AI engineer, Researcher & Founder.

At ICML 2026 — Seoul

Likelihood over Estimation: Robust Quadratic Discriminant Analysis for Heavy-Tailed Distributions

Machine learning classifiers like Quadratic Discriminant Analysis (QDA) assume your data looks like a bell curve (Gaussian). But real-world data — fraud transactions, pulsar signals, network intrusions — has heavy tails: extreme outliers are far more common than a bell curve predicts. When you apply classical QDA to this kind of data, it fails to classify outliers correctly.

The core insight

There are two ways existing QDA can go wrong with heavy-tailed data:

• Wrong model — the Gaussian likelihood underweights outliers, so the classifier is systematically wrong even with perfect knowledge of the parameters.
• Wrong estimation — the standard covariance matrix estimator becomes unreliable (effectively blows up) when data has infinite variance.

The key finding is counterintuitive: fixing the likelihood model (problem #1) matters more than fixing the estimator (problem #2). Stable-QDA replaces the Gaussian likelihood with a multivariate elliptical α-stable distribution, which naturally handles heavy tails — and this drives most of the improvement.

What is an α-stable distribution?

Think of it as a generalization of the bell curve. Just as the Central Limit Theorem says averages of many random variables converge to a Gaussian, the Generalized CLT says that sums of heavy-tailed variables converge to an α-stable distribution. So α-stable is the "right" model for heavy-tailed phenomena by the same foundational logic that makes Gaussian the right model for ordinary data.

What did the paper prove?

Stable-QDA is Bayes consistent — given enough data, it converges to the theoretically best possible classifier. It works very well even if you don't know the exact tail heaviness (α = 1.5 works universally as a default). Classical QDA under heavy tails converges to the wrong answer permanently, no matter how much data you have.

Code & reproducibility

The full implementation of Stable-QDA and all experimental code are released on GitHub — the classifier itself plus scripts to reproduce every result in the paper, so researchers can build on this work or apply Stable-QDA directly to their own heavy-tailed classification problems.