Learning Non-Gaussian SEMs with Conditional Risk, and Why the Direction of Recursion Matters
Abstract
Variance-based methods recover a Gaussian linear structural equation model (SEM) with equal or suitably constrained error variances, either by recursively identifying the source nodes with smallest conditional variance (top-down method) or sink nodes with highest conditional variance (bottom-up method). We show that the use of conditional variance is incidental: replacing it with the conditional risk, i.e., the minimal expected loss of linear prediction under the negative log-likelihood of the noise distribution, extends both methods to non-Gaussian noise. Existing methods are the least-squares special case. This yields two main results. First, exploiting the noise distribution improves structure recovery: the matched-loss procedures recover the SEM more accurately than their least-squares counterparts and the LiNGAM methods, which are designed for non-Gaussianity. Second, the two methods, interchangeable in the Gaussian setting, separate under non-Gaussian noise: the top-down method is correct for every noise density with finite differential entropy, whereas the bottom-up method requires a regularity condition and can fail without it, for instance under mean-zero exponential or Cauchy noise. Simulations confirm both results.
Related
- Formalizing Fefferman's Official Periodic Navier-Stokes Problem (B) in Lean 4: Global Linear Strain Exclusion on T3, Biot-Savart Direction Depletion Identities, and Conditional Reductions via Beale-Kato-Majda
- Coloring 3-colorable graphs with O(n4/23) colors via a Gaussian-cover recursion
- A Two-Variable Zeta Function for a Parity-Perturbed Hofstadter Q-Recursion: The Exceptional t = -1 Slice and Gaussian Boundary Layers
- The AECM Algorithm for Deterministic Maximum Likelihood Direction Finding in the Presence of Gaussian Mixture Noise
- Derivative Gaussian Processes on a Two-Direction Budget
BibTeX
@misc{shah2026learning,
title = {{Learning Non-Gaussian SEMs with Conditional Risk, and Why the Direction of Recursion Matters}},
author = {Ujas Shah and Panagiota Birmpa and Rebecca Morrison},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14246},
eprint = {2610.14246},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14246},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Learning Non-Gaussian SEMs with Conditional Risk, and Why the Direction of Recursion Matters
AU - Ujas Shah
AU - Panagiota Birmpa
AU - Rebecca Morrison
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.14246
UR - https://arxiv.org/abs/2610.14246
AB - Variance-based methods recover a Gaussian linear structural equation model (SEM) with equal or suitably constrained error variances, either by recursively identifying the source nodes with smallest conditional variance (top-down method) or sink nodes with highest conditional variance (bottom-up method). We show that the use of conditional variance is incidental: replacing it with the conditional risk, i.e., the minimal expected loss of linear prediction under the negative log-likelihood of the noise distribution, extends both methods to non-Gaussian noise. Existing methods are the least-squares special case. This yields two main results. First, exploiting the noise distribution improves structure recovery: the matched-loss procedures recover the SEM more accurately than their least-squares counterparts and the LiNGAM methods, which are designed for non-Gaussianity. Second, the two methods, interchangeable in the Gaussian setting, separate under non-Gaussian noise: the top-down method is correct for every noise density with finite differential entropy, whereas the bottom-up method requires a regularity condition and can fail without it, for instance under mean-zero exponential or Cauchy noise. Simulations confirm both results.
ER -