Bayesian evidence estimation from posterior samples with normalizing flows

Abstract

We propose a novel method ($floZ$), based on normalizing flows, to estimate the Bayesian evidence (and its numerical uncertainty) from a pre-existing set of samples drawn from the unnormalized posterior distribution. We validate it on distributions whose evidence is known analytically, up to 15 parameter space dimensions, and compare with two state-of-the-art techniques for estimating the evidence: nested sampling (which computes the evidence as its main target) and a $k$-nearest-neighbors technique that produces evidence estimates from posterior samples. Provided representative samples from the target posterior are available, our method is more robust to posterior distributions with sharp features, especially in higher dimensions. For a simple multivariate Gaussian, we demonstrate its accuracy for up to 200 dimensions with $10^5$ posterior samples. $floZ$ has wide applicability, e.g., to estimate evidence from variational inference, Markov Chain Monte Carlo samples, or any other method that delivers samples and their likelihood from the unnormalized posterior density. As a physical application, we use $floZ$ to compute the Bayes factor for the presence of the first overtone in the ringdown signal of the gravitational wave data of GW150914, finding good agreement with nested sampling.

15 pages, 8 figures, 1 table

References in corpus (41)

• MultiNest: an efficient and robust Bayesian inference tool for cosmology and particle physics
• Tests of general relativity with GW150914
• The NANOGrav 15-year Data Set: Evidence for a Gravitational-Wave Background
• Bayes in the sky: Bayesian inference and model selection in cosmology
• Search for an isotropic gravitational-wave background with the Parkes Pulsar Timing Array
• Searching for the nano-Hertz stochastic gravitational wave background with the Chinese Pulsar Timing Array Data Release I
• The second data release from the European Pulsar Timing Array III. Search for gravitational wave signals
• On gravitational-wave spectroscopy of massive black holes with the space interferometer LISA
• PolyChord: next-generation nested sampling
• GW150914: First results from the search for binary black hole coalescence with Advanced LIGO
• Quasinormal Modes, the Area Spectrum, and Black Hole Entropy
• Testing the no-hair theorem with GW150914
• Applications of Bayesian model selection to cosmological parameters
• Dynamic nested sampling: an improved algorithm for parameter estimation and evidence calculation
• The Best Inflationary Models After Planck
• Real-time gravitational-wave science with neural posterior estimation
• Observational Black Hole Spectroscopy: A time-domain multimode analysis of GW150914
• No evidence for extensions to the standard cosmological model
• Gravitational-wave parameter estimation with autoregressive neural network flows
• Analysis of Ringdown Overtones in GW150914
• HELIOS-Retrieval: An Open-source, Nested Sampling Atmospheric Retrieval Code, Application to the HR 8799 Exoplanets and Inferred Constraints for Planet Formation
• Nested Sampling with Normalising Flows for Gravitational-Wave Inference
• Testing General Relativity using Bayesian model selection: Applications to observations of gravitational waves from compact binary systems
• Neural Importance Sampling for Rapid and Reliable Gravitational-Wave Inference
• Nested Sampling Methods
• The Indian Pulsar Timing Array: First data release
• Using rational filters to uncover the first ringdown overtone in GW150914
• Reply to Comment on "Analysis of Ringdown Overtones in GW150914"
• Objective Bayesian analysis of neutrino masses and hierarchy
• Neural Posterior Estimation with guaranteed exact coverage: the ringdown of GW150914
• Comment on "Analysis of Ringdown Overtones in GW150914''
• Retrieval Study of Brown Dwarfs Across the L-T Sequence
• Accelerated Bayesian inference using deep learning
• Gravitational wave populations and cosmology with neural posterior estimation
• Bayesian inference for pulsar timing models
• Adapting to noise distribution shifts in flow-based gravitational-wave inference
• Proximal nested sampling for high-dimensional Bayesian model selection
• Bayesian model comparison for simulation-based inference
• Effect of Noise Estimation in Time-Domain Ringdown Analysis: A Case Study with GW150914
• Evidence Networks: simple losses for fast, amortized, neural Bayesian model comparison
• Improved Laplace Approximation for Marginal Likelihoods

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