  High-dimensional probability density estimation for inference suffers from
the "curse of dimensionality". For many physical inference problems, the full
posterior distribution is unwieldy and seldom used in practice. Instead, we
propose direct estimation of lower-dimensional marginal distributions,
bypassing high-dimensional density estimation or high-dimensional Markov chain
Monte Carlo (MCMC) sampling. By evaluating the two-dimensional marginal
posteriors we can unveil the full-dimensional parameter covariance structure.
We additionally propose constructing a simple hierarchy of fast neural
regression models, called Moment Networks, that compute increasing moments of
any desired lower-dimensional marginal posterior density; these reproduce exact
results from analytic posteriors and those obtained from Masked Autoregressive
Flows. We demonstrate marginal posterior density estimation using
high-dimensional LIGO-like gravitational wave time series and describe
applications for problems of fundamental cosmology.
