Title: Maximum Likelihood Training of Parametrized Diffusion Model. Abstract: Whereas the diverse variations of the diffusion model exist in image synthesis, the previous variations have not innovated the diffusing mechanism by maintaining the static linear diffusion. Meanwhile, it is intuitive that there would be more promising diffusion pattern adapted to the data distribution. This paper introduces such adaptive and nonlinear diffusion method for the score-based diffusion models. Unlike the static and linear VE-or-VP SDEs of the previous diffusion models, our parameterized diffusion model (PDM) learns the optimal diffusion process by combining the normalizing flow ahead of the diffusion process. Specifically, PDM utilizes the flow to non-linearly transform a data variable into a latent variable, and PDM applies the diffusion process to the transformed latent distribution with the linear diffusing mechanism. Subsequently, PDM enjoys the nonlinear and learned diffusion from the perspective of the data variable. This model structure is feasible because of the invertibility of the flow. We train PDM with the variational proxy of the log-likelihood, and we prove that the variational gap between the variational bound and the log-likelihood becomes tight when the normalizing flow becomes the optimal. 