Bayesian approach to the detection problem in gravitational wave astronomy

The analysis of data from gravitational wave detectors can be divided into three phases: search, characterization, and evaluation. The evaluation of the detection—determining whether a candidate event is astrophysical in origin or some artifact created by instrument noise—is a crucial step in the analysis. The ongoing analyses of data from ground-based detectors employ a frequentist approach to the detection problem. A detection statistic is chosen, for which background levels and detection efficiencies are estimated from Monte Carlo studies. This approach frames the detection problem in terms of an infinite collection of trials, with the actual measurement corresponding to some realization of this hypothetical set. Here we explore an alternative, Bayesian approach to the detection problem, that considers prior information and the actual data in hand. Our particular focus is on the computational techniques used to implement the Bayesian analysis. We find that the parallel tempered Markov chain Monte Carlo (PTMCMC) algorithm is able to address all three phases of the analysis in a coherent framework. The signals are found by locating the posterior modes, the model parameters are characterized by mapping out the joint posterior distribution, and finally, the model evidence is computed by thermodynamic integration. As a demonstration, we consider the detection problem of selecting between models describing the data as instrument noise, or instrument noise plus the signal from a single compact galactic binary. The evidence ratios, or Bayes factors, computed by the PTMCMC algorithm are found to be in close agreement with those computed using a reversible jump Markov chain Monte Carlo algorithm.

Figure 1

MAP noise parameters for ${\mathcal{M}}_0$ (red, solid) and ${\mathcal{M}}_1$ (blue, dashed). The data consist of two interferometer channels, each containing 256 Fourier bins which are divided into four subregions. Each subdivision is fitted with a unique noise parameter. The signal is injected somewhere in the second quarter of the data. Model ${\mathcal{M}}_0$ elevates the noise parameter in the second window to account for the excess power caused by the gravitational wave signal. Model ${\mathcal{M}}_1$ successfully fits to that gravitational wave leaving the noise parameter closer to unity. Notice how the noise parameters for remaining portions of the data are nearly identical between the two models.

Figure 2

Two dimensional marginalized log posterior densities for a single galactic binary with $\mathrm{SNR}=7$ as approximated by a MCMC. On the left is the $\ln A-f{T}_{\mathrm{obs}}$ plane; the right shows a Molweide projection of the sky location (parameters $\theta$ and $\varphi$). Red (white) locations have the highest log probability density while blue (black) has the least. The top panel is for a MCMC without parallel tempering where the chain was started at the injected parameters and allowed $5\times {10}^5$ burn-in iterations. The bottom panel is the same, only now with 20 parallel chains spaced geometrically in heat with a maximum temperature of 100 (${\beta }_{\max }=0.01$). The PTMCMC samples the full distribution, resolving the global structure of the PDF, while the straightforward single chain MCMC never leaves the region around injected parameter values, missing the global maximum entirely (which happens to be pushed off from the injected values by instrument noise).

Figure 3

Average likelihood for chains of different heats $\beta =1/T$. The red (solid) line is for the noise-only model ${\mathcal{M}}_0$ while the blue (dashed) line is for the signal plus noise model ${\mathcal{M}}_1$. This particular example was for data containing a $\mathrm{SNR}=8$ source.

Figure 4

$\ln p(s|\overrightarrow{\theta })$ during search (red, solid) and characterization (blue, dashed) phase of the analysis for a $\mathrm{SNR}=8$ source.

Figure 5

Marginalized log posterior densities in the $\theta -\varphi$ (left) and the $\ln A-f{T}_{\mathrm{obs}}$ (right) plane for data containing only stationary Gaussian noise. Although this data contained no gravitational wave signal the PDFs show organized locations in parameter space that look like GW signals. The evaluation step of the analysis easily discarded these potential detections returning a Bayes factor of $\sim 0.7$. The $\theta -\varphi$ PDF is shown in a Molweide projection on the sky.

Figure 6

Thermodynamic integration results for $B_{10}$ on data with signals injected at increasingly higher signal-to-noise ratio. Error bars were established by starting the chains with different random number generator seeds. The horizontal line is the Bayes factor where one would consider the result a positive detection.

Figure 7

Integrand for thermodynamic integration on data with signals injected at increasingly higher SNR. Of importance is how the curves become indistinguishable below $\beta \sim 0.01$ [inset]. The horizontal line marks the regimes where the integrand supports ${\mathcal{M}}_1$ (above) or ${\mathcal{M}}_0$ (below). The inset shows that the integrand has sufficiently vanished at the maximum temperature analyzed.

Figure 8

Thermodynamic integration and RJMCMC results for $B_{10}$ on data with signals injected at increasingly higher signal-to-noise ratio. The horizontal line is the Bayes factor where one would consider the result a positive detection.

Figure 9

Thermodynamic integration results for $B_{10}$ on data with signals injected at increasingly higher signal-to-noise ratio. The horizontal line is the Bayes factor where one would consider the result a positive detection. The red (solid) line is for the analysis done with uniform priors in the signal parameters (apart form $\dot{f}$). The blue (dashed) line is the same data analyzed with a restrictive prior on amplitude such that the minimum amplitude corresponds to a SNR of $\sim 5$. See Sec. 5a2 for more explanation.

Figure 10

Marginalized log posterior densities for $f{T}_{\mathrm{obs}}-\ln A$ (left) and sky location (right) for two runs using different random number seeds for the Markov chains. The consistency between different runs is evidence that the chains have converged to target distribution.

Figure 11

Thermodynamic integration results for $B_{10}$ on data with signals injected at increasingly higher signal-to-noise ratio. The horizontal line is the Bayes factor where one would consider the result a positive detection. Different curves represent different noise realizations but identical signal parameters. This demonstrates that the detection “threshold” is sensitive to the particulars of the noise in the detector, even when the noise characteristics are identical (stationary, Gaussian, with known spectral density).

Figure 12

Thermodynamic integration and RJMCMC results for $B_{10}$ on data with injected signals 2 and 3 (red and green, and blue and magenta, respectively) injected at increasingly higher signal-to-noise ratio (see Table . The horizontal line is the Bayes factor where one would consider the result a positive detection. This demonstrates that the detection threshold is sensitive to the details of the waveforms.