A normalizing flow maps parameters \(\theta\) to a standard-normal base
variable through a stack of invertible transforms, giving exact densities by
the change of variables. The MAF (Papamakarios et al., 2017) uses masked
autoregressive networks (MADE, Germain et al., 2015): each transform is
$$u_d = (\theta_d - \mu_d(\theta_{<d}, x)) \exp(-\alpha_d(\theta_{<d}, x)),$$
where the masks guarantee that \(\mu_d, \alpha_d\) depend only on earlier
dimensions of \(\theta\) (and freely on the conditioning data x). Density
evaluation is a single forward pass; sampling inverts the transform one
dimension at a time. Between transforms the parameter order is reversed so
every dimension gets conditioned on every other across the stack.
Details
This is sbi's default flow family, and the default estimator in neuralsbi
too. It handles non-Gaussian posteriors that the MDN struggles with. It is
selected by default, or explicitly with npe(..., density_estimator = "maf").
References
Papamakarios, G., Pavlakou, T. and Murray, I. (2017). Masked Autoregressive Flow for Density Estimation. NeurIPS. doi:10.48550/arXiv.1705.07057
Germain, M., Gregor, K., Murray, I. and Larochelle, H. (2015). MADE: Masked Autoencoder for Distribution Estimation. ICML. doi:10.48550/arXiv.1502.03509