Uncertainties in Parameters Estimated with Neural Networks: Application to Strong Gravitational Lensing
In Hezaveh et al. (2017) we showed that deep learning can be used for model parameter estimation and trained convolutional neural networks to determine the parameters of strong gravitational lensing systems. Here we demonstrate a method for obtaining the uncertainties of these parameters. We review the framework of variational inference to obtain approximate posteriors of Bayesian neural networks and apply it to a network trained to estimate the parameters of the Singular Isothermal Ellipsoid plus external shear and total flux magnification. We show that the method can capture the uncertainties due to different levels of noise in the input data, as well as training and architecturerelated errors made by the network. To evaluate the accuracy of the resulting uncertainties, we calculate the coverage probabilities of marginalized distributions for each lensing parameter. By tuning a single hyperparameter, the dropout rate, we obtain coverage probabilities approximately equal to the confidence levels for which they were calculated, resulting in accurate and precise uncertainty estimates. Our results suggest that neural networks can be a fast alternative to Monte Carlo Markov Chains for parameter uncertainty estimation in many practical applications, allowing more than seven orders of magnitude improvement in speed.
 Authors:

^{[1]}
;
^{[1]}
;
^{[1]}
 Stanford Univ. (United States). Kavli Inst. for Particle Astrophysics and Cosmology; SLAC National Accelerator Lab., Menlo Park, CA (United States)
 Publication Date:
 Grant/Contract Number:
 AC0276SF00515
 Type:
 Accepted Manuscript
 Journal Name:
 The Astrophysical Journal. Letters
 Additional Journal Information:
 Journal Volume: 850; Journal Issue: 1; Journal ID: ISSN 20418213
 Publisher:
 Institute of Physics (IOP)
 Research Org:
 SLAC National Accelerator Lab., Menlo Park, CA (United States)
 Sponsoring Org:
 USDOE
 Country of Publication:
 United States
 Language:
 English
 OSTI Identifier:
 1419653
Perreault Levasseur, Laurence, Hezaveh, Yashar D., and Wechsler, Risa H.. Uncertainties in Parameters Estimated with Neural Networks: Application to Strong Gravitational Lensing. United States: N. p.,
Web. doi:10.3847/20418213/aa9704.
Perreault Levasseur, Laurence, Hezaveh, Yashar D., & Wechsler, Risa H.. Uncertainties in Parameters Estimated with Neural Networks: Application to Strong Gravitational Lensing. United States. doi:10.3847/20418213/aa9704.
Perreault Levasseur, Laurence, Hezaveh, Yashar D., and Wechsler, Risa H.. 2017.
"Uncertainties in Parameters Estimated with Neural Networks: Application to Strong Gravitational Lensing". United States.
doi:10.3847/20418213/aa9704.
@article{osti_1419653,
title = {Uncertainties in Parameters Estimated with Neural Networks: Application to Strong Gravitational Lensing},
author = {Perreault Levasseur, Laurence and Hezaveh, Yashar D. and Wechsler, Risa H.},
abstractNote = {In Hezaveh et al. (2017) we showed that deep learning can be used for model parameter estimation and trained convolutional neural networks to determine the parameters of strong gravitational lensing systems. Here we demonstrate a method for obtaining the uncertainties of these parameters. We review the framework of variational inference to obtain approximate posteriors of Bayesian neural networks and apply it to a network trained to estimate the parameters of the Singular Isothermal Ellipsoid plus external shear and total flux magnification. We show that the method can capture the uncertainties due to different levels of noise in the input data, as well as training and architecturerelated errors made by the network. To evaluate the accuracy of the resulting uncertainties, we calculate the coverage probabilities of marginalized distributions for each lensing parameter. By tuning a single hyperparameter, the dropout rate, we obtain coverage probabilities approximately equal to the confidence levels for which they were calculated, resulting in accurate and precise uncertainty estimates. Our results suggest that neural networks can be a fast alternative to Monte Carlo Markov Chains for parameter uncertainty estimation in many practical applications, allowing more than seven orders of magnitude improvement in speed.},
doi = {10.3847/20418213/aa9704},
journal = {The Astrophysical Journal. Letters},
number = 1,
volume = 850,
place = {United States},
year = {2017},
month = {11}
}