  The planning and design of future experiments rely heavily on forecasting to
assess the potential scientific value provided by a hypothetical set of
measurements. The Fisher information matrix, due to its convenient properties
and low computational cost, provides an especially useful forecasting tool.
However, the Fisher matrix only provides a reasonable approximation to the true
likelihood when data are nearly Gaussian distributed and observables have
nearly linear dependence on the parameters of interest. Also, Fisher
forecasting techniques alone cannot be used to assess their own validity.
Thorough sampling of the exact or mock likelihood can definitively determine
whether a Fisher forecast is valid, though such sampling is often prohibitively
expensive. We propose a simple test, based on the Derivative Approximation for
LIkelihoods (DALI) technique, to determine whether the Fisher matrix provides a
good approximation to the exact likelihood. We show that the Fisher matrix
becomes a poor approximation to the true likelihood in regions where
two-dimensional slices of level surfaces of the DALI approximation to the
likelihood differ from two-dimensional slices of level surfaces of the Fisher
approximation to the likelihood. We demonstrate that our method accurately
predicts situations in which the Fisher approximation deviates from the true
likelihood for various cosmological models and several data combinations, with
only a modest increase in computational cost compared to standard Fisher
forecasts.
