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Posterior Mean Directly from Noise

MediumDiffusion Models 📚 Diffusion Models 1

Problem Statement

In practice the network predicts eps, and the posterior mean has a compact form that skips reconstructing x0 explicitly. Implement that direct eps-based mean.

Background

The DDPM reverse mean can be written directly in terms of the predicted noise:

$\mu_\theta = \frac{1}{\sqrt{\alpha_t}}\left(x_t - \frac{\beta_t}{\sqrt{1 - \bar{\alpha}_t}}\, \hat{\varepsilon}\right)$

with alpha_t = 1 - beta_t. This is the p_theta(x_{t-1}|x_t) mean used in the standard ancestral sampler (before adding the noise term).

Your Task

Implement:

def posterior_mean_from_eps(x_t, eps, beta_t, alpha_bar_t):

Return the mean as a list rounded to 4 decimals.

Input Format

x_t, eps: lists of length D.
beta_t, alpha_bar_t (float).

Output Format

A list of D floats rounded to 4 decimals.

Sample

print(posterior_mean_from_eps([1.0, 1.0], [0.5, -0.5], 0.2, 0.5))

Output:

[0.9599, 1.2761]

Example:

Input:

print(posterior_mean_from_eps([1.0, 1.0], [0.5, -0.5], 0.2, 0.5))

Output:

[0.9599, 1.2761]

Reasoning:

Compute the signal coefficient $\alpha_t$ by subtracting the noise level from 1: $\alpha_t = 1 - 0.2 = 0.8$ .
Determine the noise scaling factor by dividing $\beta_t$ by the square root of the remaining variance: $\text{coef} = \frac{0.2}{\sqrt{1 - 0.5}} = \frac{0.2}{\sqrt{0.5}} \approx 0.28284$ .
Calculate the numerator for each dimension by subtracting the scaled predicted noise from the current state $x_t$ $x_{t}$ :
- Dimension 1: $1.0 - (0.28284 \times 0.5) \approx 0.85858$
- Dimension 2: $1.0 - (0.28284 \times -0.5) \approx 1.14142$
Scale the results by the inverse square root of $\alpha_t$ $α_{t}$ to obtain the posterior mean:
- Dimension 1: $\frac{0.85858}{\sqrt{0.8}} \approx 0.9599$
- Dimension 2: $\frac{1.14142}{\sqrt{0.8}} \approx 1.2761$
The final output is [0.9599, 1.2761]

Constraints:

len(x_t) == len(eps), 0 < beta_t < 1, 0 < alpha_bar_t < 1.
alpha_t = 1 - beta_t; apply the formula above.
Round to 4 decimals; avoid -0.0.

More from Diffusion Models 1

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1. Background Knowledge

In DDPM (Denoising Diffusion Probabilistic Models), the forward process gradually adds Gaussian noise to data, while the reverse process learns to denoise. The reverse step $p_\theta(x_{t-1} \mid x_t)$ is a Gaussian whose mean depends on the predicted noise $\hat{\varepsilon}$ rather than an explicit reconstruction of $x_0$ . This "direct-from-noise" parameterization is numerically stable and avoids an extra division by $\bar{\alpha}_t$ that can amplify floating-point error.

The key schedule variables are:

$\beta_t$ : the per-step noise variance, with $\alpha_t = 1 - \beta_t$ .
$\bar{\alpha}_t = \prod_{s=1}^{t} \alpha_s$ : the cumulative product, so $x_t = \sqrt{\bar{\alpha}_t}\, x_0 + \sqrt{1-\bar{\alpha}_t}\, \varepsilon$ .

The posterior mean formula combines two terms: a scaling of the noisy observation $x_t$ by $1/\sqrt{\alpha_t}$ , and a correction proportional to the predicted noise $\hat{\varepsilon}$ , weighted by $\beta_t / \sqrt{1 - \bar{\alpha}_t}$ . Intuitively, the first term "undoes" the single-step noise, while the second term nudges the estimate toward the learned denoised direction.

2. Algorithm Approach

This is a direct formula evaluation problem. There is no iterative or search-based algorithm; you simply plug the inputs into the closed-form expression element-wise. The pattern is:

Compute the scalar coefficients that depend only on $\beta_t$ and $\bar{\alpha}_t$ .
Apply the per-element affine transformation to each component of $x_t$ and $\hat{\varepsilon}$ .
Round and return.

Because the formula is element-wise, the implementation is a simple vectorized (or loop-based) operation over the $D$ -dimensional vectors.

3. Step-by-Step Strategy

Compute $\alpha_t$ : $\alpha_t = 1 - \beta_t$ .
Compute the scaling factor: $s = 1 / \sqrt{\alpha_t}$ .
Compute the noise-correction weight: $w = \beta_t / \sqrt{1 - \bar{\alpha}_t}$ .
For each dimension $i$ :
$\mu_i = s \cdot (x_t[i] - w \cdot \varepsilon[i])$
Round each $\mu_i$ to 4 decimal places.
Return the resulting list.

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