The diagonalization method in quantum recursion theory

As quantum parallelism allows the effective co-representation of classical mutually exclusive states, the diagonalization method of classical recursion theory has to be modified. Quantum diagonalization involves unitary operators whose eigenvalues are different from one.Comment: 15 pages, completely rewritte

Coins falling in water

When a coin falls in water, its trajectory is one of four types determined by its dimensionless moment of inertia $I^\ast$ and Reynolds number Re: (A) steady; (B) fluttering; (C) chaotic; or (D) tumbling. The dynamics induced by the interaction of the water with the surface of the coin, however, makes the exact landing site difficult to predict a priori. Here, we describe a carefully designed experiment in which a coin is dropped repeatedly in water, so that we can determine the probability density functions (pdf) associated with the landing positions for each of the four trajectory types, all of which are radially symmetric about the center-drop line. In the case of the steady mode, the pdf is approximately Gaussian distributed, with variances that are small, indicating that the coin is most likely to land at the center, right below the point it is dropped from. For the other falling modes, the center is one of the least likely landing sites. Indeed, the pdf's of the fluttering, chaotic and tumbling modes are characterized by a "dip" around the center. For the tumbling mode, the pdf is a ring configuration about the center-line, with a ring width that depends on the dimensionless parameters $I^\ast$ and Re and height from which the coin is dropped. For the chaotic mode, the pdf is generally a broadband distribution spread out radially symmetrically about the center-line. For the steady and fluttering modes, the coin never flips, so the coin lands with the same side up as was dropped. For the chaotic mode, the probability of heads or tails is close to 0.5. In the case of the tumbling mode, the probability of heads or tails based on the height of the drop which determines whether the coin flips an even or odd number of times during descent

Probability theory and its models

This paper argues for the status of formal probability theory as a mathematical, rather than a scientific, theory. David Freedman and Philip Stark's concept of model based probabilities is examined and is used as a bridge between the formal theory and applications.Comment: Published in at http://dx.doi.org/10.1214/193940307000000347 the IMS Collections (http://www.imstat.org/publications/imscollections.htm) by the Institute of Mathematical Statistics (http://www.imstat.org

Hierarchical Implicit Models and Likelihood-Free Variational Inference

Implicit probabilistic models are a flexible class of models defined by a simulation process for data. They form the basis for theories which encompass our understanding of the physical world. Despite this fundamental nature, the use of implicit models remains limited due to challenges in specifying complex latent structure in them, and in performing inferences in such models with large data sets. In this paper, we first introduce hierarchical implicit models (HIMs). HIMs combine the idea of implicit densities with hierarchical Bayesian modeling, thereby defining models via simulators of data with rich hidden structure. Next, we develop likelihood-free variational inference (LFVI), a scalable variational inference algorithm for HIMs. Key to LFVI is specifying a variational family that is also implicit. This matches the model's flexibility and allows for accurate approximation of the posterior. We demonstrate diverse applications: a large-scale physical simulator for predator-prey populations in ecology; a Bayesian generative adversarial network for discrete data; and a deep implicit model for text generation.Comment: Appears in Neural Information Processing Systems, 201