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1. Robust Estimation of Tree Structured Gaussian Graphical Models

   Katiyar, Ashish; Hoffmann, Jessica; Caramanis, Constantine (June 2019, Proceedings of Machine Learning Research)

   Consider jointly Gaussian random variables whose conditional independence structure is specified by a graphical model. If we observe realizations of the variables, we can compute the covariance matrix, and it is well known that the support of the inverse covariance matrix corresponds to the edges of the graphical model. Instead, suppose we only have noisy observations. If the noise at each node is independent, we can compute the sum of the covariance matrix and an unknown diagonal. The inverse of this sum is (in general) dense. We ask: can the original independence structure be recovered? We address this question for tree structured graphical models. We prove that this problem is unidentifiable, but show that this unidentifiability is limited to a small class of candidate trees. We further present additional constraints under which the problem is identifiable. Finally, we provide an O(n^3) algorithm to find this equivalence class of trees. 

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2. Adversarially-Robust Inference on Trees via Belief Propagation

   Hopkins, Samuel B; Li, Anqi (June 2024, Conference on Learning Theory. The 37th Annual Conference on Learning Theory (COLT 2024))

   We introduce and study the problem of posterior inference on tree-structured graphical models in the presence of a malicious adversary who can corrupt some observed nodes. In the well-studied broadcasting on trees model, corresponding to the ferromagnetic Ising model on a d-regular tree with zero external field, when a natural signal-to-noise ratio exceeds one (the celebrated Kesten-Stigum threshold), the posterior distribution of the root given the leaves is bounded away from Ber(1/2), and carries nontrivial information about the sign of the root. This posterior distribution can be computed exactly via dynamic programming, also known as belief propagation. We first confirm a folklore belief that a malicious adversary who can corrupt an inverse-polynomial fraction of the leaves of their choosing makes this inference impossible. Our main result is that accurate posterior inference about the root vertex given the leaves is possible when the adversary is constrained to make corruptions at a ρ-fraction of randomly-chosen leaf vertices, so long as the signal-to-noise ratio exceeds O(logd) and ρ≤cε for some universal c>0. Since inference becomes information-theoretically impossible when ρ≫ε, this amounts to an information-theoretically optimal fraction of corruptions, up to a constant multiplicative factor. Furthermore, we show that the canonical belief propagation algorithm performs this inference. 

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3. Constraint-based Causal Discovery from a Collection of Conditioning Sets

   Lee, Kenneth; Ribeiro, Bruno; Kocaoglu, Murat (August 2025, Proceedings of the Forty-first Conference on Uncertainty in Artificial Intelligence)

   In constraint-based causal discovery, the existing algorithms systematically use a series of conditional independence (CI) relations observed in the data to recover an equivalence class of causal graphs in the large sample limit. One limitation of these algorithms is that CI tests lose statistical power as conditioning set size increases with finite samples. Recent research proposes to limit the conditioning set size for robust causal discovery. However, the existing algorithms require exhaustive testing of all CI relations with conditioning set sizes up to a certain integer k. This becomes problematic in practice when variables with large support are present, as it makes CI tests less reliable due to near-deterministic relationships, thereby violating the faithfulness assumption. To address this issue, we propose a causal discovery algorithm that only uses CI tests where the conditioning sets are restricted to a given set of conditioning sets including the empty set C. We call such set of CI relations IC conditionally closed. We define the notion of C-Markov equivalence: two causal graphs are C-Markov equivalent if they entail the same set of CI constraints from IC. We propose a graphical representation of C-Markov equivalence and characterize such equivalence between two causal graphs. Our proposed algorithm called the C-PC algorithm is sound for learning the C-Markov equivalence class. We demonstrate the utility of the proposed algorithm via synthetic and real-world experiments in scenarios where variables with large support or high correlation are present in the data. Our source code is available online at github.com/kenneth-lee-ch/cpc. 

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4. Learning nonparametric latent causal graphs with unknown interventions

   Jiang, Yibo; Aragam, Bryon (September 2024, Advances in Neural Information Processing Systems 36 (NeurIPS 2023) Main Conference Track)

   We establish conditions under which latent causal graphs are nonparametrically identifiable and can be reconstructed from unknown interventions in the latent space. Our primary focus is the identification of the latent structure in a measurement model, i.e. causal graphical models where dependence between observed variables is insignificant compared to dependence between latent representations, without making parametric assumptions such as linearity or Gaussianity. Moreover, we do not assume the number of hidden variables is known, and we show that at most one unknown intervention per hidden variable is needed. This extends a recent line of work on learning causal representations from observations and interventions. The proofs are constructive and introduce two new graphical concepts -- imaginary subsets and isolated edges -- that may be useful in their own right. As a matter of independent interest, the proofs also involve a novel characterization of the limits of edge orientations within the equivalence class of DAGs induced by unknown interventions. Experiments confirm that the latent graph can be recovered from data using our theoretical results. These are the first results to characterize the conditions under which causal representations are identifiable without making any parametric assumptions in a general setting with unknown interventions and without faithfulness. 

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5. The Lauritzen-Chen Likelihood For Graphical Models

   Ilya Shpitser (April 2023, Proceedings of The 26th International Conference on Artificial Intelligence and Statistics)

   Francisco Ruiz, Jennifer Dy (Ed.)

   Graphical models such as Markov random fields (MRFs) that are associated with undirected graphs, and Bayesian networks (BNs) that are associated with directed acyclic graphs, have proven to be a very popular approach for reasoning under uncertainty, prediction problems and causal inference. Parametric MRF likelihoods are well-studied for Gaussian and categorical data. However, in more complicated parametric and semi-parametric set- tings, likelihoods specified via clique potential functions are generally not known to be congenial (jointly well-specified) or non-redundant. Congenial and non-redundant DAG likelihoods are far simpler to specify in both parametric and semi-parametric settings by modeling Markov factors in the DAG factorization. However, DAG likelihoods specified in this way are not guaranteed to coincide in distinct DAGs within the same Markov equivalence class. This complicates likelihoods based model selection procedures for DAGs by “sneaking in” potentially un- warranted assumptions about edge orientations. In this paper we link a density function decomposition due to Chen with the clique factorization of MRFs described by Lauritzen to provide a general likelihood for MRF models. The proposed likelihood is composed of variationally independent, and non-redundant closed form functionals of the observed data distribution, and is sufficiently general to apply to arbitrary parametric and semi-parametric models. We use an extension of our developments to give a general likelihood for DAG models that is guaranteed to coincide for all members of a Markov equivalence class. Our results have direct applications for model selection and semi-parametric inference. 

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