Parameter estimation for compact binaries with ground-based gravitational-wave observations using the LALInference software library

The Advanced LIGO and Advanced Virgo gravitational-wave (GW) detectors will begin operation in the coming years, with compact binary coalescence events a likely source for the first detections. The gravitational waveforms emitted directly encode information about the sources, including the masses and spins of the compact objects. Recovering the physical parameters of the sources from the GW observations is a key analysis task. This work describes the LALInference software library for Bayesian parameter estimation of compact binary signals, which builds on several previous methods to provide a well-tested toolkit which has already been used for several studies. We show that our implementation is able to correctly recover the parameters of compact binary signals from simulated data from the advanced GW detectors. We demonstrate this with a detailed comparison on three compact binary systems: a binary neutron star, a neutron star–black hole binary and a binary black hole, where we show a cross comparison of results obtained using three independent sampling algorithms. These systems were analyzed with nonspinning, aligned spin and generic spin configurations respectively, showing that consistent results can be obtained even with the full 15-dimensional parameter space of the generic spin configurations. We also demonstrate statistically that the Bayesian credible intervals we recover correspond to frequentist confidence intervals under correct prior assumptions by analyzing a set of 100 signals drawn from the prior. We discuss the computational cost of these algorithms, and describe the general and problem-specific sampling techniques we have used to improve the efficiency of sampling the compact binary coalescence parameter space.

Figure 1

Prior probability $p(m_1,m_2|H_S)$, uniform in component masses within the bounds shown (left), and with the same distribution transformed into the $\mathcal{M}$, $q$ parametrization used for sampling.

Figure 2

The profile of the likelihood function for each of the injections in Table 2, mapped onto the fractional prior support parameter $X$ [see Eq. (28)]. The algorithm proceeds from left (sampling entire prior) to right (sampling a tiny restricted part of the prior). The values of $\log (L)$ are normalized to the likelihood of the noise model.

Figure 3

Length of MCMC subchain for nested sampling analysis of the BNS system (as in Table 2) as a function of prior scale. As the run progresses, the length of the MCMC subchain used to generate the next live point automatically adapts to the current conditions, allowing it to use fewer iterations where possible. The chain is limited to a maximum of 5000 iterations.

Figure 4

Acceptance ratio and fraction of sloppy jumps for nested sampling analysis of a BNS system. The dashed blue line shows the automatically determined fraction of proposals for which the likelihood calculation is skipped. The solid green line shows the overall acceptance rate for new live points, which thanks to the adaptive jumps remains at a healthy level despite the volume of the sampled distribution changing by 17 orders of magnitude throughout the run.

Figure 5

The integrand of the evidence integral [Eq. (41)] versus $\beta$ for the analyses of synthetic GW signals in § 5b. The evidence is given by the area under each curve. Table 1 gives the results of the integration together with the estimated error in the quadrature, following the procedure described in § 4c. The jaggedness of the curves illustrates that the temperature spacing required for convergent MCMC simulations is larger than that required for accurate quadrature to compute the evidence; the flatness at small $\beta$ illustrates that, for these simulations, the high-temperature limit is sufficient for convergence of the evidence integral.

Figure 6

Example comparing cumulative distributions for the analytic likelihood functions for each sampler for the (arbitrary) $m_1$ parameter for the three test likelihood functions. The samplers are shown as Nest: purple left hatches; MCMC: green horizontal hatches; BAMBI: blue right hatches, with the true cumulative distributions shown in red where available. (Left) Unimodal multivariate Gaussian distribution. (Middle) Bimodal distribution. (Right) Rosenbrock distribution. The different methods show good agreement with each other, and with the known analytic distributions. Vertical dashed lines indicate the 5–95% credibility interval for each method.

Figure 7

Comparison of probability density functions for the BNS signal (Table 2) as determined by each sampler. Shown are selected two-dimensional posterior density functions in greyscale, with red cross hairs indicating the true parameter values, and contours indicating the 90% credible region as estimated by each sampler. On the axes are superimposed the one-dimensional marginal distributions for each parameter, as estimated by each sampler, and the true value indicated by a vertical red line. The colors correspond to BAMBI (blue), NEST (magenta), MCMC (green). (Left) The mass posterior distribution parametrized by chirp mass and symmetric mass ratio. (Center) The location of the source on the sky. (Right) The distance $d_L$ and inclination ${\theta }_{JN}$ of the source showing the characteristic V-shaped degeneracy.

Figure 8

Comparison of probability density functions for the NSBH signal (Table 2), with the same color scheme as Fig. 7. (First row left) The mass posterior distribution parametrized by chirp mass and symmetric mass ratio. (First row center) The location of the source on the sky. (First row right) The distance $d_L$ and inclination ${\theta }_{JN}$ of the source. In this case the V-shaped degeneracy is broken, but the large correlation between $d_L$ and ${\theta }_{JN}$ remains. (Second row left) The spin magnitudes posterior distribution. (Second row center) The spin and mass of the most massive member of the binary illustrating the degeneracy between mass and spin. (Second row right) The spin and symmetric mass ratio.

Figure 9

Comparison of probability density functions for the BBH signal (Table 2), with the same color scheme as Fig. 7. (First row left) The mass posterior distribution parametrized by chirp mass and symmetric mass ratio. (First row center) The location of the source on the sky. (First row right) The distance $d_L$ and inclination ${\theta }_{JN}$ of the source showing the degeneracy is broken, as in the NSBH case. (Second row left) The spins magnitude posterior distribution. (Second row center) The spin and mass of the most massive member of the binary illustrating the degeneracy between mass and spin. (Second row right) The spin and symmetric mass ratio. (Third row left) The spins tilt posterior distribution. (Third row center) The spin tilt of the more massive member of the binary and the symmetric mass ratio. (Third row right) The spin tilt and mass of the most massive member of the binary.

Figure 10

P versus P plot for the distance parameter. On the $x$ axis is the probability $p$ contained in a credible interval, and on the $y$ axis the fraction of true values which lay inside that interval. The diagonal line indicates the ideal distribution where credible intervals perfectly reflect the frequency of recovered injections. For all three sampling algorithms the results are statistically consistent with the diagonal line, with the lowest KS statistic being 0.25.