Masked Autoregressive Flow

Many generative models can either produce realistic-looking samples (a GAN) or compute how likely a given data point is under the model (a Gaussian), but rarely both easily. A normalizing flow does both: it starts from simple noise, like a standard normal, and applies a chain of invertible transformations until the result matches a complicated real-world distribution — like the distribution of a stock's overnight return conditional on the past week.

The "masked autoregressive" part refers to how each transformation layer is built: a single neural network with connections masked out so that the k-th output can only see inputs 1 through k-1, exactly like the masking used in autoregressive language models. That structure guarantees the transformation's Jacobian is triangular, so its determinant — needed to compute exact likelihood under a change of variables — is just the product of diagonal terms, cheap to calculate even for high-dimensional data.

Stacking several such layers, each conditioning on a different ordering or permutation of variables, lets the model capture dependencies that a single autoregressive pass would miss. In quant applications, this makes MAFs attractive for modeling joint return distributions across correlated assets or scenario-generating synthetic market paths that preserve realistic tail dependence, something a plain multivariate normal cannot do.