Agreement and disagreement in a non-classical world.

The Agreement Theorem Aumann (1976 , 1236-1239. (doi:10.1214/aos/1176343654)) states that if two Bayesian agents start with a common prior, then they cannot have common knowledge that they hold different posterior probabilities of some underlying event of interest.

Rényi Entropy, Signed Probabilities, and the Qubit.

The states of the qubit, the basic unit of quantum information, are 2 × 2 positive semi-definite Hermitian matrices with trace 1. We contribute to the program to axiomatize quantum mechanics by characterizing these states in terms of an entropic uncertainty principle formulated on an eight-point phase space. We do this by employing Rényi entropy (a generalization of Shannon entropy) suitably defined for the signed phase-space probability distributions that arise in representing quantum states.

Divisive normalization is an efficient code for multivariate Pareto-distributed environments.

Divisive normalization is a canonical computation in the brain, observed across neural systems, that is often considered to be an implementation of the efficient coding principle. We provide a theoretical result that makes the conditions under which divisive normalization is an efficient code analytically precise: We show that, in a low-noise regime, encoding an -dimensional stimulus via divisive normalization is efficient if and only if its prevalence in the environment is described by a multivariate Pareto distribution. We generalize this multivariate analog of histogram equalization to allow for arbitrary metabolic costs of the representation, and show how different assumptions on costs are associated with different shapes of the distributions that divisive normalization efficiently encodes.

Observers of quantum systems cannot agree to disagree.

Is the world quantum? An active research line in quantum foundations is devoted to exploring what constraints can rule out the postquantum theories that are consistent with experimentally observed results. We explore this question in the context of epistemics, and ask whether agreement between observers can serve as a physical principle that must hold for any theory of the world. Aumann's seminal Agreement Theorem states that two observers (of classical systems) cannot agree to disagree.

Choice-theoretic foundations of the divisive normalization model.

Recent advances in neuroscience suggest that a utility-like calculation is involved in how the brain makes choices, and that this calculation may use a computation known as divisive normalization. While this tells us the brain makes choices, it is not immediately evident the brain uses this computation or exactly behavior is consistent with it. In this paper, we address both of these questions by proving a three-way equivalence theorem between the normalization model, an information-processing model, and an axiomatic characterization.