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Machine learning can benefit from causal discovery for interpretation and from causal inference for generalization. In this line of research, a few invariant learning algorithms for out-of-distribution (OOD) generalization have been proposed by using multiple training environments to find invariant relationships. Some of them are focused on causal discovery as Invariant Causal Prediction (ICP), which finds causal parents of a variable of interest, and some directly provide a causal optimal predictor that generalizes well in OOD environments as Invariant Risk Minimization (IRM). This group of algorithms works under the assumption of multiple environments that represent different interventions in the causal inference context. Those environments are not normally available when working with observational data and real-world applications. Here we propose a method to generate them in an efficient way. We assess the performance of this unsupervised learning problem by implementing ICP on simulated data. We also show how to apply ICP efficiently integrated with our method for causal discovery. Finally, we proposed an improved version of our method in combination with ICP for datasets with multiple covariates where ICP and other causal discovery methods normally degrade in performance.

Inferring the parameters of ordinary differential equations (ODEs) from noisy observations is an important problem in many scientific fields. Currently, most parameter estimation methods that bypass numerical integration tend to rely on basis functions or Gaussian processes to approximate the ODE solution and its derivatives. Due to the sensitivity of the ODE solution to its derivatives, these methods can be hindered by estimation error, especially when only sparse time-course observations are available. We present a Bayesian collocation framework that operates on the integrated form of the ODEs and also avoids the expensive use of numerical solvers. Our methodology has the capability to handle general nonlinear ODE systems. We demonstrate the accuracy of the proposed method through a simulation study, where the estimated parameters and recovered system trajectories are compared with other recent methods. A real data example is also provided.

Probabilities of Causation play a fundamental role in decision making in law, health care and public policy. Nevertheless, their point identification is challenging, requiring strong assumptions such as monotonicity. In the absence of such assumptions, existing work requires multiple observations of datasets that contain the same treatment and outcome variables, in order to establish bounds on these probabilities. However, in many clinical trials and public policy evaluation cases, there exist independent datasets that examine the effect of a different treatment each on the same outcome variable. Here, we outline how to significantly tighten existing bounds on the probabilities of causation, by imposing counterfactual consistency between SCMs constructed from such independent datasets ('causal marginal problem'). Next, we describe a new information theoretic approach on falsification of counterfactual probabilities, using conditional mutual information to quantify counterfactual influence. The latter generalises to arbitrary discrete variables and number of treatments, and renders the causal marginal problem more interpretable. Since the question of 'tight enough' is left to the user, we provide an additional method of inference when the bounds are unsatisfactory: A maximum entropy based method that defines a metric for the space of plausible SCMs and proposes the entropy maximising SCM for inferring counterfactuals in the absence of more information.

Particle flow filters solve Bayesian inference problems by smoothly transforming a set of particles into samples from the posterior distribution. Particles move in state space under the flow of an McKean-Vlasov-Ito process. This work introduces the Variational Fokker-Planck (VFP) framework for data assimilation, a general approach that includes previously known particle flow filters as special cases. The McKean-Vlasov-Ito process that transforms particles is defined via an optimal drift that depends on the selected diffusion term. It is established that the underlying probability density - sampled by the ensemble of particles - converges to the Bayesian posterior probability density. For a finite number of particles the optimal drift contains a regularization term that nudges particles toward becoming independent random variables. Based on this analysis, we derive computationally-feasible approximate regularization approaches that penalize the mutual information between pairs of particles, and avoid particle collapse. Moreover, the diffusion plays a role akin to a particle rejuvenation approach that aims to alleviate particle collapse. The VFP framework is very flexible. Different assumptions on prior and intermediate probability distributions can be used to implement the optimal drift, and localization and covariance shrinkage can be applied to alleviate the curse of dimensionality. A robust implicit-explicit method is discussed for the efficient integration of stiff McKean-Vlasov-Ito processes. The effectiveness of the VFP framework is demonstrated on three progressively more challenging test problems, namely the Lorenz '63, Lorenz '96 and the quasi-geostrophic equations.

Recent years have seen a surge of interest in learning high-level causal representations from low-level image pairs under interventions. Yet, existing efforts are largely limited to simple synthetic settings that are far away from real-world problems. In this paper, we present Causal Triplet, a causal representation learning benchmark featuring not only visually more complex scenes, but also two crucial desiderata commonly overlooked in previous works: (i) an actionable counterfactual setting, where only certain object-level variables allow for counterfactual observations whereas others do not; (ii) an interventional downstream task with an emphasis on out-of-distribution robustness from the independent causal mechanisms principle. Through extensive experiments, we find that models built with the knowledge of disentangled or object-centric representations significantly outperform their distributed counterparts. However, recent causal representation learning methods still struggle to identify such latent structures, indicating substantial challenges and opportunities for future work. Our code and datasets will be available at //sites.google.com/view/causaltriplet.

