Methods that rely on proxies, without imposing strong parametric structure, are increasingly used to deal with unobserved variables in causal inference. One influential line of this work reconstructs latent distributions used to identify the target functional by exploiting eigenvalue eigenvector structure. Within this framework, we first establish identification of the full data law in the presence of hidden outcomes, and then develop influence function based estimators for causal effects. To the best of our knowledge, this is the first work to develop influence function based estimators in this setting without relying on unbiased proxy measurements or partial observation, while achieving multiple robustness and desirable efficiency properties. We demonstrate the performance of our approach through simulation studies.

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