TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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Closed Gaussian forward marginal and conditioned posterior

Statement

Assume AC and the finite Gaussian forward chain with dimension $d \geq 1$ , horizon $T \geq 1$ , and schedule $0 < β_{t} < 1$ from the preceding definition. For each $x_{0} \in R^{d}$ and $1 \leq t \leq T$ , $q (x_{t} ∣ x_{0}) = N_{d} ⁣ (\sqrt{{\overset{ˉ}{α}}_{t}} x_{0}, (1 - {\overset{ˉ}{α}}_{t}) I_{d}) .$ For every $2 \leq t \leq T$ , the conditional law given both endpoints is $q (x_{t - 1} ∣ x_{t}, x_{0}) = N_{d} ⁣ (\frac{\sqrt{{\overset{ˉ}{α}}_{t - 1}} β_{t}}{1 - {\overset{ˉ}{α}}_{t}} x_{0} + \frac{\sqrt{α_{t}} (1 - {\overset{ˉ}{α}}_{t - 1})}{1 - {\overset{ˉ}{α}}_{t}} x_{t}, {\tilde{β}}_{t} I_{d}), {\tilde{β}}_{t} = \frac{β_{t} (1 - {\overset{ˉ}{α}}_{t - 1})}{1 - {\overset{ˉ}{α}}_{t}} > 0.$ The formula for $q (x_{t - 1} ∣ x_{t}, x_{0})$ is only asserted for $t \geq 2$ ; at $t = 1$ the conditioning on $x_{0}$ makes $q (x_{0} ∣ x_{1}, x_{0})$ a point mass, not a Lebesgue density. The Gaussian conditional above does not assert that the unconditional reverse law $q (x_{t - 1} ∣ x_{t})$ is Gaussian.

Statement references

Closed Gaussian forward marginal, posterior mean, and posterior variance: Ho, Jain, and Abbeel, Eqs. (4), (6), and (7).

Forward Gaussian composition and conditioned posterior: Prince, Chapter 18, “The diffusion kernel” and “The tractable conditional”.

Facts & Assumptions

Given: AC, a fixed $x_{0} \in R^{d}$ , finite $d, T \geq 1$ , and $0 < β_{t} < 1$ for every $t$ .

[A1]

AC is assumed by the Gaussian-stack construction and the multivariate-normal conditional-law theorem. The Axiom of Choice.

[F1]

The forward path density factors into normalized transition densities $ϕ_{d, β_{s}} (x_{s}; \sqrt{α_{s}} x_{s - 1})$ , with $α_{s} = 1 - β_{s}$ and ${\overset{ˉ}{α}}_{s} = \prod_{r = 1}^{s} α_{r}$ . Finite Gaussian forward chain and Gaussian reverse model.

[F2]

A multivariate normal law is characterized by its scalar projection laws and has the stated mean and covariance. Multivariate normal law, including singular covariance.

[F3]

A finite stack of mutually independent Gaussian vectors is jointly multivariate normal, even when a covariance is singular. A finite stack of independent Gaussian vectors is jointly Gaussian.

[F4]

If $(X, Y)$ is jointly Gaussian with positive-semidefinite block covariance and the observed block covariance $B$ is positive definite, then its conditional law has mean $a + C B^{- 1} (y - b)$ and covariance $A - C B^{- 1} C^{T}$ . Conditional law of a jointly Gaussian block with positive-definite observed covariance.

Proof

technique · standardize each transition to an independent Gaussian innovation, compute its finite affine sum, and apply the positive-definite-observation conditional-law formula to adjacent states

Given: The hypotheses in the Statement.

1.1A1F1F2givenalgebra

Under the forward path law with fixed $x_{0}$ , for each $s$ put $ξ_{s} = (X_{s} - \sqrt{α_{s}} X_{s - 1}) / \sqrt{β_{s}}$ . Changing variables $x_{s} = \sqrt{α_{s}} x_{s - 1} + \sqrt{β_{s}} z$ in the last transition density in [F1] gives the conditional density $ϕ_{d, 1} (z; 0)$ , independent of the past. Thus for Borel past events $A$ and Borel $B \subseteq R^{d}$ , $P (X_{1 : s - 1} \in A, ξ_{s} \in B) = P (X_{1 : s - 1} \in A) N_{d} (0, I_{d}) (B)$ . Each $ξ_{s}$ is standard Gaussian and independent of the previous path; since earlier innovations are measurable functions of that path, $ξ_{1}, \dots, ξ_{t}$ are mutually independent for every $t$ .

