Abstract
The Fisher Information Matrix formalism (Fisher 1935) is extended to cases where the data are divided into two parts (X, Y), where the expectation value of Y depends on X according to some theoretical model, and X and Y both have errors with arbitrary covariance. In the simplest case, (X, Y) represent data pairs of abscissa and ordinate, in which case the analysis deals with the case of data pairs with errors in both coordinates, but X can be any measured quantities on which Y depends. The analysis applies for arbitrary covariance, provided all errors are Gaussian, and provided the errors in X are small, both in comparison with the scale over which the expected signal Y changes, and with the width of the prior distribution. This generalizes the Fisher Matrix approach, which normally only considers errors in the `ordinate’ Y. In this work, we include errors in X by marginalizing over latent variables, effectively employing a Bayesian hierarchical model, and deriving the Fisher Matrix for this more general case. The methods here also extend to likelihood surfaces which are not Gaussian in the parameter space, and so techniques such as DALI (Derivative Approximation for Likelihoods) can be generalized straightforwardly to include arbitrary Gaussian data error covariances. For simple mock data and theoretical models, we compare to Markov Chain Monte Carlo experiments, illustrating the method with cosmological supernova data. We also include the new method in the FISHER4CAST software.
Author
Heavens, A. F.; Seikel, M.; Nord, B. D.; Aich, M.; Bouffanais, Y.; Bassett, B. A.; Hobson, M. P.
Journal
Monthly Notices of the Royal Astronomical Society
Paper Publication Date
January 2014
Paper Type
Astrostatistics