How to estimate Fisher information matrices from simulations

TL;DR

This paper introduces alternative and combined Monte Carlo estimators for Fisher information matrices that reduce bias and improve convergence, enabling more efficient and accurate parameter constraint forecasts in complex models.

Contribution

It proposes new estimators for Fisher information that are asymptotically unbiased and can be combined with standard methods to enhance accuracy and efficiency.

Findings

The alternative estimator underestimates Fisher information, providing bounds.

The combined estimator reduces bias and improves convergence.

Methods enable fewer simulations for accurate Fisher forecasts.

Abstract

The Fisher information matrix is a quantity of fundamental importance for information geometry and asymptotic statistics. In practice, it is widely used to quickly estimate the expected information available in a data set and guide experimental design choices. In many modern applications, it is intractable to analytically compute the Fisher information and Monte Carlo methods are used instead. The standard Monte Carlo method produces estimates of the Fisher information that can be biased when the Monte-Carlo noise is non-negligible. Most problematic is noise in the derivatives as this leads to an overestimation of the available constraining power, given by the inverse Fisher information. In this work we find another simple estimate that is oppositely biased and produces an underestimate of the constraining power. This estimator can either be used to give approximate bounds on the parameter constraints or can be combined with the standard estimator to give improved, approximately unbiased estimates. Both the alternative and the combined estimators are asymptotically unbiased so can be also used as a convergence check of the standard approach. We discuss potential limitations of these estimators and provide methods to assess their reliability. These methods accelerate the convergence of Fisher forecasts, as unbiased estimates can be achieved with fewer Monte Carlo samples, and so can be used to reduce the simulated data set size by several orders of magnitude.