Learning Bayes' theorem with a neural network for gravitational-wave inference

In the Bayesian analysis of signals immersed in noise [1], we seek a representation for the posterior probability of one or more parameters that govern the shape of the signals. Unless the parameter-to-signal map (the forward model) is very simple, the analysis (or inverse solution) comes at significant computational cost, as it requires the stochastic exploration of the likelihood surface at a large number of locations in parameter space. Such is the case, for instance, of parameter estimation for gravitational-wave sources such as the compact binaries detected by LIGO-Virgo [2, 3]; here each likelihood evaluation requires that we generate the gravitational waveform corresponding to a set of source parameters, and compute its noise-weighted correlation with detector data [4]. Waveform generation is usually the costlier operation, so gravitational-wave analysts often utilize faster, less accurate waveform models [5, 6], or accelerated surrogates of slower, more accurate models [7]. Extending the analysis from the data we have to the data we might measure (i.e., characterizing the parameter-estimation prospects of future experiments) compounds the expense, since we need to explore posteriors for many noise realizations, and across the domain of possible source parameters. For concreteness, we price the evaluation of a single Bayesian posterior at null 10 6 times the cost of generating a waveform, and the characterization of parameter-estimation prospects at null 10 6 times the cost of a posterior. With current computational resources, this means that (for instance) accurate component-mass estimates only become available hours or days after the detection of a binary black-hole coalescence [8, 9], while any extensive study of parameter-estimation prospects must rely on less reliable techniques such as the Fisher-matrix approximation [10]. In this Letter, we show how one-or two-dimensional marginalized Bayesian posteriors may be produced using deep neural networks [11] trained on large ensembles of signal noise data streams.