Use and Abuse of the Fisher Information Matrix in the Assessment of Gravitational-Wave Parameter-Estimation Prospects

The Fisher-matrix formalism is used routinely in the literature on gravitational-wave detection to characterize the parameter-estimation performance of gravitational-wave measurements, given parametrized models of the waveforms, and assuming detector noise of known colored Gaussian distribution. Unfortunately, the Fisher matrix can be a poor predictor of the amount of information obtained from typical observations, especially for waveforms with several parameters and relatively low expected signal-to-noise ratios (SNR), or for waveforms depending weakly on one or more parameters, when their priors are not taken into proper consideration. In this paper I discuss these pitfalls; show how they occur, even for relatively strong signals, with a commonly used template family for binary-inspiral waveforms; and describe practical recipes to recognize them and cope with them. Specifically, I answer the following questions: (i) What is the significance of (quasi-)singular Fisher matrices, and how must we deal with them? (ii) When is it necessary to take into account prior probability distributions for the source parameters? (iii) When is the signal-to-noise ratio high enough to believe the Fisher-matrix result? In addition, I provide general expressions for the higher-order, beyond--Fisher-matrix terms in the 1/SNR expansions for the expected parameter accuracies.Comment: 24 pages, 3 figures, previously known as "A User Manual for the Fisher Information Matrix"; final, corrected PRD versio

Prospects for ACT: simulations, power spectrum, and non-Gaussian analysis

A new generation of instruments will reveal the microwave sky at high resolution. We focus on one of these, the Atacama Cosmology Telescope, which probes scales 1000<l<10000, where both primary and secondary anisotropies are important. Including lensing, thermal and kinetic Sunyaev-Zeldovich (SZ) effects, and extragalactic point sources, we simulate the telescope's observations of the CMB in three channels, then extract the power spectra of these components in a multifrequency analysis. We present results for various cases, differing in assumed knowledge of the contaminating point sources. We find that both radio and infrared point sources are important, but can be effectively eliminated from the power spectrum given three (or more) channels and a good understanding of their frequency dependence. However, improper treatment of the scatter in the point source frequency dependence relation may introduce a large systematic bias. Even if all thermal SZ and point source effects are eliminated, the kinetic SZ effect remains and corrupts measurements of the primordial slope and amplitude on small scales. We discuss the non-Gaussianity of the one-point probability distribution function as a way to constrain the kinetic SZ effect, and we develop a method for distinguishing this effect from the CMB in a window where they overlap. This method provides an independent constraint on the variance of the CMB in that window and is complementary to the power spectrum analysis.Comment: 22 pages, 11 figures. Submitted to New Astronomy. High resolution figures provided at http://www.princeton.edu/~khuffenb/pubs/prospects-act.htm

Geometric methods for estimation of structured covariances

We consider problems of estimation of structured covariance matrices, and in particular of matrices with a Toeplitz structure. We follow a geometric viewpoint that is based on some suitable notion of distance. To this end, we overview and compare several alternatives metrics and divergence measures. We advocate a specific one which represents the Wasserstein distance between the corresponding Gaussians distributions and show that it coincides with the so-called Bures/Hellinger distance between covariance matrices as well. Most importantly, besides the physically appealing interpretation, computation of the metric requires solving a linear matrix inequality (LMI). As a consequence, computations scale nicely for problems involving large covariance matrices, and linear prior constraints on the covariance structure are easy to handle. We compare this transportation/Bures/Hellinger metric with the maximum likelihood and the Burg methods as to their performance with regard to estimation of power spectra with spectral lines on a representative case study from the literature.Comment: 12 pages, 3 figure

Quantum State Tomography of a Single Qubit: Comparison of Methods

The tomographic reconstruction of the state of a quantum-mechanical system is an essential component in the development of quantum technologies. We present an overview of different tomographic methods for determining the quantum-mechanical density matrix of a single qubit: (scaled) direct inversion, maximum likelihood estimation (MLE), minimum Fisher information distance, and Bayesian mean estimation (BME). We discuss the different prior densities in the space of density matrices, on which both MLE and BME depend, as well as ways of including experimental errors and of estimating tomography errors. As a measure of the accuracy of these methods we average the trace distance between a given density matrix and the tomographic density matrices it can give rise to through experimental measurements. We find that the BME provides the most accurate estimate of the density matrix, and suggest using either the pure-state prior, if the system is known to be in a rather pure state, or the Bures prior if any state is possible. The MLE is found to be slightly less accurate. We comment on the extrapolation of these results to larger systems.Comment: 15 pages, 4 figures, 2 tables; replaced previous figure 5 by new table I. in Journal of Modern Optics, 201

The three dimensional power spectrum of dark and luminous matter from the VIRMOS-DESCART cosmic shear survey

