2 Background

Diffusion models [53] are latent variable models of the form pθ(x0) :=(cid:82) pθ(x0:T ) dx1:T , where

x1, . . . , xT are latents of the same dimensionality as the data x0 ∼ q(x0). The joint distribution
pθ(x0:T ) is called the reverse process, and it is deﬁned as a Markov chain with learned Gaussian
transitions starting at p(xT ) = N (xT ; 0, I):

pθ(x0:T ) := p(xT )

pθ(xt−1|xt),

pθ(xt−1|xt) := N (xt−1; µθ(xt, t), Σθ(xt, t))

What distinguishes diffusion models from other types of latent variable models is that the approximate
posterior q(x1:T|x0), called the forward process or diffusion process, is ﬁxed to a Markov chain that
gradually adds Gaussian noise to the data according to a variance schedule β1, . . . , βT :

q(x1:T|x0) :=

q(xt|xt−1),

q(xt|xt−1) := N (xt;

1 − βtxt−1, βtI)

(cid:21)

(cid:20)

− log

pθ(x0:T )
q(x1:T|x0)
Denoising Diffusion Probabilistic Models