Causal Inference

The distributions we observe in the world are the outcome of complicated stochastic processes. The mechanisms which set the value of one variable inter-lock with those which set other variables. When we make a causal prediction, we want to know what would happen if the usual mechanisms controlling random variable X were suspended and it was set to x. How would this change propagate to the other variables? What distribution would result for the response variable Y? Causal inference is the undertaking of trying to answer causal questions from empirical data. One of the common examples are the casual graphical models, which is a formalism for representing causal relations using graphs.

Researchers

Prof.dr. A.J. (Annoesjka) Cabo

Prof.dr. A.W. (Aad) van der Vaart

Key Publications

2020 - Kolyan Ray, Aad van der Vaart - The Annals of Statistics

Semiparametric Bayesian causal inference

We develop a semiparametric Bayesian approach for estimating the mean response in a missing data model with binary outcomes and a nonparametrically modelled propensity score. Equivalently, we estimate the causal effect of a treatment, correcting nonparametrically for confounding. We show that standard Gaussian process priors satisfy a semiparametric Bernstein–von Mises theorem under smoothness conditions. We further propose a novel propensity score-dependent prior that provides efficient inference under strictly weaker conditions. We also show that it is theoretically preferable to model the covariate distribution with a Dirichlet process or Bayesian bootstrap, rather than modelling its density.

2017 - James M. Robins, Lingling Li, Rajarshi Mukherjee, Eric Tchetgen Tchetgen, Aad van der Vaart - The Annals of Statistics

Minimax estimation of a functional on a structured high-dimensional model

We introduce a new method of estimation of parameters in semiparametric and nonparametric models. The method employs \(U\)-statistics that are based on higher-order influence functions of the parameter of interest, which extend ordinary linear influence functions, and represent higher derivatives of this parameter. For parameters for which the representation cannot be perfect the method often leads to a bias-variance trade-off, and results in estimators that converge at a slower than \(\sqrt{n}\)-rate. In a number of examples, the resulting rate can be shown to be optimal. We are particularly interested in estimating parameters in models with a nuisance parameter of high dimension or low regularity, where the parameter of interest cannot be estimated at \(\sqrt{n}\)-rate, but we also consider efficient \(\sqrt{n}\)-estimation using novel nonlinear estimators. The general approach is applied in detail to the example of estimating a mean response when the response is not always observed.

2012 - S. A. Murphy, A. W. Van Der Vaart - Journal of the American Statistical Association

On Profile Likelihood

We show that semiparametric profile likelihoods, where the nuisance parameter has been profiled out, behave like ordinary likelihoods in that they have a quadratic expansion. In this expansion the score function and the Fisher information are replaced by the efficient score function and efficient Fisher information. The expansion may be used, among others, to prove the asymptotic normality of the maximum likelihood estimator, to derive the asymptotic chi-squared distribution of the log-likelihood ratio statistic, and to prove the consistency of the observed information as an estimator of the inverse of the asymptotic variance.

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