An empirical Bayes procedure for the selection of Gaussian graphical models

A new methodology for model determination in decomposable graphical Gaussian models is developed. The Bayesian paradigm is used and, for each given graph, a hyper inverse Wishart prior distribution on the covariance matrix is considered. This prior distribution depends on hyper-parameters. It is well-known that the models's posterior distribution is sensitive to the specification of these hyper-parameters and no completely satisfactory method is registered. In order to avoid this problem, we suggest adopting an empirical Bayes strategy, that is a strategy for which the values of the hyper-parameters are determined using the data. Typically, the hyper-parameters are fixed to their maximum likelihood estimations. In order to calculate these maximum likelihood estimations, we suggest a Markov chain Monte Carlo version of the Stochastic Approximation EM algorithm. Moreover, we introduce a new sampling scheme in the space of graphs that improves the add and delete proposal of Armstrong et al. (2009). We illustrate the efficiency of this new scheme on simulated and real datasets

Bayesian Analysis of ODE's: solver optimal accuracy and Bayes factors

In most relevant cases in the Bayesian analysis of ODE inverse problems, a numerical solver needs to be used. Therefore, we cannot work with the exact theoretical posterior distribution but only with an approximate posterior deriving from the error in the numerical solver. To compare a numerical and the theoretical posterior distributions we propose to use Bayes Factors (BF), considering both of them as models for the data at hand. We prove that the theoretical vs a numerical posterior BF tends to 1, in the same order (of the step size used) as the numerical forward map solver does. For higher order solvers (eg. Runge-Kutta) the Bayes Factor is already nearly 1 for step sizes that would take far less computational effort. Considerable CPU time may be saved by using coarser solvers that nevertheless produce practically error free posteriors. Two examples are presented where nearly 90% CPU time is saved while all inference results are identical to using a solver with a much finer time step.Comment: 28 pages, 6 figure

Implementation and Deployment of a Distributed Network Topology Discovery Algorithm

In the past few years, the network measurement community has been interested in the problem of internet topology discovery using a large number (hundreds or thousands) of measurement monitors. The standard way to obtain information about the internet topology is to use the traceroute tool from a small number of monitors. Recent papers have made the case that increasing the number of monitors will give a more accurate view of the topology. However, scaling up the number of monitors is not a trivial process. Duplication of effort close to the monitors wastes time by reexploring well-known parts of the network, and close to destinations might appear to be a distributed denial-of-service (DDoS) attack as the probes converge from a set of sources towards a given destination. In prior work, authors of this report proposed Doubletree, an algorithm for cooperative topology discovery, that reduces the load on the network, i.e., router IP interfaces and end-hosts, while discovering almost as many nodes and links as standard approaches based on traceroute. This report presents our open-source and freely downloadable implementation of Doubletree in a tool we call traceroute@home. We describe the deployment and validation of traceroute@home on the PlanetLab testbed and we report on the lessons learned from this experience. We discuss how traceroute@home can be developed further and discuss ideas for future improvements

Posterior concentration rates for empirical Bayes procedures, with applications to Dirichlet Process mixtures

In this paper we provide general conditions to check on the model and the prior to derive posterior concentration rates for data-dependent priors (or empirical Bayes approaches). We aim at providing conditions that are close to the conditions provided in the seminal paper by Ghosal and van der Vaart (2007a). We then apply the general theorem to two different settings: the estimation of a density using Dirichlet process mixtures of Gaussian random variables with base measure depending on some empirical quantities and the estimation of the intensity of a counting process under the Aalen model. A simulation study for inhomogeneous Poisson processes also illustrates our results. In the former case we also derive some results on the estimation of the mixing density and on the deconvolution problem. In the latter, we provide a general theorem on posterior concentration rates for counting processes with Aalen multiplicative intensity with priors not depending on the data.Comment: With supplementary materia

The Specialty Coffee Quality Rating as a Measure of Product Differentiation and Price Signal to Growers: an Entropy Analysis of E-Auction Data

Demand and Price Analysis,