We present the first optimal power spectrum estimation and three dimensional deprojections for the dark and luminous matter and their cross correlations. The results are obtained using a new optimal fast estimator (Pen 2003) deprojected using minimum variance and SVD techniques. We show the resulting 3-D power spectra for dark matter and galaxies, and their covariance for the VIRMOS-DESCART weak lensing shear and galaxy data. The survey is most sensitive to nonlinear scales k_NL ~ 1 h Mpc^-1. On these scales, our 3-D power spectrum of dark matter is in good agreement with the RCS 3-D power spectrum found by (Hoekstra et al 2002). Our galaxy power is similar to that found by the 2MASS survey, and larger than that of SDSS, APM and RCS, consistent with the expected difference in galaxy population. We find an average bias b=1.24+/-0.18 for the I selected galaxies, and a cross correlation coefficient r=0.75+/-0.23. Together with the power spectra, these results optimally encode the entire two point information about dark matter and galaxies, including galaxy-galaxy lensing. We address some of the implications regarding galaxy halos and mass-to-light ratios. The best fit ``halo'' parameter h=r/b=0.57+/-0.16, suggesting that dynamical masses estimated using galaxies systematically underestimate total mass. Ongoing surveys, such as the Canada-France-Hawaii-Telescope-Legacy survey will significantly improve on the dynamic range, and future photometric redshift catalogs will allow tomography along the same principles.Comment: 17 pages, 19 figures, submitted to mnra

Efficient semiparametric estimation and model selection for multidimensional mixtures

In this paper, we consider nonparametric multidimensional finite mixture models and we are interested in the semiparametric estimation of the population weights. Here, the i.i.d. observations are assumed to have at least three components which are independent given the population. We approximate the semiparametric model by projecting the conditional distributions on step functions associated to some partition. Our first main result is that if we refine the partition slowly enough, the associated sequence of maximum likelihood estimators of the weights is asymptotically efficient, and the posterior distribution of the weights, when using a Bayesian procedure, satisfies a semiparametric Bernstein von Mises theorem. We then propose a cross-validation like procedure to select the partition in a finite horizon. Our second main result is that the proposed procedure satisfies an oracle inequality. Numerical experiments on simulated data illustrate our theoretical results

Myths and Truths Concerning Estimation of Power Spectra

It is widely believed that maximum likelihood estimators must be used to provide optimal estimates of power spectra. Since such estimators require require of order N_d^3 operations they are computationally prohibitive for N_d greater than a few tens of thousands. Because of this, a large and inhomogeneous literature exists on approximate methods of power spectrum estimation. These range from manifestly sub-optimal, but computationally fast methods, to near optimal but computationally expensive methods. Furthermore, much of this literature concentrates on the power spectrum estimates rather than the equally important problem of deriving an accurate covariance matrix. In this paper, I consider the problem of estimating the power spectrum of cosmic microwave background (CMB) anisotropies from large data sets. Various analytic results on power spectrum estimators are derived, or collated from the literature, and tested against numerical simulations. An unbiased hybrid estimator is proposed that combines a maximum likelihood estimator at low multipoles and pseudo-C_\ell estimates at high multipoles. The hybrid estimator is computationally fast, nearly optimal over the full range of multipoles, and returns an accurate and nearly diagonal covariance matrix for realistic experimental configurations (provided certain conditions on the noise properties of the experiment are satisfied). It is argued that, in practice, computationally expensive methods that approximate the N_d^3 maximum likelihood solution are unlikely to improve on the hybrid estimator, and may actually perform worse. The results presented here can be generalised to CMB polarization and to power spectrum estimation using other types of data, such as galaxy clustering and weak gravitational lensing.Comment: 27 pages, 15 figures, MNRAS in press. Resubmission matches accepted versio

On the non-existence of unbiased estimators in constrained estimation problems

We address the problem of existence of unbiased constrained parameter estimators. We show that if the constrained set of parameters is compact and the hypothesized distributions are absolutely continuous with respect to one another, then there exists no unbiased estimator.Weaker conditions for the absence of unbiased constrained estimators are also specified. We provide several examples which demonstrate the utility of these conditions

Exploring Estimator Bias-Variance Tradeoffs Using the Uniform CR Bound

We introduce a plane, which we call the delta-sigma plane, that is indexed by the norm of the estimator bias gradient and the variance of the estimator. The norm of the bias gradient is related to the maximum variation in the estimator bias function over a neighborhood of parameter space. Using a uniform Cramer-Rao (CR) bound on estimator variance, a delta-sigma tradeoff curve is specified that defines an “unachievable region” of the delta-sigma plane for a specified statistical model. In order to place an estimator on this plane for comparison with the delta-sigma tradeoff curve, the estimator variance, bias gradient, and bias gradient norm must be evaluated. We present a simple and accurate method for experimentally determining the bias gradient norm based on applying a bootstrap estimator to a sample mean constructed from the gradient of the log-likelihood. We demonstrate the methods developed in this paper for linear Gaussian and nonlinear Poisson inverse problems.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/86001/1/Fessler98.pd