I'm currently reading Pearl's piece (Pearl, 2009, 2nd edition) on causality and struggle to establish the link between nonparametric identification of a model and actual estimation. Unfortunately, Pearl himself is very silent on this topic.
To give an example, I have a simple model in mind with a causal path, $x \rightarrow z \rightarrow y$, and a confounder that affects all variables $w \rightarrow x$, $w \rightarrow z$ and $w \rightarrow y$. In addition, $x$ and $y$ are related by unobserved influences, $x \leftarrow \rightarrow y$. By the rules of do-calculus I now know that the post-intervention (discrete) probability distribution is given by:
$$ P(y \mid do(x)) = \sum_{w,z}\bigl[P(z\mid w,x)P(w)\sum_{x}\bigl[P(y\mid w,x,z)P(x\mid w)\bigr]\bigr]. $$
I know wonder how I can estimate this quantity (non-parametrically or by introducing parametric assumptions)? Especially for the case when $w$ is a set of several confounding variables and quantities of interest are continuous. To estimate the joint pre-intervention distribution of the data appears to be very impractical in this case. Does somebody know an application of Pearl's methods that deals with these problems? I would be very happy for a pointer.