Obtaining guarantees on the convergence of the minimizers of empirical risks to the ones of the true risk is a fundamental matter in statistical learning. Instead of deriving guarantees on the usual estimation error, the goal of this paper is to provide concentration inequalities on the distance between the sets of minimizers of the risks for a broad spectrum of estimation problems. In particular, the risks are defined on metric spaces through probability measures that are also supported on metric spaces. A particular attention will therefore be given to include unbounded spaces and non-convex cost functions that might also be unbounded. This work identifies a set of assumptions allowing to describe a regime that seem to govern the concentration in many estimation problems, where the empirical minimizers are stable. This stability can then be leveraged to prove parametric concentration rates in probability and in expectation. The assumptions are verified, and the bounds showcased, on a selection of estimation problems such as barycenters on metric space with positive or negative curvature, subspaces of covariance matrices, regression problems and entropic-Wasserstein barycenters.

Contrastive loss has been increasingly used in learning representations from multiple modalities. In the limit, the nature of the contrastive loss encourages modalities to exactly match each other in the latent space. Yet it remains an open question how the modality alignment affects the downstream task performance. In this paper, based on an information-theoretic argument, we first prove that exact modality alignment is sub-optimal in general for downstream prediction tasks. Hence we advocate that the key of better performance lies in meaningful latent modality structures instead of perfect modality alignment. To this end, we propose three general approaches to construct latent modality structures. Specifically, we design 1) a deep feature separation loss for intra-modality regularization; 2) a Brownian-bridge loss for inter-modality regularization; and 3) a geometric consistency loss for both intra- and inter-modality regularization. Extensive experiments are conducted on two popular multi-modal representation learning frameworks: the CLIP-based two-tower model and the ALBEF-based fusion model. We test our model on a variety of tasks including zero/few-shot image classification, image-text retrieval, visual question answering, visual reasoning, and visual entailment. Our method achieves consistent improvements over existing methods, demonstrating the effectiveness and generalizability of our proposed approach on latent modality structure regularization.

Causal discovery and causal reasoning are classically treated as separate and consecutive tasks: one first infers the causal graph, and then uses it to estimate causal effects of interventions. However, such a two-stage approach is uneconomical, especially in terms of actively collected interventional data, since the causal query of interest may not require a fully-specified causal model. From a Bayesian perspective, it is also unnatural, since a causal query (e.g., the causal graph or some causal effect) can be viewed as a latent quantity subject to posterior inference -- other unobserved quantities that are not of direct interest (e.g., the full causal model) ought to be marginalized out in this process and contribute to our epistemic uncertainty. In this work, we propose Active Bayesian Causal Inference (ABCI), a fully-Bayesian active learning framework for integrated causal discovery and reasoning, which jointly infers a posterior over causal models and queries of interest. In our approach to ABCI, we focus on the class of causally-sufficient, nonlinear additive noise models, which we model using Gaussian processes. We sequentially design experiments that are maximally informative about our target causal query, collect the corresponding interventional data, and update our beliefs to choose the next experiment. Through simulations, we demonstrate that our approach is more data-efficient than several baselines that only focus on learning the full causal graph. This allows us to accurately learn downstream causal queries from fewer samples while providing well-calibrated uncertainty estimates for the quantities of interest.

This paper focuses on the expected difference in borrower's repayment when there is a change in the lender's credit decisions. Classical estimators overlook the confounding effects and hence the estimation error can be magnificent. As such, we propose another approach to construct the estimators such that the error can be greatly reduced. The proposed estimators are shown to be unbiased, consistent, and robust through a combination of theoretical analysis and numerical testing. Moreover, we compare the power of estimating the causal quantities between the classical estimators and the proposed estimators. The comparison is tested across a wide range of models, including linear regression models, tree-based models, and neural network-based models, under different simulated datasets that exhibit different levels of causality, different degrees of nonlinearity, and different distributional properties. Most importantly, we apply our approaches to a large observational dataset provided by a global technology firm that operates in both the e-commerce and the lending business. We find that the relative reduction of estimation error is strikingly substantial if the causal effects are accounted for correctly.