Abstract
Causal id (for discrete random variables) is not possible in the presence of a latent global confounder. But this can change if it is assumed that the confounder ranges over only k values. For binary observables that are independent conditional on the confounder, it is known that the number of observables must scale linearly in k. In the current work we consider the situation where there are m>1 independent latent global confounders, with log-linear effect on the observables. (This is closely related to ``restricted Boltzmann machines'' that have been of interest in learning theory and algebraic statistics.) The identification problem can be solved by reduction to the single-confounder case, but this requires ~ k^m observables. We show---in the very simplest case of this problem---that the exponential in m blow-up is unnecessary: km observables suffice. The proof relies on a root interlacing phenomenon. The general case remains the subject of ongoing work. Joint work with Spencer Gordon