Likelihood-Free Cosmological Inference with Type Ia Supernovae:
  Approximate Bayesian Computation for a Complete Treatment of Uncertainty
  Cosmological inference becomes increasingly difficult when complex
data-generating processes cannot be modeled by simple probability
distributions. With the ever-increasing size of data sets in cosmology, there
is increasing burden placed on adequate modeling; systematic errors in the
model will dominate where previously these were swamped by statistical errors.
For example, Gaussian distributions are an insufficient representation for
errors in quantities like photometric redshifts. Likewise, it can be difficult
to quantify analytically the distribution of errors that are introduced in
complex fitting codes. Without a simple form for these distributions, it
becomes difficult to accurately construct a likelihood function for the data as
a function of parameters of interest. Approximate Bayesian computation (ABC)
provides a means of probing the posterior distribution when direct calculation
of a sufficiently accurate likelihood is intractable. ABC allows one to bypass
direct calculation of the likelihood but instead relies upon the ability to
simulate the forward process that generated the data. These simulations can
naturally incorporate priors placed on nuisance parameters, and hence these can
be marginalized in a natural way. We present and discuss ABC methods in the
context of supernova cosmology using data from the SDSS-II Supernova Survey.
Assuming a flat cosmology and constant dark energy equation of state we
demonstrate that ABC can recover an accurate posterior distribution. Finally we
show that ABC can still produce an accurate posterior distribution when we
contaminate the sample with Type IIP supernovae.
