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1. Adaptive Bayesian Estimation of Discrete‐Continuous Distributions Under Smoothness and Sparsity

   https://doi.org/10.3982/ECTA17884  

   Norets, Andriy; Pelenis, Justinas (January 2022, Econometrica)

   We consider nonparametric estimation of a mixed discrete‐continuous distribution under anisotropic smoothness conditions and a possibly increasing number of support points for the discrete part of the distribution. For these settings, we derive lower bounds on the estimation rates. Next, we consider a nonparametric mixture of normals model that uses continuous latent variables for the discrete part of the observations. We show that the posterior in this model contracts at rates that are equal to the derived lower bounds up to a log factor. Thus, Bayesian mixture of normals models can be used for (up to a log factor) optimal adaptive estimation of mixed discrete‐continuous distributions. The proposed model demonstrates excellent performance in simulations mimicking the first stage in the estimation of structural discrete choice models. 

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2. Towards Characterizing Domain Counterfactuals for Invertible Latent Causal Models

   Zhou, Zeyu; Bai, Ruqi; Kulinski, Sean; Kocaoglu, Murat; Inouye, David I (May 2024, International Conference on Learning Representations)

   Answering counterfactual queries has important applications such as explainability, robustness, and fairness but is challenging when the causal variables are unobserved and the observations are non-linear mixtures of these latent variables, such as pixels in images. One approach is to recover the latent Structural Causal Model (SCM), which may be infeasible in practice due to requiring strong assumptions, e.g., linearity of the causal mechanisms or perfect atomic interventions. Meanwhile, more practical ML-based approaches using naive domain translation models to generate counterfactual samples lack theoretical grounding and may construct invalid counterfactuals. In this work, we strive to strike a balance between practicality and theoretical guarantees by analyzing a specific type of causal query called domain counterfactuals, which hypothesizes what a sample would have looked like if it had been generated in a different domain (or environment). We show that recovering the latent SCM is unnecessary for estimating domain counterfactuals, thereby sidestepping some of the theoretic challenges. By assuming invertibility and sparsity of intervention, we prove domain counterfactual estimation error can be bounded by a data fit term and intervention sparsity term. Building upon our theoretical results, we develop a theoretically grounded practical algorithm that simplifies the modeling process to generative model estimation under autoregressive and shared parameter constraints that enforce intervention sparsity. Finally, we show an improvement in counterfactual estimation over baseline methods through extensive simulated and image-based experiments. 

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3. Causal Diffusion Autoencoders: Toward Counterfactual Generation via Diffusion Probabilistic Models

   https://doi.org/10.3233/FAIA240780  

   Komanduri, Aneesh; Zhao, Chen; Chen, Feng; Wu, Xintao (October 2024, IOS Press)

   Diffusion probabilistic models (DPMs) have become the state-of-the-art in high-quality image generation. However, DPMs have an arbitrary noisy latent space with no interpretable or controllable semantics. Although there has been significant research effort to improve image sample quality, there is little work on representation-controlled generation using diffusion models. Specifically, causal modeling and controllable counterfactual generation using DPMs is an underexplored area. In this work, we propose CausalDiffAE, a diffusion-based causal representation learning framework to enable counterfactual generation according to a specified causal model. Our key idea is to use an encoder to extract high-level semantically meaningful causal variables from high-dimensional data and model stochastic variation using reverse diffusion. We propose a causal encoding mechanism that maps high-dimensional data to causally related latent factors and parameterize the causal mechanisms among latent factors using neural networks. To enforce the disentanglement of causal variables, we formulate a variational objective and leverage auxiliary label information in a prior to regularize the latent space. We propose a DDIM-based counterfactual generation procedure subject to do-interventions. Finally, to address the limited label supervision scenario, we also study the application of CausalDiffAE when a part of the training data is unlabeled, which also enables granular control over the strength of interventions in generating counterfactuals during inference. We empirically show that CausalDiffAE learns a disentangled latent space and is capable of generating high-quality counterfactual images. 

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4. Semiparametric Counterfactual Density Estimation

   https://doi.org/10.1093/biomet/asad017  

   Kennedy, Edward; Balakrishnan, Sivaraman; Wasserman, Larry (January 2023, Biometrika)

   Causal effects are often characterized with averages, which can give an incomplete picture of the underlying counterfactual distributions. Here we consider estimating the entire counterfactual density and generic functionals thereof. We focus on two kinds of target parameters. The first is a density approximation, defined by a projection onto a finite-dimensional model using a generalized distance metric, which includes f-divergences as well as Lp norms. The second is the distance between counterfactual densities, which can be used as a more nuanced effect measure than the mean difference, and as a tool for model selection. We study nonparametric efficiency bounds for these targets, giving results for smooth but otherwise generic models and distances. Importantly, we show how these bounds connect to means of particular non-trivial functions of counterfactuals, linking the problems of density and mean estimation. We go on to propose doubly robust-style estimators for the density approximations and distances, and study their rates of convergence, showing they can be optimally efficient in large nonparametric models. We also give analogous methods for model selection and aggregation, when many models may be available and of interest. Our results all hold for generic models and distances, but throughout we highlight what happens for particular choices, such as L2 projections on linear models, and KL projections on exponential families. Finally we illustrate by estimating the density of CD4 count among patients with HIV, had all been treated with combination therapy versus zidovudine alone, as well as a density effect. Our results suggest combination therapy may have increased CD4 count most for high-risk patients. Our methods are implemented in the freely available R package npcausal on GitHub. 

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5. Statistical Guarantees for Transformation Based Models with applications to Implicit Variational Inference

   Plummer, Sean; Zhou, Shuang; Bhattacharya, Anirban; Dunson, David; Pati, Debdeep (April 2021, Proceedings of Machine Learning Research)

   null (Ed.)

   Transformation-based methods have been an attractive approach in non-parametric inference for problems such as unconditional and conditional density estimation due to their unique hierarchical structure that models the data as flexible transformation of a set of common latent variables. More recently, transformation-based models have been used in variational inference (VI) to construct flexible implicit families of variational distributions. However, their use in both nonparametric inference and variational inference lacks theoretical justification. We provide theoretical justification for the use of non-linear latent variable models (NL-LVMs) in non-parametric inference by showing that the support of the transformation induced prior in the space of densities is sufficiently large in the L1 sense. We also show that, when a Gaussian process (GP) prior is placed on the transformation function, the posterior concentrates at the optimal rate up to a logarithmic factor. Adopting the flexibility demonstrated in the non-parametric setting, we use the NL-LVM to construct an implicit family of variational distributions, deemed GP-IVI. We delineate sufficient conditions under which GP-IVI achieves optimal risk bounds and approximates the true posterior in the sense of the Kullback–Leibler divergence. To the best of our knowledge, this is the first work on providing theoretical guarantees for implicit variational inference. 

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