Uncertainties in Parameters Estimated with Neural Networks: Application to Strong Gravitational Lensing

1. INTRODUCTION

• Neural networks learn relationships from examples by applying weighted transformations and nonlinear functions across layers, with training optimizing weights against target outputs.Source: §1 · p. 1
• Bayesian neural networks replace fixed weights with probability distributions, allowing output posterior probabilities and well-defined uncertainty estimates.Inferring the model posterior is difficult, motivating approximate methods.Source: §1 · p. 1
• The authors extend convolutional neural networks for strong-lens parameter estimation to uncertainty estimation, drawing on variational-inference methods for Bayesian neural networks.The paper applies the reviewed framework to strong-lensing systems.Source: §1 · p. 1

2.2. Aleatoric Uncertainties

• Aleatoric uncertainty represents input corruption, and a network can predict an observation-noise parameter for each input when noise levels vary across data.The network’s final layer predicts both parameters of interest and their associated noise estimates, learned by optimizing the likelihood without uncertainty labels.Source: §2 · p. 1, §2.2 · p. 2

2. OBTAINING MODEL UNCERTAINTIES IN NEURAL NETWORKS

• Neural-network parameter uncertainty has aleatoric and epistemic sources: input corruption limits the former, while training or network errors contribute to the latter.More training or network flexibility can reduce epistemic uncertainty; the section reviews variational inference as an efficient approximation framework.Source: §2 · p. 1

2.1. Epistemic Uncertainties in Neural Networks

• Epistemic uncertainty is approximated by replacing the intractable weight posterior with a variational distribution and optimizing it through the log-evidence lower bound.A Bernoulli variational distribution makes Monte Carlo sampling equivalent to dropout; repeated dropout forward passes provide approximate posterior samples at test time.Source: §2.1 · p. 2

2.3. Combining Aleatoric and Epistemic Uncertainties

• Total uncertainty combines epistemic posterior samples with aleatoric noise by adding Gaussian draws with variance σ_k^2 to each prediction.The resulting parameter samples include both uncertainty sources; Figure 1 presents predicted 68.3% uncertainties for flux magnification against its true value.Source: §2.3 · p. 3, Figure 1 · p. 3

3. APPLICATION TO STRONG GRAVITATIONAL LENSING

• AlexNet predicts eight SIE-plus-shear lens parameters and log-variance outputs, using dropout before weight layers on simulated data with shear amplitudes up to 0.3.The model also predicts total flux magnification; log-variance is used for numerical stability and to avoid potential division by zero.Source: §3 · p. 3

3.1. Tests on the accuracy of the combined uncertainties

• For each of 1,000 unseen simulated examples, the network makes 2,000 dropout-based predictions, and Gaussian noise is added to combine aleatoric and epistemic uncertainty samples.Coverage is assessed by checking how often true parameter values fall within the corresponding confidence intervals.Source: §3 · p. 3, §3.1 · p. 3
• With a 97% dropout keep rate, average coverage probabilities closely match their confidence levels; lower keep rates overestimate errors, while 99% yields mildly permissive uncertainties.The authors recommend tuning dropout rate as a model hyperparameter for accurate uncertainty estimates.Source: §3.1 · p. 4, Figure 2 · p. 4
• Individual-parameter coverage probabilities are sufficiently close to ideal values for most practical uses, although individual parameters can deviate from their targets.Table 1 reports coverage probabilities by lensing parameter and median standard deviation as a measure of precision.Source: §3.1 · p. 4, Table 1 · p. 4
• Estimated uncertainties increase as input noise grows, including at noise levels above the training range of rms less than 0.1.The evaluation also reports Einstein-radius coverage probabilities of 74.7, 96.8, and 99.5% at maximum per-pixel SNR 100, versus 71.0, 95.9, and 99.4% at SNR 10.Source: §3.1 · p. 4, Figure 3 · p. 5
• At similar noise levels, configurations close to Einstein rings have more precise estimates than configurations with only a pair of compact images.Figure 4 shows representative test images with increasing average uncertainty across parameters.Source: §3.1 · p. 4, Figure 4 · p. 5

• Table 1 demonstrates accurate interval estimates for lensing parameters with precision comparable to traditional lens modeling methods.Source: §4 · p. 5
• A few hundred posterior samples take seconds on one GPU, yielding more than seven orders of magnitude improvement in speed over traditional modeling.Source: §4 · p. 5
• Different dropout rates define different variational distributions and prior strengths, so tuning the rate can produce more accurate uncertainty estimates.Source: §4 · p. 5
• Observation noise σ also adjusts to network errors, while epistemic uncertainty captures error in the network’s own uncertainty estimation.Source: §4 · p. 5
• Even when averaged coverage matches confidence levels, individual parameter probabilities may deviate slightly; separate branches and dropout rates may improve marginalized uncertainties.Source: §4 · p. 5
• Although aleatoric uncertainties are Gaussian, epistemic contributions can make total distributions highly non-Gaussian; diagonal σ yields marginals, while joint distributions require off-diagonal elements.Source: §4 · p. 5
• Neural networks suit large data samples and complex models where exploring parameter space with maximum-likelihood methods could be slow or intractable.Source: §4 · p. 6