t I), we can write:

Lt−1 = Eq

(cid:20) 1

+ C

2σ2

(8)

where C is a constant that does not depend on θ. So, we see that the most straightforward parameteri-
zation of µθ is a model that predicts ˜µt, the forward process posterior mean. However, we can expand
Eq. (8) further by reparameterizing Eq. (4) as xt(x0, (cid:15)) = √¯αtx0 + √1 − ¯αt(cid:15) for (cid:15) ∼ N (0, I) and

applying the forward process posterior formula (7):
Lt−1 − C = Ex0,(cid:15)

xt(x0, (cid:15)),

1
√¯αt

1
2σ2
t

(xt(x0, (cid:15)) − √1 − ¯αt(cid:15))

− µθ(xt(x0, (cid:15)), t)

(cid:19)

(cid:13)(cid:13)(cid:13)(cid:13)2(cid:35)

(cid:13)(cid:13)(cid:13)(cid:13)2(cid:35)

(9)

(10)

(cid:34)
(cid:34)

(cid:13)(cid:13)(cid:13)(cid:13) ˜µt
(cid:18)
(cid:13)(cid:13)(cid:13)(cid:13) 1

√αt

(cid:18)

= Ex0,(cid:15)

1
2σ2
t

xt(x0, (cid:15)) −

βt√1 − ¯αt

(cid:15)

− µθ(xt(x0, (cid:15)), t)

(cid:19)

3

(cid:17)
(cid:18)

(cid:17)

(cid:19)
Denoising Diffusion Probabilistic Models