7,288 research outputs found
Uniform existence of the integrated density of states for models on \ZZ^d
We provide an ergodic theorem for certain Banach-space valued functions on
structures over \ZZ^d, which allow for existence of frequencies of finite
patterns. As an application we obtain existence of the integrated density of
states for associated finite-range operators in the sense of convergence of the
distributions with respect to the supremum norm. These results apply to various
examples including periodic operators, percolation models and nearest-neighbour
hopping on the set of visible points. Our method gives explicit bounds on the
speed of convergence in terms of the speed of convergence of the underlying
frequencies. It uses neither von Neumann algebras nor a framework of random
operators on a probability space.Comment: 15 page
Bayesian Inference for Latent Biologic Structure with Determinantal Point Processes (DPP)
We discuss the use of the determinantal point process (DPP) as a prior for
latent structure in biomedical applications, where inference often centers on
the interpretation of latent features as biologically or clinically meaningful
structure. Typical examples include mixture models, when the terms of the
mixture are meant to represent clinically meaningful subpopulations (of
patients, genes, etc.). Another class of examples are feature allocation
models. We propose the DPP prior as a repulsive prior on latent mixture
components in the first example, and as prior on feature-specific parameters in
the second case. We argue that the DPP is in general an attractive prior model
for latent structure when biologically relevant interpretation of such
structure is desired. We illustrate the advantages of DPP prior in three case
studies, including inference in mixture models for magnetic resonance images
(MRI) and for protein expression, and a feature allocation model for gene
expression using data from The Cancer Genome Atlas. An important part of our
argument are efficient and straightforward posterior simulation methods. We
implement a variation of reversible jump Markov chain Monte Carlo simulation
for inference under the DPP prior, using a density with respect to the unit
rate Poisson process
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