160 research outputs found
Dynamic temperature selection for parallel-tempering in Markov chain Monte Carlo simulations
Modern problems in astronomical Bayesian inference require efficient methods
for sampling from complex, high-dimensional, often multi-modal probability
distributions. Most popular methods, such as Markov chain Monte Carlo sampling,
perform poorly on strongly multi-modal probability distributions, rarely
jumping between modes or settling on just one mode without finding others.
Parallel tempering addresses this problem by sampling simultaneously with
separate Markov chains from tempered versions of the target distribution with
reduced contrast levels. Gaps between modes can be traversed at higher
temperatures, while individual modes can be efficiently explored at lower
temperatures. In this paper, we investigate how one might choose the ladder of
temperatures to achieve more efficient sampling, as measured by the
autocorrelation time of the sampler. In particular, we present a simple,
easily-implemented algorithm for dynamically adapting the temperature
configuration of a sampler while sampling. This algorithm dynamically adjusts
the temperature spacing to achieve a uniform rate of exchanges between chains
at neighbouring temperatures. We compare the algorithm to conventional
geometric temperature configurations on a number of test distributions and on
an astrophysical inference problem, reporting efficiency gains by a factor of
1.2-2.5 over a well-chosen geometric temperature configuration and by a factor
of 1.5-5 over a poorly chosen configuration. On all of these problems a sampler
using the dynamical adaptations to achieve uniform acceptance ratios between
neighbouring chains outperforms one that does not.Comment: 21 pages, 21 figure
Using spin to understand the formation of LIGO's black holes
With the detection of four candidate binary black hole (BBH) mergers by the
Advanced LIGO detectors thus far, it is becoming possible to constrain the
properties of the BBH merger population in order to better understand the
formation of these systems. Black hole (BH) spin orientations are one of the
cleanest discriminators of formation history, with BHs in dynamically formed
binaries in dense stellar environments expected to have spins distributed
isotropically, in contrast to isolated populations where stellar evolution is
expected to induce BH spins preferentially aligned with the orbital angular
momentum. In this work we propose a simple, model-agnostic approach to
characterizing the spin properties of LIGO's BBH population. Using measurements
of the effective spin of the binaries, which is LIGO's best constrained spin
parameter, we introduce a simple parameter to quantify the fraction of the
population that is isotropically distributed, regardless of the spin magnitude
distribution of the population. Once the orientation characteristics of the
population have been determined, we show how measurements of effective spin can
be used to directly constrain the underlying BH spin magnitude distribution.
Although we find that the majority of the current effective spin measurements
are too small to be informative, with LIGO's four BBH candidates we find a
slight preference for an underlying population with aligned spins over one with
isotropic spins (with an odds ratio of 1.1). We argue that it will be possible
to distinguish symmetric and anti-symmetric populations at high confidence with
tens of additional detections, although mixed populations may take
significantly more detections to disentangle. We also derive preliminary spin
magnitude distributions for LIGO's black holes, under the assumption of aligned
or isotropic populations
A more effective coordinate system for parameter estimation of precessing compact binaries from gravitational waves
Ground-based gravitational wave detectors are sensitive to a narrow range of
frequencies, effectively taking a snapshot of merging compact-object binary
dynamics just before merger. We demonstrate that by adopting analysis
parameters that naturally characterize this 'picture', the physical parameters
of the system can be extracted more efficiently from the gravitational wave
data, and interpreted more easily. We assess the performance of MCMC parameter
estimation in this physically intuitive coordinate system, defined by (a) a
frame anchored on the binary's spins and orbital angular momentum and (b) a
time at which the detectors are most sensitive to the binary's gravitational
wave emission. Using anticipated noise curves for the advanced-generation LIGO
and Virgo gravitational wave detectors, we find that this careful choice of
reference frame and reference time significantly improves parameter estimation
efficiency for BNS, NS-BH, and BBH signals.Comment: 11 pages, 5 figures, submitted to Phys. Rev.
An Efficient Interpolation Technique for Jump Proposals in Reversible-Jump Markov Chain Monte Carlo Calculations
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.Comment: Minor revision to match published versio
Efficient method for measuring the parameters encoded in a gravitational-wave signal
Once upon a time, predictions for the accuracy of inference on
gravitational-wave signals relied on computationally inexpensive but often
inaccurate techniques. Recently, the approach has shifted to actual inference
on noisy signals with complex stochastic Bayesian methods, at the expense of
significant computational cost. Here, we argue that it is often possible to
have the best of both worlds: a Bayesian approach that incorporates prior
information and correctly marginalizes over uninteresting parameters, providing
accurate posterior probability distribution functions, but carried out on a
simple grid at a low computational cost, comparable to the inexpensive
predictive techniques.Comment: 17 pages, 5 figure
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