfies general lattice symmetries. With other applications in mind, our derivations assume minimal symmetries, relying on neither complementarity nor continuity or differentiability. Each relevant symmetry corresponds to an axiom of quantification, and these axioms are used to derive a unique set of rules governing quantification of the lattice. These rules form the familiar probability calculus. We also derive a unique quantification of divergence and information. Taken together these results form a simple and clear foundation for the quantification of inference.
For those of us who find ourselves compelled by the view of probability as a generalization of logic that is isomorphic to (algorithmic, as if there were any other kind) information theory, here is some recent i-dotting and t-crossing. The connection to Curry-Howard or, if you prefer, Krivine's classical realizability is something I hope to explore in the near future.
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