Parameter inference and model comparison using theoretical predictions from noisy simulations

TL;DR

This paper introduces a Bayesian hierarchical framework to correct likelihood estimates when using noisy simulated data for parameter inference and model comparison, improving accuracy and reducing simulation needs.

Contribution

It presents a novel correction to likelihood functions that accounts for variance in estimated summary statistics, especially for Gaussian likelihoods with simulated covariance.

Findings

Corrects likelihood estimates in noisy simulation scenarios.

Demonstrates impact on Bayesian evidence with JLA data.

Reduces the number of simulations needed for accurate inference.

Abstract

When inferring unknown parameters or comparing different models, data must be compared to underlying theory. Even if a model has no closed-form solution to derive summary statistics, it is often still possible to simulate mock data in order to generate theoretical predictions. For realistic simulations of noisy data, this is identical to drawing realizations of the data from a likelihood distribution. Though the estimated summary statistic from simulated data vectors may be unbiased, the estimator has variance which should be accounted for. We show how to correct the likelihood in the presence of an estimated summary statistic by marginalizing over the true summary statistic in the framework of a Bayesian hierarchical model. For Gaussian likelihoods where the covariance must also be estimated from simulations, we present an alteration to the Sellentin-Heavens corrected likelihood. We show that excluding the proposed correction leads to an incorrect estimate of the Bayesian evidence with JLA data. The correction is highly relevant for cosmological inference that relies on simulated data for theory (e.g. weak lensing peak statistics and simulated power spectra) and can reduce the number of simulations required.