How do Energy Based Models solve Multiple possible outputs given one input

think of a GAN… the discriminator is just asked if a specific $x$ works for a specific $y$, and thus you can think of the discriminator as an energy model

indeed, this way, you don’t have to average over possible “answers”, but just to learn a possible completion

Indeed the usual probabilistic models such as VAE whose main drawback is that samples from the model trained on images tend to be somewhat blurry possibly due to your mentioned inherent effect of maximum likelihood or equivalently KL-Divergence between true input data generating and model distributions.

Energy-based models (EBMs) offer a different approach to handling situations where there are multiple possible outputs for a given input compared to probabilistic models. In EBMs, instead of directly modeling the probability distribution over outputs, the model learns a scalar-valued energy function that assigns low energy to plausible outputs and high energy to implausible ones. This energy function captures the compatibility between inputs and outputs and allows the model to generate diverse outputs without the usual noisy blurred effect.

One example architecture that uses EBMs to address this problem is the Generative Adversarial Network (GAN). The generator network in a GAN can be viewed as an energy-based model. It learns to generate outputs that have low energy (i.e., are plausible) according to the discriminator's judgment and usually with better output effect in terms of noisy blurriness compared to VAE's.