Bayesian exoplanet tests of a new method for MCMC sampling in highly correlated model parameter spaces

Physics and Astronomy Department, University of British Columbia, 6224 Agricultural Road, Vancouver, BC V6T 1Z1, Canada
Monthly Notices of the Royal Astronomical Society (Impact Factor: 5.11). 12/2010; 410(1):94 - 110. DOI: 10.1111/j.1365-2966.2010.17428.x


The Markov chain Monte Carlo (MCMC) method is a powerful technique for facilitating Bayesian non-linear model fitting. In many cases, the MCMC exploration of the parameter space is very inefficient, because the model parameters are highly correlated. Differential evolution MCMC is one technique that addresses this problem by employing multiple parallel chains. We present a new method that automatically achieves efficient MCMC sampling in highly correlated parameter spaces, which does not require additional chains to accomplish this. It was designed to work with an existing hybrid MCMC (HMCMC) algorithm, which incorporates parallel tempering, simulated annealing and genetic cross-over operations. These features, together with the new correlated parameter sampler, greatly facilitate the detection of a global minimum in χ2. The new HMCMC algorithm is very general in scope. Two tests of the algorithm are described employing (a) exoplanet precision radial velocity (RV) data and (b) simulated space astrometry data. The latter test explores the accuracy of parameter estimates obtained with the Bayesian HMCMC algorithm on the assumed astrometric noise.

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    • "Note that this problem is also very closely related to the problem of detecting exoplanet signals in radial velocity data, which has attracted a lot of research attention in recent years (e.g. Gregory, 2011; Hou et al., 2014; Feroz et al., 2011). "
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    ABSTRACT: Many inference problems involve inferring the number $N$ of objects in some region, along with their properties $\{\mathbf{x}_i\}_{i=1}^N$, from a dataset $\mathcal{D}$. A common statistical example is finite mixture modelling. In the Bayesian framework, these problems are typically solved using one of the following two methods: i) by executing a Monte Carlo algorithm (such as Nested Sampling) once for each possible value of $N$, and calculating the marginal likelihood or evidence as a function of $N$; or ii) by doing a single run that allows the model dimension $N$ to change (such as Markov Chain Monte Carlo with birth/death moves), and obtaining the posterior for $N$ directly. In this paper we present a general approach to this problem that uses trans-dimensional MCMC embedded {\it within} a Nested Sampling algorithm, allowing us to explore the posterior distribution and calculate the marginal likelihood (summed over $N$) even if the problem contains a phase transition or other difficult features such as multimodality. We present two example problems, finding sinusoidal signals in noisy data, and finding and measuring galaxies in a noisy astronomical image. Both of the examples demonstrate phase transitions in the relationship between the likelihood and the cumulative prior mass.
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    • "Some of the other challenges in time domain astronomy that I have no room to discuss include high-precision pulsar timing (Liu et al. 2011), astroseismology (Appourchaux 2011) and planet detection (Gregory 2011; Ford et al. 2011). "
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    ABSTRACT: Progress in astronomy comes from interpreting the signals encoded in the light received from distant objects: the distribution of light over the sky (images), over photon wavelength (spectrum), over polarization angle and over time (usually called light curves by astronomers). In the time domain, we see transient events such as supernovae, gamma-ray bursts and other powerful explosions; we see periodic phenomena such as the orbits of planets around nearby stars, radio pulsars and pulsations of stars in nearby galaxies; and we see persistent aperiodic variations ('noise') from powerful systems such as accreting black holes. I review just a few of the recent and future challenges in the burgeoning area of time domain astrophysics, with particular attention to persistently variable sources, the recovery of reliable noise power spectra from sparsely sampled time series, higher order properties of accreting black holes, and time delays and correlations in multi-variate time series.
    Philosophical Transactions of The Royal Society A Mathematical Physical and Engineering Sciences 02/2013; 371(1984):20110549. DOI:10.1098/rsta.2011.0549 · 2.15 Impact Factor
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    • "Independently of earlier work, Cumming (2004) explored the connections between conventional periodogram methods and the Bayesian approach (albeit hampered by an incorrect marginal likelihood calculation); Cumming and Dragomir (2010) later extended this work, reproducing the Kepler periodogram of LC00. Ford (2005) (2008) and Gregory (2005) applied classic posterior sampling techniques (Metropolis random walk (MRW) and parallel tempering Markov chain Monte Carlo (MCMC), respectively) to orbit modeling; Gregory's approach uses a control system to tune the proposal parameters in a pilot run, and he has recently augmented his algorithm to include parameter updates based on genetic algorithms (Gregory 2011). Balan & Lahav (2008) applied an early, approximate adaptive MRW algorithm to the problem. "
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    ABSTRACT: We describe work in progress by a collaboration of astronomers and statisticians developing a suite of Bayesian data analysis tools for extrasolar planet (exoplanet) detection, planetary orbit estimation, and adaptive scheduling of observations. Our work addresses analysis of stellar reflex motion data, where a planet is detected by observing the "wobble" of its host star as it responds to the gravitational tug of the orbiting planet. Newtonian mechanics specifies an analytical model for the resulting time series, but it is strongly nonlinear, yielding complex, multimodal likelihood functions; it is even more complex when multiple planets are present. The parameter spaces range in size from few-dimensional to dozens of dimensions, depending on the number of planets in the system, and the type of motion measured (line-of-sight velocity, or position on the sky). Since orbits are periodic, Bayesian generalizations of periodogram methods facilitate the analysis. This relies on the model being linearly separable, enabling partial analytical marginalization, reducing the dimension of the parameter space. Subsequent analysis uses adaptive Markov chain Monte Carlo methods and adaptive importance sampling to perform the integrals required for both inference (planet detection and orbit measurement), and information-maximizing sequential design (for adaptive scheduling of observations). We present an overview of our current techniques and highlight directions being explored by ongoing research.
    Statistical Methodology 07/2011; 9(1):101-114. DOI:10.1016/j.stamet.2011.07.005 · 0.64 Impact Factor
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