This paper proposes a confidence interval construction for heterogeneous treatment effects in the context of multi-stage experiments with $N$ samples and high-dimensional, $d$, confounders. Our focus is on the case of $d\gg N$, but the results obtained also apply to low-dimensional cases. We showcase that the bias of regularized estimation, unavoidable in high-dimensional covariate spaces, is mitigated with a simple double-robust score. In this way, no additional bias removal is necessary, and we obtain root-$N$ inference results while allowing multi-stage interdependency of the treatments and covariates. Memoryless property is also not assumed; treatment can possibly depend on all previous treatment assignments and all previous multi-stage confounders. Our results rely on certain sparsity assumptions of the underlying dependencies. We discover new product rate conditions necessary for robust inference with dynamic treatments.
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We study nonparametric Bayesian models for reversible multi-dimensional diffusions with periodic drift. For continuous observation paths, reversibility is exploited to prove a general posterior contraction rate theorem for the drift gradient vector field under approximation-theoretic conditions on the induced prior for the invariant measure. The general theorem is applied to Gaussian priors and $p$-exponential priors, which are shown to converge to the truth at the minimax optimal rate over Sobolev smoothness classes in any dimension
We present a new finite-sample analysis of M-estimators of locations in $\mathbb{R}^d$ using the tool of the influence function. In particular, we show that the deviations of an M-estimator can be controlled thanks to its influence function (or its score function) and then, we use concentration inequality on M-estimators to investigate the robust estimation of the mean in high dimension in a corrupted setting (adversarial corruption setting) for bounded and unbounded score functions. For a sample of size $n$ and covariance matrix $\Sigma$, we attain the minimax speed $\sqrt{Tr(\Sigma)/n}+\sqrt{\|\Sigma\|_{op}\log(1/\delta)/n}$ with probability larger than $1-\delta$ in a heavy-tailed setting. One of the major advantages of our approach compared to others recently proposed is that our estimator is tractable and fast to compute even in very high dimension with a complexity of $O(nd\log(Tr(\Sigma)))$ where $n$ is the sample size and $\Sigma$ is the covariance matrix of the inliers. In practice, the code that we make available for this article proves to be very fast.
This paper deals with robust inference for parametric copula models. Estimation using Canonical Maximum Likelihood might be unstable, especially in the presence of outliers. We propose to use a procedure based on the Maximum Mean Discrepancy (MMD) principle. We derive non-asymptotic oracle inequalities, consistency and asymptotic normality of this new estimator. In particular, the oracle inequality holds without any assumption on the copula family, and can be applied in the presence of outliers or under misspecification. Moreover, in our MMD framework, the statistical inference of copula models for which there exists no density with respect to the Lebesgue measure on $[0,1]^d$, as the Marshall-Olkin copula, becomes feasible. A simulation study shows the robustness of our new procedures, especially compared to pseudo-maximum likelihood estimation. An R package implementing the MMD estimator for copula models is available.
When are inferences (whether Direct-Likelihood, Bayesian, or Frequentist) obtained from partial data valid? This paper answers this question by offering a new asymptotic theory about inference with missing data that is more general than existing theories. By using more powerful tools from real analysis and probability theory than those used in previous research, it proves that as the sample size increases and the extent of missingness decreases, the mean-loglikelihood function generated by partial data and that ignores the missingness mechanism will almost surely converge uniformly to that which would have been generated by complete data; and if the data are Missing at Random, this convergence depends only on sample size. Thus, inferences from partial data, such as posterior modes, uncertainty estimates, confidence intervals, likelihood ratios, test statistics, and indeed, all quantities or features derived from the partial-data loglikelihood function, will be consistently estimated. They will approximate their complete-data analogues. This adds to previous research which has only proved the consistency and asymptotic normality of the posterior mode, and developed separate theories for Direct-Likelihood, Bayesian, and Frequentist inference. Practical implications of this result are discussed, and the theory is verified using a previous study of International Human Rights Law.
I propose kernel ridge regression estimators for long term causal inference, where a short term experimental data set containing randomized treatment and short term surrogates is fused with a long term observational data set containing short term surrogates and long term outcomes. I propose estimators of treatment effects, dose responses, and counterfactual distributions with closed form solutions in terms of kernel matrix operations. I allow covariates, treatment, and surrogates to be discrete or continuous, and low, high, or infinite dimensional. For long term treatment effects, I prove $\sqrt{n}$ consistency, Gaussian approximation, and semiparametric efficiency. For long term dose responses, I prove uniform consistency with finite sample rates. For long term counterfactual distributions, I prove convergence in distribution.
