Abstract
Fisher matrices play an important role in experimental design and in data analysis. Their primary role is to make predictions for the inference of model parameters—both their errors and covariances. In this short review, I outline a number of extensions to the simple Fisher matrix formalism, covering a number of recent developments in the field. These are: (a) situations where the data (in the form of ( x , y ) pairs) have errors in both x and y; (b) modifications to parameter inference in the presence of systematic errors, or through fixing the values of some model parameters; (c) Derivative Approximation for LIkelihoods (DALI) - higher-order expansions of the likelihood surface, going beyond the Gaussian shape approximation; (d) extensions of the Fisher-like formalism, to treat model selection problems with Bayesian evidence.
Highlights
Fisher information matrices are widely used for making predictions for the errors and covariances of parameter estimates
Fisher matrices have been extensively used in cosmology, where future experiments have been designed in order to deduce as precisely as possible the parameters of the standard cosmological model, so-called ΛCDM (Cold Dark Matter, with a cosmological constant Λ), and are routinely used to give “figures-of-merit” [1] for the power of each experiment
At the root of the Fisher matrix formalism is the Laplace approximation, i.e., the assumption that the likelihood surface is a multivariate Gaussian when viewed as a function of the model parameters
Summary
Fisher information matrices are widely used for making predictions for the errors and covariances of parameter estimates. Fisher matrices have been extensively used in cosmology, where future experiments have been designed in order to deduce as precisely as possible the parameters of the standard cosmological model, so-called ΛCDM (Cold Dark Matter, with a cosmological constant Λ), and are routinely used to give “figures-of-merit” [1] for the power of each experiment. These studies are standard applications of Fisher matrix theory, often simplified by an approximation (which is very good for observations of the Early Universe) that the data are Gaussian-distributed.
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