# Confounder Identification

Consider the following discrete mixture models with mixture-source variable $U \in {0, \ldots, k}$:

1. The $k$-MixIID problem handles a discrete mixture in which we are allowed to sample a single variables multiple times without re-sampling $U$. Alternatively, we can think of this as multiple i.i.d. samples of the within-source (constant $U$) distribution. This is sometimes called the $k$-coin problem because it corresponds to selecting a random biased coin and flipping it multiple times (and then repeating this process with another randomly selected coin).
2. The $k$-MixProd problem handles a discrete mixture with random variables that are independent from one another within each source (when conditioned on $U$). Graphically, this can be modeled as a single vertex $U$ with edges to vertices $\mathbf{X} = X_1, \ldots, X_n$ with no edges within $\mathbf{X}$.
3. The $k$-MixBND problem relaxes the independence assumption, allowing for an arbitrary dependence structure among the vertices, modeled by a Bayesian Netork DAG $\mathcal{G} = (\mathbf{V}, \mathbf{E})$.

The $k$-MixBND problem is fundamental to causal inference, where $U$ models an unobserved confounding effect of multiple populations, obscuring the causal relationships in the observable DAG $\mathcal{G}$. By solving the mixture problem and recovering the joint probability distribution on $\mathbf{V}, U$, graphically unidentifiable causal relationships become identifiable. We have developed the first algorithm to learn mixtures of non-empty DAGs. The algorithm involves a two-step reduction from $k$-MixIID $\rightarrow$ $k$-MixProd $\rightarrow$ $k$-MixBND.

We have contributed to complexity improvements on all three steps of this reduction. We have also contributed the first known algorithm for identifying causal structure in the mixture setting, which we refer to as “latent class confounding.”