Experimental data does not suffer from confounding but is usually limited in both scope and scale.

While SE CE, it is not generally true thatSE = CE. Consider now an intervention onX0. In the example, this only happens when we set the weights, means and variances to very particular values. Here we present a slightly adapted version of Invariant Causal Prediction [27]. Under this approach, the complexity oftesting asingle setofpredictors ofsize k is the cost of performing a least-squares regression and computing the residuals (O(k2N)) and the cost of performing the t-test and F-test over each split of thee environments (O(eN)).

We aim at using a large observational sample and a possibly much smaller experimental sample.

We develop new methods to integrate experimental and observational data in causal inference. While randomized controlled trials offer strong internal validity, they are often costly and therefore limited in sample size. Observational data, though cheaper and often with larger sample sizes, are prone to biases due to unmeasured confounders. To harness their complementary strengths, we propose a systematic framework that formulates causal estimation as an empirical risk minimization (ERM) problem. A full model containing the causal parameter is obtained by minimizing a weighted combination of experimental and observational losses--capturing the causal parameter's validity and the full model's fit, respectively. The weight is chosen through cross-validation on the causal parameter across experimental folds. Our experiments on real and synthetic data show the efficacy and reliability of our method. We also provide theoretical non-asymptotic error bounds.

While traditional program evaluations typically rely on surveys to measure outcomes, certain economic outcomes such as living standards or environmental quality may be infeasible or costly to collect. As a result, recent empirical work estimates treatment effects using remotely sensed variables (RSVs), such mobile phone activity or satellite images, instead of ground-truth outcome measurements. Common practice predicts the economic outcome from the RSV, using an auxiliary sample of labeled RSVs, and then uses such predictions as the outcome in the experiment. We prove that this approach leads to biased estimates of treatment effects when the RSV is a post-outcome variable. We nonparametrically identify the treatment effect, using an assumption that reflects the logic of recent empirical research: the conditional distribution of the RSV remains stable across both samples, given the outcome and treatment. Our results do not require researchers to know or consistently estimate the relationship between the RSV, outcome, and treatment, which is typically mis-specified with unstructured data. We form a representation of the RSV for downstream causal inference by predicting the outcome and predicting the treatment, with better predictions leading to more precise causal estimates. We re-evaluate the efficacy of a large-scale public program in India, showing that the program's measured effects on local consumption and poverty can be replicated using satellite

Experimental and observational studies often lack validity due to untestable assumptions. We propose a double machine learning approach to combine experimental and observational studies, allowing practitioners to test for assumption violations and estimate treatment effects consistently. Our framework tests for violations of external validity and ignorability under milder assumptions. When only one assumption is violated, we provide semi-parametrically efficient treatment effect estimators. However, our no-free-lunch theorem highlights the necessity of accurately identifying the violated assumption for consistent treatment effect estimation. We demonstrate the applicability of our approach in three real-world case studies, highlighting its relevance for practical settings.

The unit selection problem is to identify a group of individuals who are most likely to exhibit a desired mode of behavior, for example, selecting individuals who would respond one way if incentivized and a different way if not. The unit selection problem consists of evaluation and search subproblems. Li and Pearl defined the "benefit function" to evaluate the average payoff of selecting a certain individual with given characteristics. The search subproblem is then to design an algorithm to identify the characteristics that maximize the above benefit function. The hardness of the search subproblem arises due to the large number of characteristics available for each individual and the sparsity of the data available in each cell of characteristics. In this paper, we present a machine learning framework that uses the bounds of the benefit function that are estimable from the finite population data to learn the bounds of the benefit function for each cell of characteristics. Therefore, we could easily obtain the characteristics that maximize the benefit function.

The probabilities of causation are commonly used to solve decision-making problems. Tian and Pearl derived sharp bounds for the probability of necessity and sufficiency (PNS), the probability of sufficiency (PS), and the probability of necessity (PN) using experimental and observational data. The assumption is that one is in possession of a large enough sample to permit an accurate estimation of the experimental and observational distributions. In this study, we present a method for determining the sample size needed for such estimation, when a given confidence interval (CI) is specified. We further show by simulation that the proposed sample size delivered stable estimations of the bounds of PNS.

Unobserved confounding is one of the main challenges when estimating causal effects. We propose a novel causal reduction method that replaces an arbitrary number of possibly high-dimensional latent confounders with a single latent confounder that lives in the same space as the treatment variable without changing the observational and interventional distributions entailed by the causal model. After the reduction, we parameterize the reduced causal model using a flexible class of transformations, so-called normalizing flows. We propose a learning algorithm to estimate the parameterized reduced model jointly from observational and interventional data. This allows us to estimate the causal effect in a principled way from combined data. We perform a series of experiments on data simulated using nonlinear causal mechanisms and find that we can often substantially reduce the number of interventional samples when adding observational training samples without sacrificing accuracy. Thus, adding observational data may help to more accurately estimate causal effects even in the presence of unobserved confounders.