Diffusion Models are Kelly Gamblers

We draw a connection between diffusion models and the Kelly criterion for maximizing returns in betting games. We find that conditional diffusion models store additional information to bind the signal $X$ with the conditioning information $Y$, equal to the mutual information between them. Classifier-free guidance effectively boosts the mutual information between $X$ and $Y$ at sampling time. This is especially helpful in image models, since the mutual information between images and their labels is low, a fact which is intimately connected to the manifold hypothesis. Finally, we point out some nuances in the popular perspective that diffusion models are infinitely deep autoencoders. In doing so, we relate the denoising loss to the Fermi Golden Rule from quantum mechanics.