"I got lost here (and in the following equations..."

https://arbital.com/p/5by

by Eric Rogstad Jul 13 2016 updated Jul 13 2016

A causal model goes beyond the graph by including specific probability functions $\\mathbb P(X\_i | \\mathbf{pa}\_i)$ for how to calculate the probability of each node $X\_i$ taking on the value $x\_i$ given the values $\\mathbf {pa}\_i$ of $x\_i$'s immediate ancestors\. It is implicitly assumed that the causal model factorizes, so that the probability of any value assignment $\\mathbf x$ to the whole graph can be calculated using the product:

I got lost here (and in the following equations). I think it's a combination of needing the "factorizes" redlink filled in, and not understanding the do() syntax.

Ah, one additional thing I'm confused about -- what do $X_i$ and $x_i$ refer to? I thought $X_i$ referred to the node (so that SEASON would be $X_0$, {RAINING, SPRINKLER} $X_1$, {SIDEWALK} $X_2$, and {SLIPPERY} $X_3$), but then I'm not sure what lowercase $x_i$ would refer to…