---
title: "Diffusion forward-process marginal"
description: "Samples a noised version at any diffusion time directly from the clean sample without simulating every intermediate step."
canonical_url: "https://fanout.sh/labs/math-decoder/formula/diffusion-closed-form-forward-process"
md_url: "https://fanout.sh/labs/math-decoder/formula/diffusion-closed-form-forward-process.md"
last_updated: "2026-08-09"
access: "public"
---

# Diffusion forward-process marginal

Samples a noised version at any diffusion time directly from the clean sample without simulating every intermediate step.

## Public overview

q of x t given x zero is a normal distribution with mean square root alpha bar t times x zero and covariance one minus alpha bar t times the identity.

At time t, keep a scheduled fraction of the clean signal and fill the remaining variance with Gaussian noise.

The normal-distribution constructor is the main verb: it defines the complete conditional law of x t.

This closed form lets training choose random time steps directly; the learned reverse process is trained to undo the scheduled corruption.

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