Are Bayesian networks typically faithful?

• URL: http://arxiv.org/abs/2410.16004v2
• Date: Mon, 20 Jan 2025 09:40:00 GMT
• Title: Are Bayesian networks typically faithful?
• Authors: Philip Boeken, Patrick Forré, Joris M. Mooij,
• Abstract summary: We show that among all Bayesian networks over a given DAG, the faithful networks are indeed typical' We consider the class of Bayesian networks parametrised by conditional exponential families.
• Score: 12.27570686178551
• License:
• Abstract: Faithfulness is a ubiquitous 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. However, this does not imply that faithfulness is typical in restricted classes of Bayesian networks, as 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 mild regularity conditions, the faithful parameters constitute a dense, open set and the unfaithful parameters have Lebesgue measure zero, extending the existing results for linear Gaussian and discrete Bayesian networks. Finally, we show that the aforementioned results also hold for Bayesian networks with latent variables.

Related papers

• A Note on Bayesian Networks with Latent Root Variables [56.86503578982023]
  We show that the marginal distribution over the remaining, manifest, variables also factorises as a Bayesian network, which we call empirical. A dataset of observations of the manifest variables allows us to quantify the parameters of the empirical Bayesian net.
  arXiv  Detail & Related papers  (2024-02-26T23:53:34Z)
• Wide Deep Neural Networks with Gaussian Weights are Very Close to Gaussian Processes [1.0878040851638]
  We show that the distance between the network output and the corresponding Gaussian approximation scales inversely with the width of the network, exhibiting faster convergence than the naive suggested by the central limit theorem. We also apply our bounds to obtain theoretical approximations for the exact posterior distribution of the network, when the likelihood is a bounded Lipschitz function of the network output evaluated on a (finite) training set.
  arXiv  Detail & Related papers  (2023-12-18T22:29:40Z)
• Numerically assisted determination of local models in network scenarios [55.2480439325792]
  We develop a numerical tool for finding explicit local models that reproduce a given statistical behaviour. We provide conjectures for the critical visibilities of the Greenberger-Horne-Zeilinger (GHZ) and W distributions. The developed codes and documentation are publicly available at281.com/mariofilho/localmodels.
  arXiv  Detail & Related papers  (2023-03-17T13:24:04Z)
• On the Effective Number of Linear Regions in Shallow Univariate ReLU Networks: Convergence Guarantees and Implicit Bias [50.84569563188485]
  We show that gradient flow converges in direction when labels are determined by the sign of a target network with $r$ neurons. Our result may already hold for mild over- parameterization, where the width is $tildemathcalO(r)$ and independent of the sample size.
  arXiv  Detail & Related papers  (2022-05-18T16:57:10Z)
• Robust Estimation for Nonparametric Families via Generative Adversarial Networks [92.64483100338724]
  We provide a framework for designing Generative Adversarial Networks (GANs) to solve high dimensional robust statistics problems. Our work extend these to robust mean estimation, second moment estimation, and robust linear regression. In terms of techniques, our proposed GAN losses can be viewed as a smoothed and generalized Kolmogorov-Smirnov distance.
  arXiv  Detail & Related papers  (2022-02-02T20:11:33Z)
• Uncertainty Reasoning for Probabilistic Petri Nets via Bayesian Networks [1.471992435706872]
  We exploit extended Bayesian networks for uncertainty reasoning on Petri nets. In particular, Bayesian networks are used as symbolic representations of probability distributions. We show how to derive information from a modular Bayesian net.
  arXiv  Detail & Related papers  (2020-09-30T17:40:54Z)
• Being Bayesian, Even Just a Bit, Fixes Overconfidence in ReLU Networks [65.24701908364383]
  We show that a sufficient condition for a uncertainty on a ReLU network is "to be a bit Bayesian calibrated" We further validate these findings empirically via various standard experiments using common deep ReLU networks and Laplace approximations.
  arXiv  Detail & Related papers  (2020-02-24T08:52:06Z)
• Bayesian Deep Learning and a Probabilistic Perspective of Generalization [56.69671152009899]
  We show that deep ensembles provide an effective mechanism for approximate Bayesian marginalization. We also propose a related approach that further improves the predictive distribution by marginalizing within basins of attraction.
  arXiv  Detail & Related papers  (2020-02-20T15:13:27Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.