Bayesian approach to the study of white dwarf binaries in LISA data: The application of a reversible jump Markov chain Monte Carlo method

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.

Figure 1

The flow chart of our RJ-MCMC sampler. For details see the text.

Figure 2

Scenario A: Adaptation statistics for verification binary AM CVn (number 1); acceptance rates. We find convergence towards the desired acceptance rate of 0.25 in all the parameters but the noise level $S_0$, with the latter ongoing adaptation still yielding a reliable sampling of the posterior as seen e.g. in Table .

Figure 3

Scenario A: Adaptation statistics for verification binary AM CVn (number 1); standard deviations. We see the standard deviations to converge towards a single optimal value in all the parameters but the noise level $S_0$. Nevertheless, even with varying standard deviations in the noise level we sample robustly from the posterior.

Figure 4

Scenario A: Marginalized PDFs for verification binary AM CVn (number 1). Red lines denote the true value of the parameter. We observe Gaussian distributions in all the parameters with the peculiarity of a bimodal distribution in ${\Phi }_0$ indicating a possible degeneracy of $\Psi$ modulo $\pi /2$ in the LISA response function. We observe the true value always to be covered by the recovered marginal posterior distributions, a result that will be elaborated in Table .

Figure 5

Scenario B1 (left panel) and Scenario B2 (right panel). Adaptation statistics for the handpicked verification binary; acceptance rates. We find convergence towards the desired acceptance rate of 0.25 in all the parameters but the noise level $S_0$, where we see the standard deviation of the noise level essentially underestimated all the time. This leads to almost always accepted states as the chain does not move strongly enough. Nevertheless the trend slowly points toward the optimal acceptance, therefore does not show a runaway effect but a very slow convergence. This yields a stable sampling, as seen e.g. in Table .

Figure 6

Scenario B1 (left panel) and Scenario B2 (right panel). Adaptation statistics for the handpicked verification binary; standard deviations. We see the standard deviations to converge towards a single optimal value in all the parameters but the noise level $S_0$. Nevertheless, even with strongly varying standard deviations in the noise level we sample robustly from the posterior.

Figure 7

Scenario B1 (left panel) and Scenario B2 (right panel). Marginalized PDFs for the handpicked verification binary. Red lines denote the true values of the parameter. The true noise level is unknown as it contains contributions from the unknown galactic DWD population (Scenario B1, Scenario B2), BH mergers (Scenario B2) and EMRIs (Scenario B2)

Figure 8

Scenario C: The marginalized posterior densities for our RJ-MCMC example for the correctly identified model to contain the actual amount of signals in the data, here model 3 with 3 signals. We show the posteriors of the signals in order from top to bottom. The noise level $S_0$ estimated besides the three signals is shown in the bottom right panel. Red lines denote the true value of the parameter.

Figure 9

Scenario C: The evolution of the RJ-MCMC model indicator proposal density for three signals. The probabilities for each model were set equivalent at the beginning of the RJ-MCMC main sampling stage (after 10.000 iterations burn-in), and are adapted towards the experienced PDF of the model indicator. We see this adaptation to reset 26 times, a fail-safe to ensure that adaptation is performed on the intrinsic PDF of interest and not towards a model in which the RJ sampler got stuck. After 26 resets (105.000 iterations) the sampler reaches the equilibrium distribution and the adaptation towards the posterior PDF finalizes.