  We present a comparison of simulation-based inference to full, field-based
analytical inference in cosmological data analysis. To do so, we explore
parameter inference for two cases where the information content is calculable
analytically: Gaussian random fields whose covariance depends on parameters
through the power spectrum; and correlated lognormal fields with cosmological
power spectra. We compare two inference techniques: i) explicit field-level
inference using the known likelihood and ii) implicit likelihood inference with
maximally informative summary statistics compressed via Information Maximising
Neural Networks (IMNNs). We find that a) summaries obtained from convolutional
neural network compression do not lose information and therefore saturate the
known field information content, both for the Gaussian covariance and the
lognormal cases, b) simulation-based inference using these maximally
informative nonlinear summaries recovers nearly losslessly the exact posteriors
of field-level inference, bypassing the need to evaluate expensive likelihoods
or invert covariance matrices, and c) even for this simple example, implicit,
simulation-based likelihood incurs a much smaller computational cost than
inference with an explicit likelihood. This work uses a new IMNNs
implementation in $\texttt{Jax}$ that can take advantage of
fully-differentiable simulation and inference pipeline. We also demonstrate
that a single retraining of the IMNN summaries effectively achieves the
theoretically maximal information, enhancing the robustness to the choice of
fiducial model where the IMNN is trained.
