Neural Ratio Estimation

Abstract

The central topic of this project is the application of Neural Ratio Estimation (NRE) in the search for axion-like particles (ALPs) using cosmic gamma-rays. The existence of ALPs would induce distortions in the gamma-ray spectra of astrophysical sources, such as spectral attenuation or irregular features caused by photon-ALP oscillations in external magnetic fields. By comparing experimental data with theoretical models, researchers infer model parameters that, in turn, describe the properties of the ALPs under study. Traditional likelihood-based methods struggle with this problem due to the high-dimensional parameter space and the computationally expensive evaluation of likelihood functions. NRE circumvents this challenge by leveraging neural networks to estimate the likelihood ratio, enabling efficient Bayesian inference of ALP parameters. This research was conducted by Heidi Sandaker¡¯s group with dSAG providing expertise with HPC implementation and troubleshooting as well as feedback on designing, fine-tuning and evaluating the machine learning approach.

Background

For more background information on this project visit the following links:

Cherenkov Telescope Array - project and project draft.

Methodology

Neural Ratio Estimation (NRE) is a class of likelihood-free inference techniques that approximates the likelihood ratio between two probability distributions using a neural network with the aim of estimating the posterior probability density function (pdf). Let  x¡ÊX be the observed data, sampled from a model with parameters ¦È¡ÊRd. Given the prior pdf of the parameters p(¦È) the posterior pdf p(¦È|x) can be calculated from Bayes¡¯ theorem:

p(¦Èx= p(¦È)p(x|¦È)p(x) 

However, in complex high-dimensional problems, evaluating the likelihood function p(¦È|x) directly becomes computationally infeasible due to expensive forward simulations. Likelihood-free inference methods like NRE address this by estimating the likelihood ratio instead of the likelihood:

r(x|¦È)= p(x|¦È)p(x) 

This allows posterior inference without ever computing the full likelihood function:

p(¦È|x)=p(¦È) r(x|¦È)  

Practically, the likelihood ratio r(x|¦È) can be estimated using a binary classification, where the goal is to distinguish correctly paired (data, parameter) samples from randomly paired ones, effectively learning the likelihood ratio. This can be achieved by training a logistic regression or a neural network classifier on the sampled data. The classifier learns the likelihood ratio and therefore is able to approximate the posterior odds.

For further reading on this topic Likelihood-free inference by ratio estimation (arXiv) or A neural network approach to likelihood-free inference (Semantic Scholar).

In this project, the neural network was trained on simulated gamma-ray spectra, represented as binned energy distributions, with ALP parameters ?, (mass), ? (photon coupling), and additional nuisance parameters. The NRE and neural network implementation were performed using SWYFT.

 The figure below (taken from here) illustrates the convergence of the posterior estimation of the parameters with the true values that were used for simulation of the observed data.

The convergence of the posterior pdf around the true values demonstrates that NRE can successfully infer ALP parameters from simulated gamma-ray spectra, offering a promising approach for likelihood-free inference in high-dimensional problems. A particular achievement is that the necessary computations can be performed without neglecting the large uncertainties of the model¡¯s many nuisance parameters, thus avoiding a significant compromise on the analysis¡¯ credibility. Systematic uncertainties were neglected for this study but are, in principle, straight-forward to include in future work. The resulting posteriors can be validated using coverage tests, which indicate no significant discrepancy between empirical and expected coverage. However, this does not conclusively demonstrate the correctness of the inferred posteriors, and we encourage more research in the direction of validating the results of likelihood-free inference approaches such as these