  Likelihood-free inference provides a framework for performing rigorous
Bayesian inference using only forward simulations, properly accounting for all
physical and observational effects that can be successfully included in the
simulations. The key challenge for likelihood-free applications in cosmology,
where simulation is typically expensive, is developing methods that can achieve
high-fidelity posterior inference with as few simulations as possible.
Density-estimation likelihood-free inference (DELFI) methods turn inference
into a density estimation task on a set of simulated data-parameter pairs, and
give orders of magnitude improvements over traditional Approximate Bayesian
Computation approaches to likelihood-free inference. In this paper we use
neural density estimators (NDEs) to learn the likelihood function from a set of
simulated datasets, with active learning to adaptively acquire simulations in
the most relevant regions of parameter space on-the-fly. We demonstrate the
approach on a number of cosmological case studies, showing that for typical
problems high-fidelity posterior inference can be achieved with just
$\mathcal{O}(10^3)$ simulations or fewer. In addition to enabling efficient
simulation-based inference, for simple problems where the form of the
likelihood is known, DELFI offers a fast alternative to MCMC sampling, giving
orders of magnitude speed-up in some cases. Finally, we introduce
\textsc{pydelfi} -- a flexible public implementation of DELFI with NDEs and
active learning -- available at \url{https://github.com/justinalsing/pydelfi}.
