# Are Bayesian networks typically faithful?

arXiv: https://arxiv.org/abs/2410.16004 · HTML: https://mlsanity.com/paper/2410.16004/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 46 references that are in the map.
- First posted on arXiv 21 Oct 2024.

## Builds on (most cited first, 9 of 9)

- [Foundations of Structural Causal Models with Cycles and Latent Variables](https://mlsanity.com/paper/1611.06221/) (2016)
- [Causal Networks: Semantics and Expressiveness](https://mlsanity.com/paper/1304.2379/) (2013)
- [Strong Completeness and Faithfulness in Bayesian Networks](https://mlsanity.com/paper/1302.4973/) (2013)
- [A Discovery Algorithm for Directed Cyclic Graphs](https://mlsanity.com/paper/1302.3599/) (2013)
- [Constraint-Based Causal Discovery using Partial Ancestral Graphs in the presence of Cycles](https://mlsanity.com/paper/2005.00610/) (2020)
- [A Constraint-Based Algorithm For Causal Discovery with Cycles, Latent Variables and Selection Bias](https://mlsanity.com/paper/1805.02087/) (2018)
- [A Topological Perspective on Causal Inference](https://mlsanity.com/paper/2107.08558/) (2021)
- [On Learning Causal Structures from Non-Experimental Data without Any Faithfulness Assumption](https://mlsanity.com/paper/1802.07051/) (2018)
- [The Topology of Statistical Verifiability](https://mlsanity.com/paper/1707.09378/) (2017)

## Built on by (most cited first, 6 of 6)

- [Since Faithfulness Fails: The Performance Limits of Neural Causal Discovery](https://mlsanity.com/paper/2502.16056/) (2025)
- [Conditional Forecasts and Proper Scoring Rules for Reliable and Accurate Performative Predictions](https://mlsanity.com/paper/2510.21335/) (2025)
- [Robust Weighted Triangulation of Causal Effects Under Model Uncertainty](https://mlsanity.com/paper/2603.01119/) (2026)
- [Causal Learning with the Invariance Principle](https://mlsanity.com/paper/2605.13589/) (2026)
- [Verifying formulas for interventional distributions](https://mlsanity.com/paper/2607.13883/) (2026)
- [High-Order Markov Blanket Discovery via a k-Order Relaxation of the Faithfulness Assumption](https://mlsanity.com/paper/2607.26357/) (2026)

## Abstract (arXiv)

Faithfulness is a common assumption in causal inference, often motivated by the fact that the faithful parameters of linear Gaussian and discrete Bayesian networks are typical, and the folklore belief that this should also hold for other classes of Bayesian networks. We address this open question by showing that among all Bayesian networks over a given DAG, the faithful Bayesian networks are indeed `typical': they constitute a dense, open set with respect to the total variation metric. This does not directly imply that faithfulness is typical in restricted classes of Bayesian networks that are often considered in statistical applications. To this end we consider the class of Bayesian networks parametrised by conditional exponential families, for which we show that under regularity conditions, the faithful parameters constitute a dense and open set, the unfaithful parameters have Lebesgue measure zero, and the induced faithful distributions are open and dense in the weak topology. This extends the existing results for linear Gaussian and discrete Bayesian networks. We also show for nonparametric classes of Bayesian networks with uniformly equicontinuous and uniformly bounded conditional densities that the faithful Bayesian networks are open and dense in the weak topology. All these results also hold for Bayesian networks with latent variables, if faithfulness is only required to hold with respect to the latent projection. Finally, for the considered conditional exponential family parametrisations and nonparametric conditional density models, the topological properties of conditional independence imply the existence of a consistent conditional independence test. Together with the topological properties of faithfulness, this implies that sound constraint-based causal discovery algorithms like PC and FCI are consistent on an open and dense -- and hence `typical' -- set of Bayesian networks.