2.1A1F1F2F3givenstep 1.1algebra

Iterating $X_{t} = \sqrt{α_{t}} X_{t - 1} + \sqrt{β_{t}} ξ_{t}$ gives $X_{t} = \sqrt{{\overset{ˉ}{α}}_{t}} x_{0} + \sum_{s = 1}^{t} \sqrt{β_{s} \prod_{r = s + 1}^{t} α_{r}} ξ_{s}$ . By step 1.1 and [F3], the finite innovation stack is jointly Gaussian. Thus each scalar projection of $X_{t}$ is normal with mean $u \cdot \sqrt{{\overset{ˉ}{α}}_{t}} x_{0}$ and, by independence, variance $u^{T} (\sum_{s = 1}^{t} β_{s} \prod_{r = s + 1}^{t} α_{r} I_{d}) u$ ; [F2] identifies its law as multivariate normal with that mean and covariance. Since $β_{s} \prod_{r = s + 1}^{t} α_{r} = \prod_{r = s + 1}^{t} α_{r} - \prod_{r = s}^{t} α_{r}$ , the covariance sum telescopes to $(1 - {\overset{ˉ}{α}}_{t}) I_{d}$ .

3.1A1F1F2F3step 1.1step 2.1algebra

Fix $t \geq 2$ . By step 2.1, $X_{t - 1}$ is Gaussian, and by step 1.1, $ξ_{t}$ is independent of the past and standard Gaussian. The finite-stack result [F3] makes $(X_{t - 1}, ξ_{t})$ jointly Gaussian. For any $u, v \in R^{d}$ , the projection of $(X_{t - 1}, X_{t})$ is $(u + \sqrt{α_{t}} v) \cdot X_{t - 1} + \sqrt{β_{t}} v \cdot ξ_{t}$ , a scalar projection of that Gaussian stack; hence [F2] makes the pair jointly Gaussian. Its means are $\sqrt{{\overset{ˉ}{α}}_{t - 1}} x_{0}$ and $\sqrt{{\overset{ˉ}{α}}_{t}} x_{0}$ , and its covariance blocks are $A = (1 - {\overset{ˉ}{α}}_{t - 1}) I_{d}$ , $C = \sqrt{α_{t}} (1 - {\overset{ˉ}{α}}_{t - 1}) I_{d}$ , and $B = (1 - {\overset{ˉ}{α}}_{t}) I_{d}$ . The full block covariance is positive semidefinite because each of its quadratic forms is the variance of a scalar projection of $(X_{t - 1}, X_{t})$ . Since $0 < α_{s} < 1$ , we have $0 < {\overset{ˉ}{α}}_{t} < 1$ , so $B$ is positive definite.

4.1A1F1F4step 3.1algebra∎

Applying [F4] with these blocks gives conditional covariance $A - C B^{- 1} C^{T} = ((1 - {\overset{ˉ}{α}}_{t - 1}) - \frac{α_{t} (1 - {\overset{ˉ}{α}}_{t - 1})^{2}}{1 - {\overset{ˉ}{α}}_{t}}) I_{d} = \frac{β_{t} (1 - {\overset{ˉ}{α}}_{t - 1})}{1 - {\overset{ˉ}{α}}_{t}} I_{d}$ and conditional mean $\sqrt{{\overset{ˉ}{α}}_{t - 1}} x_{0} + \frac{\sqrt{α_{t}} (1 - {\overset{ˉ}{α}}_{t - 1})}{1 - {\overset{ˉ}{α}}_{t}} (x_{t} - \sqrt{{\overset{ˉ}{α}}_{t}} x_{0}) = \frac{\sqrt{{\overset{ˉ}{α}}_{t - 1}} β_{t}}{1 - {\overset{ˉ}{α}}_{t}} x_{0} + \frac{\sqrt{α_{t}} (1 - {\overset{ˉ}{α}}_{t - 1})}{1 - {\overset{ˉ}{α}}_{t}} x_{t}$ . For $t \geq 2$ , ${\overset{ˉ}{α}}_{t - 1} < 1$ and $β_{t} > 0$ , so ${\tilde{β}}_{t} > 0$ . This is the asserted conditional law; no density for the degenerate $t = 1$ conditional was used.

Source notes

Joint Gaussian conditional law and its positive-definite observed covariance hypothesis: Conditional law of a jointly Gaussian block with positive-definite observed covariance.

Finite independent Gaussian stack: A finite stack of independent Gaussian vectors is jointly Gaussian.

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Sources

Jonathan Ho, Ajay Jain, and Pieter Abbeel, Denoising Diffusion Probabilistic Models (2020) (standard reference, not scraped)
Simon J. D. Prince, Understanding Deep Learning, Chapter 18: Diffusion models (2026), pp. 363–387 (standard reference, not scraped)