Lemma 3. Let X be a random variable taking values in X and let F be a family of measurable functions with f 2F: X Let G be a family of measurable functions with g 2G: X! [ M,M], Let ˆR Corollary 4. The inequalities in Lemma 3 can be strengthened to the following: X Using these sharper bounds in the expressions (obtained from McDiarmid's inequality) in Lemma 3 (and using 2 in place of) yields the first pair of equations. Lemma 5. Let h 2 H: A W X! [ M,M] such that if h 2 H, h 2 H, Y [ M,M], k: (A Z X) We analyze each of these four terms separately. A, X, Z! 0, so i Strictly Positive Definite, implies that E [h We tuned the architectures of the Naive Net and NMMR models on both the Demand and dSprite experiments. Within each experiment, the Naive Net and NMMR models used similar architectures. In the Demand experiment, both models consisted of 2-5 ("Network depth" in Table S1) fully connected layers with a variable number ("Network width") of hidden units.

Lemma 3. Let X be a random variable taking values in X and let F be a family of measurable functions with f 2F: X Let G be a family of measurable functions with g 2G: X! [ M,M], Let ˆR Corollary 4. The inequalities in Lemma 3 can be strengthened to the following: X Using these sharper bounds in the expressions (obtained from McDiarmid's inequality) in Lemma 3 (and using 2 in place of) yields the first pair of equations. Lemma 5. Let h 2 H: A W X! [ M,M] such that if h 2 H, h 2 H, Y [ M,M], k: (A Z X) We analyze each of these four terms separately. A, X, Z! 0, so i Strictly Positive Definite, implies that E [h We tuned the architectures of the Naive Net and NMMR models on both the Demand and dSprite experiments. Within each experiment, the Naive Net and NMMR models used similar architectures. In the Demand experiment, both models consisted of 2-5 ("Network depth" in Table S1) fully connected layers with a variable number ("Network width") of hidden units.

The No Unmeasured Confounding Assumption is widely used to identify causal effects in observational studies. Recent work on proximal inference has provided alternative identification results that succeed even in the presence of unobserved confounders, provided that one has measured a sufficiently rich set of proxy variables, satisfying specific structural conditions. However, proximal inference requires solving an ill-posed integral equation. Previous approaches have used a variety of machine learning techniques to estimate a solution to this integral equation, commonly referred to as the bridge function. However, prior work has often been limited by relying on pre-specified kernel functions, which are not data adaptive and struggle to scale to large datasets. In this work, we introduce a flexible and scalable method based on a deep neural network to estimate causal effects in the presence of unmeasured confounding using proximal inference. Our method achieves state of the art performance on two well-established proximal inference benchmarks. Finally, we provide theoretical consistency guarantees for our method.