Analytic methods for cosmological likelihoods

We present general, analytic methods for cosmological likelihood analysis and solve the ‘many parameters’ problem in cosmology. Maxima are found by Newton's method, while marginalization over nuisance parameters, and parameter errors and covariances are estimated by analytic marginalization of an arbitrary likelihood function, expanding the log-likelihood to second order, with flat or Gaussian priors. We show that information about remaining parameters is preserved by marginalization. Marginalizing over all parameters, we find an analytic expression for the Bayesian evidence for model selection. We apply these methods to data described by Gaussian likelihoods with parameters in the mean and covariance. These methods can speed up conventional likelihood analysis by orders of magnitude when combined with Markov chain Monte Carlo methods, while Bayesian model selection becomes effectively instantaneous.