In the book: Bayesian Networks With Examples in R, the author does this independence test:
As I see it, this works both ways, we test if travel is independent of education likewise if education is independent of travel. (there is no order, we test independence)
This is the DAG we are working with:
What I don't understand is why only O,R were given (only the parents of travel and not the parents of education, why?)
That is just a function of the question being asked here.
You want to know if you can say Travel is dependent on Education to a degree not explained by the Occupation and Residence factors.
You're not asking what Education is dependent on, nor are you asking whether Age and Sex are or are not a factors in Travel. You either know or assume the conditional independence structure of the rest of the DAG, that's a separate problem altogether.
Under these assumptions, Travel is conditionally independent of Age and Sex, and thus $p(T|O,R,E,A,S) = p(T|O,R,E)$ by the very definition of conditional independence.
