• Uncertainty quantification: Motivation and background
• Uncertainty quantification in image regression
• The villain: Distribution shift

Solution #1: The principled baseline

• The method
• Scattering transforms
• Discussion and benchmarking

Solution #2: Foundation model-based

• The method
• Discussion and benchmarking

Summary & future

Uncertainty quantification ≈ predicting magnitude/probability of error

$^*$Actual guarantees require assumptions on the model & data distribution

Probabilistic models: input ⟶ distribution (prediction + uncertainty)

Allows computing queries like mutual information: $\!\alpha(x) \!=\! I(y(x); f \given \mathcal{D})$

Can help selecting the optimal point to add to data [Active learning]

For example:

Predict skin lesion size

$ \rightarrow $

$ \mathcal{N}(\mu, \sigma^2) $

Predict molecule properties

$ \rightarrow $

$ \mathcal{N}(\mu, \sigma^2) $

Default approach — deep ensembles:

• Retrain one architecture $K$ times, get $f_1, \ldots, f_K$
• Prediction: $f(x) = \tfrac{1}{K} \sum_{k=1}^K f_k(x)$
• Uncertainty: $\sigma(x) = \bigl(\tfrac{1}{K} \sum_{k=1}^K (f_k(x) - f(x))^2\bigr)^{1/2}$

$^*$Usually $f_i$ are also trained to predict variance (formulas adjust)

For example:

In distribution
(training set)

Out of distribution
(change in equipment & population)

You cannot expect to maintain quality of prediction under distribution shift

But: you would hope that uncertainty remains meaningful

It's usually not the case (e.g., for ensembles — Ovadia et al. 2019) 🙁

(It gets even worse if there is not a lot of data — common in biomed)

The method is called Bayesian Scattering (Fichera et al. 2026)

Key idea:

• Fixed features cannot overfit to train distribution
• If we use a simple probabilistic head on top, might avoid it altogether

Where to find such fixed universal features?

Go back 15 years. CNNs are the best.
Can we achieve same performance with little to no learning?

Yann LeCun

VS

Stephane Mallat

Mallat tried and... almost did it,
creating wavelet scattering transforms along the way.

Problem: Find a feature map that transforms image $\rightarrow$ features,
with geometric properties applicable universally across all image data:

Translation (in,equi)variance

Stability to additive noise

Stability to smooth deformations

All this + preserving the bulk of information (invertibility in the limit)

Problem: Find a feature map that transforms image $\rightarrow$ features,
with universal geometric properties + preserving information

Turns out it is not trivial at all

Mallat (2011), Bruna and Mallat (2012):
— solution based on repeated application of wavelet transforms

By the way, it generalizes to 3D and allows for rotation (in,equi)variance too!

• Fourier transform (simpler) + nonlinearity – not stable under deformations
• Wavelet transform + nonlinearity – stable, but not translation-invariant

• Averaging gives translation-invariance but loses information
• Depth is the only missing ingredient...

Determined by

• Depth parameter $m \geq 1$. Usually $m=2$
• Rotational resolution $L \geq 1$ (the higher the better but saturates quickly)
• Parameter $J$, with $2 \leq 2^J \leq $ image size
  Roughly, interpolates between translation equivariance and invariance

• Training set size: 1000 (small data)
• Distribution shifts:
  • SLP: different equipment and skin type distribution (Austria vs Australia)
  • HP: different institution with inherent variations in staining and sample preparation
  • AW: training and test sets are drawn from disjoint lists of nations

Key outcome: Bayesian Scattering has vastly better NLL

Key idea — same as before

Features — embeddings of some image foundation model (e.g., DINO, JEPA)

• Features encode very broad knowledge from massive pretraining
• If not fine-tuned on labeled training data, features remain unbiased

• Training set size: 1000 (small data)
• Distribution shifts:
  • SLP: different equipment and skin type distribution (Austria vs Australia)
  • HP: different institution with inherent variations in staining and sample preparation
  • AW: training and test sets are drawn from disjoint lists of nations

But we should be able to do much more:

• Domain-specific foundation models
• Fine-tuning foundation models:
  • Improve predictive power
  • Remain unbiased: Avoid overfitting to the training distribution

Problem: optimize some property of interest over molecules (e.g. energy).

Represent molecules as approximate electronic density images:

Both Solution #1 and #2 are probabilistic models.
Bayesian optimization: choose molecules with largest expected improvement
(optimal with respect to the model, so the better the model, the faster optimization convergence)

Bayesian Scattering on par with UMOL-based approach for planar molecules

Bayesian Scattering outperforms UMOL-based approach for 3D molecules