Symbolic Density Estimators for Unnormalized Distributions
Abstract
Estimating the symbolic or analytical form of probability density functions (PDFs) from observed samples is a fundamental challenge in statistical and computational modelling. This process is critical for deriving interpretable and generalizable relationships characterizing the underlying phenomenon. Traditionally, this estimation depends strongly on domain expertise and prior field-specific knowledge, with experts selecting appropriate functional forms or parametric families based on empirical evidence and theoretical understanding. The coefficients of these forms are then typically determined through parameter estimation. In this paper, we develop a framework for estimating symbolic expressions of unnormalized distributions from observed samples using domain-specific prior knowledge, such as the range of interactions and a predefined set of primitive functions. We integrate deep generative models with symbolic regression (SR), incorporating inductive biases, such as factorizing large distributions, to keep the problem tractable. The deep generative models we examine include likelihood-based models, viz., flow models, and score-based models. Experiments show the effectiveness of the proposed framework for estimating density functions for multivariate toy distributions as well as lattices from computational physics, namely, XY model and ϕ4 theory. When applied to the renormalization problem in ϕ4 theory, the proposed framework estimates compact symbolic approximations of the hamiltonian function at different scales directly from samples, yielding expressions that may be challenging to derive using traditional perturbative or analytic approaches in nonperturbative settings.
Related
- SIMANF: Sample Free Learning of Unnormalized Distributions via Simulated Annealing in Normalizing Flows
- HOMO-LUMO Intervals from Modern Density Functionals as Fundamental Gap Estimators
- FUND: Density Flow for Sampling Unnormalised Distributions
- sRQA: an integrative pipeline for symbolic recurrence quantification analysis
- Measurement speed limit and Cramér-Rao bound for time-averaged estimators
BibTeX
@misc{kanaujia2026symbolic,
title = {{Symbolic Density Estimators for Unnormalized Distributions}},
author = {Vikas Kanaujia and Riyansha Singh and Shashank Sharma and Vipul Arora},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10807},
eprint = {2610.10807},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10807},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Symbolic Density Estimators for Unnormalized Distributions
AU - Vikas Kanaujia
AU - Riyansha Singh
AU - Shashank Sharma
AU - Vipul Arora
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.10807
UR - https://arxiv.org/abs/2610.10807
AB - Estimating the symbolic or analytical form of probability density functions (PDFs) from observed samples is a fundamental challenge in statistical and computational modelling. This process is critical for deriving interpretable and generalizable relationships characterizing the underlying phenomenon. Traditionally, this estimation depends strongly on domain expertise and prior field-specific knowledge, with experts selecting appropriate functional forms or parametric families based on empirical evidence and theoretical understanding. The coefficients of these forms are then typically determined through parameter estimation. In this paper, we develop a framework for estimating symbolic expressions of unnormalized distributions from observed samples using domain-specific prior knowledge, such as the range of interactions and a predefined set of primitive functions. We integrate deep generative models with symbolic regression (SR), incorporating inductive biases, such as factorizing large distributions, to keep the problem tractable. The deep generative models we examine include likelihood-based models, viz., flow models, and score-based models. Experiments show the effectiveness of the proposed framework for estimating density functions for multivariate toy distributions as well as lattices from computational physics, namely, XY model and ϕ^4 theory. When applied to the renormalization problem in ϕ^4 theory, the proposed framework estimates compact symbolic approximations of the hamiltonian function at different scales directly from samples, yielding expressions that may be challenging to derive using traditional perturbative or analytic approaches in nonperturbative settings.
ER -