Factorizable Normalizing Flows for parameter-dependent density morphing

In many scientific fields, such as high energy physics, researchers must model how a probability density function (PDF) changes in response to continuous parameters like systematic uncertainties or calibration constants. Traditional methods often rely on binned histograms and low-order polynomial interpolation, which struggle in high-dimensional spaces and fail to capture complex correlations. The authors seek a scalable, differentiable, and interpretable way to perform this density morphing for unbinned likelihood fits.

The authors introduce Factorizable Normalizing Flows (FNFs). Instead of training a new generative model for every possible parameter configuration—which is computationally intractable—they decouple the problem. First, they train a high-fidelity Normalizing Flow on a fixed reference configuration (e.g., the nominal simulation). Second, they learn a separate, lightweight transformation that acts as a 'warp' to pull observed data back to this reference frame. This transformation is designed to be polynomial in the parameters and factorized across them. Because each parameter's effect is learned independently using only its own variation samples, the model scales linearly with the number of parameters rather than exponentially.

FNFs successfully reproduce complex, parameter-dependent deformations in a controlled toy dataset. The model accurately captures how probability mass migrates under different systematic shifts and squeezes. By summing the individual contributions of each parameter at inference, the FNF recovers the joint density without ever needing to sample the full, high-dimensional joint parameter space. The authors demonstrate that while the factorized assumption is highly accurate, they can also include optional pairwise interaction terms to capture residual correlations when multiple parameters vary strongly at once, providing a flexible trade-off between computational cost and modeling precision.