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 $y$.
- 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.
- 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.