Normalizing Flows

Imagine starting with a simple, well-understood distribution — a standard normal — and squeezing and warping it through a sequence of transformations until it matches a complicated real-world distribution, like the joint distribution of several correlated asset returns with fat tails and skew. A normalizing flow is exactly that: a chain of neural-network transformations, each required to be invertible, applied to samples from a simple base distribution.

The invertibility requirement is the whole trick. Because each transformation can be run forward (simple to complex, for generating new samples) and backward (complex to simple, for evaluating a likelihood), and because there's a known formula — the change-of-variables rule with a Jacobian determinant — for how probability density transforms under an invertible map, a flow gives you an exact likelihood for any data point, not just an approximate one. That's the feature GANs lack (they can sample but can't tell you the probability of a given point) and that plain autoencoder-based models only approximate.

The cost is architectural: every layer has to be built specifically to be easily invertible with an easily computable Jacobian, which rules out ordinary neural network layers and requires specialized constructions. In finance, flows have been explored for modeling the joint distribution of returns and for generating synthetic market scenarios where an exact, tractable likelihood is genuinely useful, for example in risk models.