# On the Identifiability of Mixed Ordinal and Exponential Family Causal DAGs under Linear Parametric Models

arXiv: https://arxiv.org/abs/2609.17942 · HTML: https://mlsanity.com/paper/2609.17942/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 23 of its 65 references that are in the map.
- First posted on arXiv 16 Sep 2026.

## Builds on (most cited first, 23 of 23)

- [Causal Discovery with Continuous Additive Noise Models](https://mlsanity.com/paper/1309.6779/) (2013)
- [Causal Bandits: Learning Good Interventions via Causal Inference](https://mlsanity.com/paper/1606.03203/) (2016)
- [Causal Autoregressive Flows](https://mlsanity.com/paper/2011.02268/) (2020)
- [Characterizing and Learning Equivalence Classes of Causal DAGs under Interventions](https://mlsanity.com/paper/1802.06310/) (2018)
- [On the Identifiability and Estimation of Causal Location-Scale Noise Models](https://mlsanity.com/paper/2210.09054/) (2022)
- [Learning Sparse Causal Models is not NP-hard](https://mlsanity.com/paper/1309.6824/) (2013)
- [Identifying Patient-Specific Root Causes with the Heteroscedastic Noise Model](https://mlsanity.com/paper/2205.13085/) (2022)
- [Learning Identifiable Gaussian Bayesian Networks in Polynomial Time and Sample Complexity](https://mlsanity.com/paper/1703.01196/) (2017)
- [Causal Bandits for Linear Structural Equation Models](https://mlsanity.com/paper/2208.12764/) (2022)
- [Optimal estimation of Gaussian DAG models](https://mlsanity.com/paper/2201.10548/) (2022)
- [Causal Reasoning: Charting a Revolutionary Course for Next-Generation AI-Native Wireless Networks](https://mlsanity.com/paper/2309.13223/) (2023)
- [Structure learning in polynomial time: Greedy algorithms, Bregman information, and exponential families](https://mlsanity.com/paper/2110.04719/) (2021)
- [Causal Inference on Multivariate and Mixed-Type Data](https://mlsanity.com/paper/1702.06385/) (2017)
- [Ordinal Causal Discovery](https://mlsanity.com/paper/2201.07396/) (2022)
- [Identifiability of causal graphs under nonadditive conditionally parametric causal models](https://mlsanity.com/paper/2303.15376/) (2023)
- [Structural restrictions in local causal discovery: identifying direct causes of a target variable](https://mlsanity.com/paper/2307.16048/) (2023)
- [Identifiability of an Integer Modular Acyclic Additive Noise Model and its Causal Structure Discovery](https://mlsanity.com/paper/1401.5625/) (2014)
- [Causal Link Discovery with Unequal Edge Error Tolerance](https://mlsanity.com/paper/2507.21570/) (2025)
- [Causal Discovery with Generalized Linear Models through Peeling Algorithms](https://mlsanity.com/paper/2310.16698/) (2023)
- [Asymmetric Graph Error Control with Low Complexity in Causal Bandits](https://mlsanity.com/paper/2408.11240/) (2024)
- [Causal Discovery in Equal Variance Linear Gaussian DAGs via SURE-Tuned Ridge Regression](https://mlsanity.com/paper/2608.17132/) (2026)
- [Density Ratio-based Causal Discovery from Bivariate Continuous-Discrete Data](https://mlsanity.com/paper/2505.08371/) (2025)
- [Skewness-Robust Causal Discovery in Location-Scale Noise Models](https://mlsanity.com/paper/2511.14441/) (2025)

## Abstract (arXiv)

The problem of identifiability in linear parametric models (LPMs) whose nodes follow either an ordered logit model or a regular one-parameter exponential family is evaluated. The results go beyond classical structural equation models as well as results for nodes with observations from a homogeneous family of distributions. The main result establishes that the orientation of every edge joining an ordinal node to an exponential-family node is identifiable from the joint distribution alone at every parameter value, provided the ordinal node has at least three categories and the exponential-family node at least three points of support, with no restriction on the sufficient statistic. Converses show that both requirements are necessary: the three-category requirement is binding only for affine sufficient statistics, and the three-point requirement is binding under the canonical link. The guarantee extends to orienting every such mixed ordinal-exponential family edge of a given d-node undirected skeleton. Numerical experiments illustrate the theoretical results by successfully separating orientations within a Markov equivalence class, which are indistinguishable by conditional independence alone.