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