Methane is a strong greenhouse gas, with a higher radiative forcing per unit mass and shorter atmospheric lifetime than carbon dioxide. The remote sensing of methane in regions of industrial activity is a key step toward the accurate monitoring of emissions that drive climate change. Whilst the TROPOspheric Monitoring Instrument (TROPOMI) on board the Sentinal-5P satellite is capable of providing daily global measurement of methane columns, data are often compromised by cloud cover. Here, we develop a statistical model which uses nitrogen dioxide concentration data from TROPOMI to accurately predict values of methane columns, expanding the average daily spatial coverage of observations of the Permian Basin from 16% to 88% in the year 2019. The addition of predicted methane abundances at locations where direct observations are not available will support inversion methods for estimating methane emission rates at shorter timescales than is currently possible.
Unobserved confounding is one of the main challenges when estimating causal effects. We propose a causal reduction method that, given a causal model, replaces an arbitrary number of possibly high-dimensional latent confounders with a single latent confounder that takes values in the same space as the treatment variable, without changing the observational and interventional distributions the causal model entails. This allows us to estimate the causal effect in a principled way from combined data without relying on the common but often unrealistic assumption that all confounders have been observed. We apply our causal reduction in three different settings. In the first setting, we assume the treatment and outcome to be discrete. The causal reduction then implies bounds between the observational and interventional distributions that can be exploited for estimation purposes. In certain cases with highly unbalanced observational samples, the accuracy of the causal effect estimate can be improved by incorporating observational data. Second, for continuous variables and assuming a linear-Gaussian model, we derive equality constraints for the parameters of the observational and interventional distributions. Third, for the general continuous setting (possibly nonlinear or non-Gaussian), we parameterize the reduced causal model using normalizing flows, a flexible class of easily invertible nonlinear transformations. We perform a series of experiments on synthetic data and find that in several cases the number of interventional samples can be reduced when adding observational training samples without sacrificing accuracy.
Counterfactual explanations are usually generated through heuristics that are sensitive to the search's initial conditions. The absence of guarantees of performance and robustness hinders trustworthiness. In this paper, we take a disciplined approach towards counterfactual explanations for tree ensembles. We advocate for a model-based search aiming at "optimal" explanations and propose efficient mixed-integer programming approaches. We show that isolation forests can be modeled within our framework to focus the search on plausible explanations with a low outlier score. We provide comprehensive coverage of additional constraints that model important objectives, heterogeneous data types, structural constraints on the feature space, along with resource and actionability restrictions. Our experimental analyses demonstrate that the proposed search approach requires a computational effort that is orders of magnitude smaller than previous mathematical programming algorithms. It scales up to large data sets and tree ensembles, where it provides, within seconds, systematic explanations grounded on well-defined models solved to optimality.
Generative models (GMs) such as Generative Adversary Network (GAN) and Variational Auto-Encoder (VAE) have thrived these years and achieved high quality results in generating new samples. Especially in Computer Vision, GMs have been used in image inpainting, denoising and completion, which can be treated as the inference from observed pixels to corrupted pixels. However, images are hierarchically structured which are quite different from many real-world inference scenarios with non-hierarchical features. These inference scenarios contain heterogeneous stochastic variables and irregular mutual dependences. Traditionally they are modeled by Bayesian Network (BN). However, the learning and inference of BN model are NP-hard thus the number of stochastic variables in BN is highly constrained. In this paper, we adapt typical GMs to enable heterogeneous learning and inference in polynomial time.We also propose an extended autoregressive (EAR) model and an EAR with adversary loss (EARA) model and give theoretical results on their effectiveness. Experiments on several BN datasets show that our proposed EAR model achieves the best performance in most cases compared to other GMs. Except for black box analysis, we've also done a serial of experiments on Markov border inference of GMs for white box analysis and give theoretical results.
Amortized inference has led to efficient approximate inference for large datasets. The quality of posterior inference is largely determined by two factors: a) the ability of the variational distribution to model the true posterior and b) the capacity of the recognition network to generalize inference over all datapoints. We analyze approximate inference in variational autoencoders in terms of these factors. We find that suboptimal inference is often due to amortizing inference rather than the limited complexity of the approximating distribution. We show that this is due partly to the generator learning to accommodate the choice of approximation. Furthermore, we show that the parameters used to increase the expressiveness of the approximation play a role in generalizing inference rather than simply improving the complexity of the approximation.
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