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.

• The paper introduces the novel floZ method that employs normalizing flows to directly compute Bayesian evidence from unnormalized posterior samples.
• It utilizes neural network transformations and a sequence of loss functions to accurately model the posterior and reduce variance in evidence estimates.
• Validation against benchmarks up to 15 dimensions demonstrates enhanced computational efficiency and robust performance over traditional sampling methods.

Enhancing Bayesian Evidence Estimation Using Normalizing Flows

Introduction to Bayesian Evidence Estimation Challenges

Bayesian evidence plays a pivotal role in model selection and hypothesis testing within the scientific data analysis framework. Calculating this evidence, however, remains computationally intensive, especially in high-dimensional scenarios where traditional methods like nested sampling or analytical approximations struggle either due to the curse of dimensionality or the complexity of the distribution function.

Overview of the Proposed Method "floZ"

The paper presents a novel approach, termed "floZ," leveraging normalizing flows for the estimation of Bayesian evidence from samples drawn from the unnormalized posterior distribution. This method builds on the idea of transforming a complex posterior distribution into a simpler base distribution using a bijective and differentiable transformation modeled by a neural network, facilitating a direct computation of the Bayesian evidence.

Normalizing Flows and Evidence Calculation

The core concept behind normalizing flows involves the mapping of a complicated target probability function (the product of a likelihood and a prior) into a simpler probability distribution using a neural network. The Jacobian of the transformation encapsulates the alteration in density, which, imperatively, includes the Bayesian evidence.

Algorithmic Implementation and Loss Function Optimization

The neural network employs a sequence of loss functions to encourage an accurate modeling of the posterior distribution and to minimize the variance of evidence estimates:

• General feature learning of the posterior distribution: Initially focused on matching the target and model distributions.
• Reduction of variance in evidence estimation: This phase corrects the variability across different sample estimations of the evidence.
• Optimization phases: Includes transitions between loss functions to smooth out the learning process and improve robustness and convergence.

Validation and Benchmarking Results

The method was thoroughly validated against known benchmarks, including multivariate Gaussians and Gaussian mixtures, across various dimensional settings up to fifteen dimensions. Comparisons drawn against traditional nested sampling and a $k$-nearest neighbors method showcased "floZ's" superior ability to reliably compute the Bayesian evidence, particularly highlighting its robustness in handling complex, multi-modal distributions.

Computational Efficiency

Although "floZ" requires pre-calculated posterior samples which presupposes an existing sampling process like MCMC, it offers a computationally efficient post-processing step to compute Bayesian evidence. This is particularly advantageous when the direct nested sampling is computationally expensive.

Conclusions and Future Work

The introduction of "floZ" paves the way for more efficient Bayesian model comparison, especially in scenarios dealing with high-dimensional parameter spaces where standard methods falter. The method's reliance on previously generated samples also makes it a versatile tool in scenarios where sample generation is feasible but direct evidence computation is not. Further research would explore extending this method to even higher dimensions leveraging advances in optimization and neural network architectures, potentially expanding its applicability across various scientific domains requiring robust evidence computation for model selection.