Energy-Based Models

Most generative models are built to output a valid probability — numbers that are positive and sum to one over all possible outcomes, which forces the model's architecture to be constrained in specific ways (like the triangular Jacobians normalizing flows need). An energy-based model sidesteps this: it learns a single scalar "energy" for any input, where lower energy means more plausible, and never requires that energy function to integrate to one.

Formally, the model defines a probability only implicitly, as $p(x) \propto e^{-E(x)}$ for some learned energy function $E$, but the proportionality constant — the normalizing sum over every possible $x$ — is usually impossible to compute directly for high-dimensional data like images or return paths. Training therefore can't use ordinary maximum likelihood directly; instead it relies on techniques like contrastive divergence, which nudges energy down on real data and up on samples drawn from the model itself, or score-based methods that sidestep the normalizing constant entirely by matching gradients of the energy rather than the energy itself.

In quant contexts, energy-based models have been explored for scenario generation and anomaly scoring — flagging a trade or a market state as implausible without needing a fully calibrated probability, since a low-energy region can represent "normal" market conditions and a sudden jump into high-energy territory can flag a regime that looks unlike anything the model has seen.