Statistical hypothesis testing is a central problem in empirical inference. Observing data from a distribution P * , one is interested in the hypothesis P * ∈ H0 and requires any test to control the probability of false rejections. In this work, we introduce statistical testing under distributional shifts. We are still interested in a target hypothesis P * ∈ H0, but observe data from a distribution Q * in an observational domain. We assume that P * is related to Q * through a known shift τ and formally introduce a framework for hypothesis testing in this setting. We propose a general testing procedure that first resamples from the n observed data points to construct an auxiliary data set (mimicking properties of P * ) and then applies an existing test in the target domain. We prove that this procedure holds pointwise asymptotic level -if the target test holds pointwise asymptotic level, the size of the resample is at most o( √ n), and the resampling weights are well-behaved. We further show that if the map τ is unknown, it can, under mild conditions, be estimated from data, maintaining level guarantees. Testing under distributional shifts allows us to tackle a diverse set of problems. We argue that it may prove useful in reinforcement learning, we show how it reduces conditional to unconditional independence testing and we provide example applications in causal inference. Code is easy-to-use and will be available online.
Instrumental variable models allow us to identify a causal function between covariates X and a response Y , even in the presence of unobserved confounding. Most of the existing estimators assume that the error term in the response Y and the hidden confounders are uncorrelated with the instruments Z. This is often motivated by a graphical separation, an argument that also justifies independence. Posing an independence condition, however, leads to strictly stronger identifiability results. We connect to existing literature in econometrics and provide a practical method for exploiting independence that can be combined with any gradient-based learning procedure. We see that even in identifiable settings, taking into account higher moments may yield better finite sample results. Furthermore, we exploit the independence for distribution generalization. We prove that the proposed estimator is invariant to distributional shifts on the instruments and worstcase optimal whenever these shifts are sufficiently strong. These results hold even in the under-identified case where the instruments are not sufficiently rich to identify the causal function.
In this paper the computational aspects of probability calculations for dynamical partial sum expressions are discussed. Such dynamical partial sum expressions have many important applications, and examples are provided in the fields of reliability, product quality assessment, and stochastic control. While these probability calculations are ostensibly of a high dimension, and consequently intractable in general, it is shown how a recursive integration methodology can be implemented to obtain exact calculations as a series of two-dimensional calculations. The computational aspects of the implementaion of this methodology, with the adoption of Fast Fourier Transforms, are discussed.
Counterfactual inference has become a ubiquitous tool in online advertisement, recommendation systems, medical diagnosis, and finance. An accurate modelling of outcome distributions associated with different interventions-known as counterfactual distributions-is crucial for the success of these applications. In this work, we propose to model counterfactual distributions using a novel Hilbert space representation called counterfactual mean embedding (CME). The CME embeds the associated counterfactual distribution into a reproducing kernel Hilbert space (RKHS) endowed with a positive definite kernel, which allows us to perform causal inference over the entire landscape of the counterfactual distribution. Based on this representation, we propose a distributional treatment effect (DTE) which can quantify the causal effect over entire outcome distributions. Our approach is nonparametric as the CME can be estimated consistently from observational data without requiring any parametric assumption about the underlying distributions. We also establish a rate of convergence of the proposed estimator which depends on the smoothness of the conditional mean and the Radon-Nikodym derivative of the underlying marginal distributions. Furthermore, our framework also allows for more complex outcomes such as images, sequences, and graphs. Lastly, our experimental results on synthetic data and off-policy evaluation tasks demonstrate the advantages of the proposed estimator.
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