Abstract:
The Laser Interferometer Space Antenna (LISA) defines new demands on data analysis efforts in its all-sky gravitational wave survey, recording simultaneously thousands of galactic compact object binary foreground sources and tens to hundreds of background sources like binary black hole mergers and extreme mass ratio inspirals. We approach this problem with an adaptive and fully automatic Reversible Jump Markov Chain Monte Carlo sampler, able to sample from the joint posterior density function (as established by Bayes theorem) for a given mixture of signals "out of the box'', handling the total number of signals as an additional unknown parameter beside the unknown parameters of each individual source and the noise floor. We show in examples from the LISA Mock Data Challenge implementing the full response of LISA in its TDI description that this sampler is able to extract monochromatic Double White Dwarf signals out of colored instrumental noise and additional foreground and background noise successfully in a global fitting approach. We introduce 2 examples with fixed number of signals (MCMC sampling), and 1 example with unknown number of signals (RJ-MCMC), the latter further promoting the idea behind an experimental adaptation of the model indicator proposal densities in the main sampling stage. We note that the experienced runtimes and degeneracies in parameter extraction limit the shown examples to the extraction of a low but realistic number of signals.

Abstract:
We propose a novel reversible jump Markov chain Monte Carlo (MCMC) simulated annealing algorithm to optimize radial basis function (RBF) networks. This algorithm enables us to maximize the joint posterior distribution of the network parameters and the number of basis functions. It performs a global search in the joint space of the parameters and number of parameters, thereby surmounting the problem of local minima. We also show that by calibrating a Bayesian model, we can obtain the classical AIC, BIC and MDL model selection criteria within a penalized likelihood framework. Finally, we show theoretically and empirically that the algorithm converges to the modes of the full posterior distribution in an efficient way.

Abstract:
The Reversible Jump algorithm is one of the most widely used Markov chain Monte Carlo algorithms for Bayesian estimation and model selection. A generalized multiple-try version of this algorithm is proposed. The algorithm is based on drawing several proposals at each step and randomly choosing one of them on the basis of weights (selection probabilities) that may be arbitrary chosen. Among the possible choices, a method is employed which is based on selection probabilities depending on a quadratic approximation of the posterior distribution. Moreover, the implementation of the proposed algorithm for challenging model selection problems, in which the quadratic approximation is not feasible, is considered. The resulting algorithm leads to a gain in efficiency with respect to the Reversible Jump algorithm, and also in terms of computational effort. The performance of this approach is illustrated for real examples involving a logistic regression model and a latent class model.

Abstract:
In the context of nonparametric Bayesian estimation a Markov chain Monte Carlo algorithm is devised and implemented to sample from the posterior distribution of the drift function of a continuously or discretely observed one-dimensional diffusion. The drift is modeled by a scaled linear combination of basis functions with a Gaussian prior on the coefficients. The scaling parameter is equipped with a partially conjugate prior. The number of basis function in the drift is equipped with a prior distribution as well. For continuous data, a reversible jump Markov chain algorithm enables the exploration of the posterior over models of varying dimension. Subsequently, it is explained how data-augmentation can be used to extend the algorithm to deal with diffusions observed discretely in time. Some examples illustrate that the method can give satisfactory results. In these examples a comparison is made with another existing method as well.

Abstract:
In this paper we present an extension of population-based Markov chain Monte Carlo (MCMC) to the trans-dimensional case. One of the main challenges in MCMC-based inference is that of simulating from high and trans-dimensional target measures. In such cases, MCMC methods may not adequately traverse the support of the target; the simulation results will be unreliable. We develop population methods to deal with such problems, and give a result proving the uniform ergodicity of these population algorithms, under mild assumptions. This result is used to demonstrate the superiority, in terms of convergence rate, of a population transition kernel over a reversible jump sampler for a Bayesian variable selection problem. We also give an example of a population algorithm for a Bayesian multivariate mixture model with an unknown number of components. This is applied to gene expression data of 1000 data points in six dimensions and it is demonstrated that our algorithm out performs some competing Markov chain samplers.

Abstract:
A method is presented for Bayesian model selection without explicitly computing evidences, by using a combined likelihood and introducing an integer model selection parameter $n$ so that Bayes factors, or more generally posterior odds ratios, may be read off directly from the posterior of $n$. If the total number of models under consideration is specified a priori, the full joint parameter space $(\theta, n)$ of the models is of fixed dimensionality and can be explored using standard MCMC or nested sampling methods, without the need for reversible jump MCMC techniques. The posterior on $n$ is then obtained by straightforward marginalisation. We demonstrate the efficacy of our approach by application to several toy models. We then apply it to constraining the dark energy equation-of-state using a free-form reconstruction technique. We show that $\Lambda$CDM is significantly favoured over all extensions, including the simple $w(z){=}{\rm constant}$ model.

Abstract:
Selection among alternative theoretical models given an observed data set is an important challenge in many areas of physics and astronomy. Reversible-jump Markov chain Monte Carlo (RJMCMC) is an extremely powerful technique for performing Bayesian model selection, but it suffers from a fundamental difficulty: it requires jumps between model parameter spaces, but cannot efficiently explore both parameter spaces at once. Thus, a naive jump between parameter spaces is unlikely to be accepted in the MCMC algorithm and convergence is correspondingly slow. Here we demonstrate an interpolation technique that uses samples from single-model MCMCs to propose inter-model jumps from an approximation to the single-model posterior of the target parameter space. The interpolation technique, based on a kD-tree data structure, is adaptive and efficient in modest dimensionality. We show that our technique leads to improved convergence over naive jumps in an RJMCMC, and compare it to other proposals in the literature to improve the convergence of RJMCMCs. We also demonstrate the use of the same interpolation technique as a way to construct efficient "global" proposal distributions for single-model MCMCs without prior knowledge of the structure of the posterior distribution, and discuss improvements that permit the method to be used in higher-dimensional spaces efficiently.

Abstract:
Regional flood frequency analysis is a convenient way to reduce estimation uncertainty when few data are available at the gauging site. In this work, a model that allows a non-null probability to a regional fixed shape parameter is presented. This methodology is integrated within a Bayesian framework and uses reversible jump techniques. The performance on stochastic data of this new estimator is compared to two other models: a conventional Bayesian analysis and the index flood approach. Results show that the proposed estimator is absolutely suited to regional estimation when only a few data are available at the target site. Moreover, unlike the index flood estimator, target site index flood error estimation seems to have less impact on Bayesian estimators. Some suggestions about configurations of the pooling groups are also presented to increase the performance of each estimator.

Abstract:
The large deviations principle for the empirical measure for both continuous and discrete time Markov processes is well known. Various expressions are available for the rate function, but these expressions are usually as the solution to a variational problem, and in this sense not explicit. An interesting class of continuous time, reversible processes was identified in the original work of Donsker and Varadhan for which an explicit expression is possible. While this class includes many (reversible) processes of interest, it excludes the case of continuous time pure jump processes, such as a reversible finite state Markov chain. In this paper, we study the large deviations principle for the empirical measure of pure jump Markov processes and provide an explicit formula of the rate function under reversibility.