# Unbiasedness of confounding variable models

This is a question based on work in this paper: https://dl4physicalsciences.github.io/files/nips_dlps_2017_14.pdf

I am interested in causality and representations using probabilistic graphical models in general. You can see figure 1 of the paper for a graphical model of the relationship.

Suppose we have a confounding latent variable, $z$, that has a directional arc to $x$ and to $y$. $z$ is the latent variable and can have many dimensions and be an arbitrary functional form. Thus it is a stand in for all latent variables of this form. I’m interested in determining the causal impact of $x$ on $y$.

1. Will modeling with maximum likelihood yield consistent or unbiased estimates of the probability weight relating $x\rightarrow y$ assuming $x\rightarrow y$ is linear and I use a linear model? If it's nonlinear? Note that $z\rightarrow x$? and $z\rightarrow y$ can still have arbitrary functional forms and modeled by a neural network for example.
2. Will the estimates (i.e. the entire distribution) be unbiased in the full information Variational Inference this will be the case as well? I assume it will be unbiased because Bayes rule is the optimal thing to do, but curious if there are caveats.
• I tried my best to parse this question into an understandable form. In the paper it looks the "z" is not fully unobserved but has a set of indicator variables for the population structure... something I don't totally understand. You refer to "flexible nonlinear forms"... but of what? The indicator variables for "z" or other confounders? – AdamO Apr 12 '18 at 20:58
• Thanks for the sorely needed edits! I attempted to clarify your edits to explain what I wanted. – www3 Apr 12 '18 at 21:51