Bayesian bounds on parameter estimation accuracy for compact coalescing binary gravitational wave signals

A global network of very sensitive large-scale laser interferometric gravitational wave detectors is projected to be in operation by around the turn of the century. The network is anticipated to bring a range of new astrophysical information — relating to neutron stars, black holes, and the very early universe — and also new fundamental physics information, relating to the nature of gravity in the strongly nonlinear regime for example. This information is borne by gravitational waves that will typically be very much weaker than the level of intrinsic strain noise in the detectors. Sophisticated signal extraction methods will therefore be required to analyze the network’s data. Here, the noisy output of a single laser interferometric detector is examined. A gravitational wave is assumed to have been detected in the data. This paper is concerned only with the subsequent problem of parameter estimation. Specifically, we investigate theoretical lower bounds on the minimum mean-square errors (MSE) associated with measuring the parameters that characterize the waveform. The pre-merger inspiral waveform generated by an orbiting system of neutron stars or black holes is ideal for this study. Monte Carlo measurements of the parameters of noisy inspiral waveforms have been performed elsewhere, and the results must now confront statistical signal processing theory. Three theoretical lower bounds on parameter estimation accuracy are considered here: the Cramer-Rao bound (CRB); the Weiss-Weinstein bound (WWB); and the Ziv-Zakai bound (ZZB). The CRB is the simplest and most well-known of these bounds, but suffers from a number of limitations. It has been applied a number of times already to bound gravitational wave measurement errors. The WWB and ZZB on the other hand are computationally less simple, and we apply them here to gravitational wave parameter estimation for the very first time. The CRB is known as a local bound because it assumes that the parameters one seeks to estimate are deterministic, and provides bounds on their MSE for every possible set of intrinsic parameter values. The WWB and ZZB are known as global (Bayesian) bounds because they assume that the parameter set is random, of known prior distribution. They bound the global MSE averaged over this prior distribution. We first set up a model problem in order to develop intuition about the conditions under which global bounds are more appropriate than their local counterparts. Then we obtain the WWB and ZZB for the Newtonian-form of the coalescing binary waveform, and compare them with published analytic CRB and numerical Monte Carlo results. At large signal-to-noise ratio (SNR), we find that the theoretical bounds are all identical and are attained by the Monte Carlo results. As SNR gradually drops below $\sim 10$, the WWB and ZZB are both found to provide increasingly tighter lower bounds than the CRB. However, at these levels of moderate SNR, there is a significant departure between all the bounds and the numerical Monte Carlo results. We argue that the WWB and ZZB are probably within a few percent of the theoretical minimum MSE attainable for this problem. The implication is that the maximum likelihood method of parameter estimation used by the Monte Carlo simulations is not the optimal estimator for this problem at low-to-moderate SNR. In fact, it is well-known that the optimal parameter estimator is the conditional mean estimator. This, unfortunately, is notoriously difficult to compute in general. We therefore advance a strategy for implementing this method efficiently, as a post-processor to the maximum likelihood estimator, in order to achieve improved accuracy in parameter estimation.