Source-linked AI summary
Bayesian parameter estimation using conditional variational autoencoders for gravitational-wave astronomy
Hunter Gabbard, Chris Messenger, Ik Siong Heng, Francesco Tonolini, Roderick Murray-Smith
TL;DR
The paper addresses obtaining comprehensive joint posterior samples for gravitational-wave signals without explicitly calculating the likelihood. It uses a conditional variational autoencoder and evaluates posterior calibration and sampler similarity, while noting parameterization limits for higher-order modes.
Problem
Standard sampling algorithms require explicit likelihood calculations and an analytic noise and signal model, motivating alternatives that can generate comprehensive joint posterior samples.
Method
A conditional variational autoencoder uses a GW time-series observation, recognition function, and convolutional neural networks to model and sample the posterior.
Results
Posterior estimates are assessed against Bilby using p-p plots, JS-divergence, and convergence of the training cost function.
Takeaways & Limitations
The approach requires sampling from the likelihood rather than calculating it explicitly, potentially allowing training with real non-Gaussian detector noise.
Takeaways & Limitations
The chosen parameterization is not appropriate for higher-order modes because the φ0, ψ degeneracy is broken.
Abstract
from arXiv · showhide
Gravitational wave (GW) detection is now commonplace and as the sensitivity of the global network of GW detectors improves, we will observe $\mathcal{O}(100)$s of transient GW events per year. The current methods used to estimate their source parameters employ optimally sensitive but computationally costly Bayesian inference approaches where typical analyses have taken between 6 hours and 5 days. For binary neutron star and neutron star black hole systems prompt counterpart electromagnetic (EM) signatures are expected on timescales of 1 second -- 1 minute and the current fastest method for alerting EM follow-up observers, can provide estimates in $\mathcal{O}(1)$ minute, on a limited range of key source parameters. Here we show that a conditional variational autoencoder pre-trained on binary black hole signals can return Bayesian posterior probability estimates. The training procedure need only be performed once for a given prior parameter space and the resulting trained machine can then generate samples describing the posterior distribution $\sim 6$ orders of magnitude faster than existing techniques.
METHODS
A conditional variational autoencoder conditions on GW observations and uses encoder–decoder networks to learn a lower-dimensional latent representation for posterior inference.
- METHODS: A CVAE conditions a variational autoencoder on a one-dimensional GW time-series signal y.Its encoder and decoder learn a distilled representation through a latent-space bottleneck.
- METHODS: The encoder maps inputs into a typically lower-dimensional latent space, while the decoder reconstructs the original input representation.Training teaches both networks to represent the dataset efficiently in latent space.
- METHODS: Unlike a standard autoencoder, the variational encoder outputs distribution parameters and samples latent locations before decoding.This introduces variation into the latent representation.
- METHODS: The VItamin implementation uses deep convolutional neural networks for both encoder and decoder components.
Cost function derivation
The method trains neural networks to approximate the true conditional posterior by minimizing a computable evidence lower bound based on reconstruction and KL-divergence terms.
- Cost function derivation: The target is the true posterior p(x|y), approximated by a trainable conditional distribution rθ(x|y).The parameters θ are trainable neural-network parameters.
- Cost function derivation: Minimizing cross-entropy makes the parametric distribution rθ(x|y) similar to the target posterior for possible measurements y.
- Cost function derivation: The model combines encoder rθ1(z|y) and decoder rθ2(x|z,y) distributions, marginalized over the latent variable z.The encoder maps data to a latent distribution, and the decoder maps latent samples plus data to a physical-parameter distribution.
- Cost function derivation: Direct optimization is intractable, so a recognition network qφ(z|x,y) is introduced to derive an evidence lower bound.
- Cost function derivation: The cost is approximated stochastically using draws from the prior, likelihood, and recognition function, requiring likelihood sampling rather than explicit likelihood evaluation.This permits future training with real non-Gaussian detector noise.
Network design
The network design uses three convolutional neural networks, specialized latent and output distributions, and parameter transformations tailored to GW signal geometry.
- Network design: The CVAE comprises three neural networks modeling rθ1, rθ2, and qφ, with convolutional layers shared across the networks.
- Network design: The rθ1 encoder maps detector time-series data to a 15-dimensional latent Gaussian mixture with 32 components.
- Network design: The qφ network receives y and true parameters x, appending x after convolutional processing, and outputs a single-modal 15-dimensional Gaussian.Conditioning on x removes ambiguity about the latent mode for each time-series.
- Network design: The decoder uses truncated Gaussians for bounded physical parameters and specialized distributions for cyclic and sky-location parameters.These choices respect predefined parameter boundaries and periodic geometry.
- Network design: The phase and polarization angle are reparameterized to reduce the number of modes seen by the network, then converted back after sampling.
- Network design: The phase–polarization reparameterization is unsuitable for higher-order modes because their degeneracy is broken.
Training procedure
Training feeds paired noisy GW time-series and true parameters through three networks, optimizes reconstruction and KL losses, and uses annealing and large-scale augmentation to stabilize learning.
- Training procedure: Each training batch contains time-series y paired with corresponding true GW parameters x.The batch is processed through the encoder, latent sampling, decoder, and loss calculations.
- Training procedure: The qφ encoder uses y and x to define a diagonal latent Gaussian, while rθ1 uses only y to define a Gaussian mixture.
- Training procedure: A latent sample from qφ and its y data are passed to rθ2, whose output defines a distribution in physical parameter space.
- Training procedure: The reconstruction loss evaluates the decoder density at the true training parameters, while KL divergence compares qφ and rθ1.
- Training procedure: The KL term is gradually increased from 0 to 1 because early training can trap the network with decreasing reconstruction cost and nearly zero KL divergence.
- Training procedure: Approximately 3×10^4 epochs, batch size 1500, learning rate 10^-4, and 10^7 training samples were used for BBH parameter-space coverage.Unique noise realizations and parameter shifts augment the finite waveform set.
The testing procedure
After training, VItamin converts new data into posterior samples by repeatedly sampling a latent representation and decoding it into physical parameters.
- A new data sample y is passed through the trained encoder to obtain a latent-space distribution description.
- A latent sample is randomly drawn from that distribution and combined with the original data as decoder input.
- The decoder returns a multivariate physical-parameter distribution, from which a random parameter realization x is drawn.
- Repeating the procedure produces samples from the entire joint posterior represented by the latent model.
- Across 250 test datasets, VItamin’s best- and worst-case p-values were 0.918 and 0.047.
Additional tests
Additional tests assess posterior calibration with p-p plots and distributional similarity with JS-divergence, while documenting benchmark sampler configurations.
- P-p plots: P-p plots compare cumulative posterior probabilities at true simulation values against the ideal diagonal across 250 test datasets.Curves consistent with the diagonal indicate Bayesian distributions consistent with the corresponding frequentist coverage interpretation.
- Benchmark configurations: Benchmark sampler configuration values were selected using recommended defaults and private communication with the Bilby development team.
- JS-divergence: JS-divergence compares one-dimensional posterior distributions produced by every sampler on the same test data.Identical distributions have JS-divergence zero in the ideal finite-sample limit, although finite samples yield nonzero values.
- JS-divergence: Dynesty-versus-VItamin JS values were competitive with Dynesty-versus-ptemcee for nearly all parameters, except φ12 and ψ.