# Counterfactual inference in sequential experimental design

Wednesday, March 2, 2022 - 4:00pm to 4:30pm

LIDS & Stats Tea

Raaz Dwivedi

LIDS & Harvard

LIDS Lounge

### Abstract

We consider the problem of counterfactual inference in sequentially designed experiments wherein a collection of $\mathbf{N}$ units each undergo a sequence of interventions for $\mathbf{T}$ time periods, based on policies that sequentially adapt over time. Our goal is counterfactual inference, i.e., estimate what would have happened if alternate policies were used, a problem that is inherently challenging due to the heterogeneity in the outcomes across units and time. To tackle this task, we introduce a suitable latent factor model where the potential outcomes are determined by exogenous unit and time level latent factors. Under suitable conditions, we show that it is possible to estimate the missing (potential) outcomes using a simple variant of nearest neighbors. First, assuming a bilinear latent factor model and allowing for an arbitrary adaptive sampling policy, we establish a distribution-free non-asymptotic guarantee for estimating the missing outcome of \emph{any} unit at \emph{any} time; under suitable regularity condition, this guarantee implies that our estimator is consistent. Second, for a generic non-parametric latent factor model, we establish that the estimate for the missing outcome of any unit at time $\mathbf{T}$ satisfies a central limit theorem as $\mathbf{T} \to \infty$, under suitable regularity conditions. Finally, en route to establishing this central limit theorem, we establish a non-asymptotic mean-squared-error bound for the estimate of the missing outcome of any unit at time $\mathbf{T}$. Our work extends the recently growing literature on inference with adaptively collected data by allowing for policies that pool across units, and also compliments the matrix completion literature when the entries are revealed sequentially in an arbitrarily dependent manner based on prior observed data.

### Biography

Raaz Dwivedi is currently a FODSI postdoc fellow with Prof. Susan Murphy in the Departments of Computer Science and Statistics at Harvard, and Prof. Devavrat Shah in the Laboratory of Information Decision and Systems (LIDS), Department of EECS at MIT. He finished his Ph.D. in the Department of EECS at UC Berkeley where he was co-advised by Prof. Martin Wainwright and Prof. Bin Yu, and bachelors in Electrical Engineering from IIT Bombay where he was advised by Prof. Vivek Borkar. His research interests are broadly in both theoretical and applied statistical machine learning, covering topics in random sampling, improving sample quality, and more recently, reinforcement learning and causal inference. He is a recipient of President of India Gold Medal, Institute Silver Medal, and Best Dissertation award at IIT Bombay, Berkeley Fellowship, and Outstanding graduate student instructor award at UC Berkeley, and a best student paper award for his work on optimal compression in near-linear time.