Our Unfolding Approach

Table of Links

Abstract and 1. Introduction

2. Unfolding

   2.1 Posing the Unfolding Problem

   2.2 Our Unfolding Approach

3. Denoising Diffusion Probabilistic Models

   3.1 Conditional DDPM

4. Unfolding with cDDPMs

5. Results

   5.1 Toy models

   5.2 Physics Results

6. Discussion, Acknowledgments, and References

Appendices

A. Conditional DDPM Loss Derivation

B. Physics Simulations

C. Detector Simulation and Jet Matching

D. Toy Model Results

E. Complete Physics Results

2.2 Our Unfolding Approach

Although we cannot achieve an ideal universal unfolder, we can seek an approach that will enhance the inductive bias of the unfolding method to improve generalization to cover various posteriors pertaining to different physics data distributions. From eq. (2) we can see that the posteriors for two different physics processes i and j are related by a ratio of the probability density functions of each process,

Assuming we can learn the posterior for a given physics process, we note that we could extrapolate to unseen posteriors if the priors ftrue(x) and detector distributions fdet(y) can be approximated or written in a closed form. Although these functions have no analytical form, we can approximate key features using the first moments of these distributions. By making use of these moments, we can have a more flexible unfolder that is not strictly tied to a selected prior distribution, and enables it to interpolate and extrapolate to unseen posteriors based on the provided moments. Consequently, this unfolding tool gains the ability to handle a wider range of physics processes and enhances the generalization capabilities, making it a more versatile tool for unfolding in various high energy physics applications.

In practice, one can use a training dataset of pairs {x, y} to train a machine learning model to learn a posterior P(x|y). To implement our approach and improve the inductive bias, we define a training dataset consisting of multiple prior distributions and incorporate the moments of these distributions to the data pairs. The moments are therefore included in the conditioning and generative aspects of the machine learning model such that it may be able to model multiple posteriors. As a result, we establish an unfolding tool as a posterior sampler that, when trained with sufficient priors within a family of distributions, is “universal” in the sense that it has a strong inductive bias to allow generalization towards estimating the prior distribution of unseen datasets. Further details and a technical description of this method are provided in section 4.

Our proposed approach calls for a flexible generative model, and denoising diffusion probabilistic models (DDPMs) [13] lend themselves naturally to this task. DDPMs learn via a reversible generative process that can be conditioned directly on the moments of the distribution fdet(y) and on the detector values themselves, providing a natural way to model P(x|y) for unfolding. In particular, the various conditioning methods available for DDPMs offer the flexibility to construct a model that can adapt to different detector data distributions and physics processes. Further details on DDPMs are provided in section 3.