Massive optimal data compression and density estimation for scalable,
  likelihood-free inference in cosmology
  Many statistical models in cosmology can be simulated forwards but have
intractable likelihood functions. Likelihood-free inference methods allow us to
perform Bayesian inference from these models using only forward simulations,
free from any likelihood assumptions or approximations. Likelihood-free
inference generically involves simulating mock data and comparing to the
observed data; this comparison in data-space suffers from the curse of
dimensionality and requires compression of the data to a small number of
summary statistics to be tractable. In this paper we use massive
asymptotically-optimal data compression to reduce the dimensionality of the
data-space to just one number per parameter, providing a natural and optimal
framework for summary statistic choice for likelihood-free inference. Secondly,
we present the first cosmological application of Density Estimation
Likelihood-Free Inference (\textsc{delfi}), which learns a parameterized model
for joint distribution of data and parameters, yielding both the parameter
posterior and the model evidence. This approach is conceptually simple,
requires less tuning than traditional Approximate Bayesian Computation
approaches to likelihood-free inference and can give high-fidelity posteriors
from orders of magnitude fewer forward simulations. As an additional bonus, it
enables parameter inference and Bayesian model comparison simultaneously. We
demonstrate Density Estimation Likelihood-Free Inference with massive data
compression on an analysis of the joint light-curve analysis supernova data, as
a simple validation case study. We show that high-fidelity posterior inference
is possible for full-scale cosmological data analyses with as few as $\sim
10^4$ simulations, with substantial scope for further improvement,
demonstrating the scalability of likelihood-free inference to large and complex
cosmological datasets.
