Likelihood-Free Inference in Cosmology: Potential for the Estimation of Luminosity Functions

Statistical inference of cosmological quantities of interest is complicated by significant observational limitations, including heteroscedastic measurement error and irregular selection effects. These observational difficulties exacerbate challenges posed by the often-complex relationship between estimands and the distribution of observables; indeed, in some situations it is only possible to simulate realizations of observations under various assumed cosmological theories. When faced with these challenges, one is naturally led to consider utilizing repeated simulations of the full data generation process, and then comparing observed and simulated data sets to constrain the parameters. In such a scenario, one would not have a likelihood function relating the parameters to the observable data. This paper will present an overview of methods that allow a likelihood-free approach to inference, with emphasis on approximate Bayesian computation, a class of procedures originally motivated by similar inference problems in population genetics.