Generalised Fisher Matrices
  The Fisher Information Matrix formalism is extended to cases where the data
is 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 generalises the Fisher Matrix approach, which
normally only considers errors in the `ordinate' Y. In this work, we include
errors in X by marginalising 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 generalised 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.
