  Simulation-based inference enables learning the parameters of a model even
when its likelihood cannot be computed in practice. One class of methods uses
data simulated with different parameters to infer models of the
likelihood-to-evidence ratio, or equivalently the posterior function. Here we
frame the inference task as an estimation of an energy function parametrized
with an artificial neural network. We present an intuitive approach where the
optimal model of the likelihood-to-evidence ratio is found by maximizing the
likelihood of simulated data. Within this framework, the connection between the
task of simulation-based inference and mutual information maximization is
clear, and we show how several known methods of posterior estimation relate to
alternative lower bounds to mutual information. These distinct objective
functions aim at the same optimal energy form and therefore can be directly
benchmarked. We compare their accuracy in the inference of model parameters,
focusing on four dynamical systems that encompass common challenges in time
series analysis: dynamics driven by multiplicative noise, nonlinear
interactions, chaotic behavior, and high-dimensional parameter space.
