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Register a new userA Bayesian approach to the study of white dwarf binaries in LISA data: The application of a reversible jump Markov chain Monte Carlo method
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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.
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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.
A number of problems in a variety of fields are characterised by target distributions with a multimodal structure in which the presence of several isolated local maxima dramatically reduces the efficiency of Markov Chain Monte Carlo sampling algorithms. Several solutions, such as simulated tempering or the use of parallel chains, have been proposed to facilitate the exploration of the relevant parameter space. They provide effective strategies in the cases in which the dimension of the parameter space is small and/or the computational costs are not a limiting factor. These approaches fail however in the case of high-dimensional spaces where the multimodal structure is induced by degeneracies between regions of the parameter space. In this paper we present a fully Markovian way to efficiently sample this kind of distribution based on the general Delayed Rejection scheme with an arbitrary number of steps, and provide details for an efficient numerical implementation of the algorithm.
We are developing a Bayesian approach based on Markov chain Monte Carlo techniques to search for and extract information about white dwarf binary systems with the Laser Interferometer Space Antenna (LISA). Here we present results obtained by applying an initial implementation of this method to some of the data sets released in Round 1B of the Mock LISA Data Challenges. For Challenges 1B.1.1a and 1b the signals were recovered with parameters lying within the 95.5% posterior probability interval and the correlation between the true and recovered waveform is in excess of 99%. Results were not submitted for Challenge 1B.1.1c due to some convergence problems of the algorithms, despite this, the signal was detected in a search over a 2 mHz band.
In this article we consider computing expectations w.r.t.~probability laws associated to a certain class of stochastic systems. In order to achieve such a task, one must not only resort to numerical approximation of the expectation, but also to a biased discretization of the associated probability. We are concerned with the situation for which the discretization is required in multiple dimensions, for instance in space and time. In such contexts, it is known that the multi-index Monte Carlo (MIMC) method can improve upon i.i.d.~sampling from the most accurate approximation of the probability law. Indeed by a non-trivial modification of the multilevel Monte Carlo (MLMC) method and it can reduce the work to obtain a given level of error, relative to the afore mentioned i.i.d.~sampling and relative even to MLMC. In this article we consider the case when such probability laws are too complex to sampled independently. We develop a modification of the MIMC method which allows one to use standard Markov chain Monte Carlo (MCMC) algorithms to replace independent and coupled sampling, in certain contexts. We prove a variance theorem which shows that using our MIMCMC method is preferable, in the sense above, to i.i.d.~sampling from the most accurate approximation, under assumptions. The method is numerically illustrated on a problem associated to a stochastic partial differential equation (SPDE).
We present orbital elements and mass sums for eighteen visual binary stars of spectral types B to K (five of which are new orbits) with periods ranging from 20 to more than 500 yr. For two double-line spectroscopic binaries with no previous orbits, the individual component masses, using combined astrometric and radial velocity data, have a formal uncertainty of ~0.1 MSun. Adopting published photometry, and trigonometric parallaxes, plus our own measurements, we place these objects on an H-R diagram, and discuss their evolutionary status. These objects are part of a survey to characterize the binary population of stars in the Southern Hemisphere, using the SOAR 4m telescope+HRCAM at CTIO. Orbital elements are computed using a newly developed Markov Chain Monte Carlo algorithm that delivers maximum likelihood estimates of the parameters, as well as posterior probability density functions that allow us to evaluate the uncertainty of our derived parameters in a robust way. For spectroscopic binaries, using our approach, it is possible to derive a self-consistent parallax for the system from the combined astrometric plus radial velocity data (orbital parallax), which compares well with the trigonometric parallaxes. We also present a mathematical formalism that allows a dimensionality reduction of the feature space from seven to three search parameters (or from ten to seven dimensions - including parallax - in the case of spectroscopic binaries with astrometric data), which makes it possible to explore a smaller number of parameters in each case, improving the computational efficiency of our Markov Chain Monte Carlo code.
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