  Score-based generative modeling has recently emerged as a promising
alternative to traditional likelihood-based or implicit approaches. Learning in
score-based models involves first perturbing data with a continuous-time
stochastic process, and then matching the time-dependent gradient of the
logarithm of the noisy data density - or score function - using a continuous
mixture of score matching losses. In this note, we show that such an objective
is equivalent to maximum likelihood for certain choices of mixture weighting.
This connection provides a principled way to weight the objective function, and
justifies its use for comparing different score-based generative models. Taken
together with previous work, our result reveals that both maximum likelihood
training and test-time log-likelihood evaluation can be achieved through
parameterization of the score function alone, without the need to explicitly
parameterize a density function.
