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.