  In many fields of science, generalized likelihood ratio tests are established
tools for statistical inference. At the same time, it has become increasingly
common that a simulator (or generative model) is used to describe complex
processes that tie parameters $\theta$ of an underlying theory and measurement
apparatus to high-dimensional observations $\mathbf{x}\in \mathbb{R}^p$.
However, simulator often do not provide a way to evaluate the likelihood
function for a given observation $\mathbf{x}$, which motivates a new class of
likelihood-free inference algorithms. In this paper, we show that likelihood
ratios are invariant under a specific class of dimensionality reduction maps
$\mathbb{R}^p \mapsto \mathbb{R}$. As a direct consequence, we show that
discriminative classifiers can be used to approximate the generalized
likelihood ratio statistic when only a generative model for the data is
available. This leads to a new machine learning-based approach to
likelihood-free inference that is complementary to Approximate Bayesian
Computation, and which does not require a prior on the model parameters.
Experimental results on artificial problems with known exact likelihoods
illustrate the potential of the proposed method.
