Philosophers are territorial when it comes to the expression "begs the question." They insist that it has only one meaning, namely: to assume the point at issue. Though I'm a philosopher, I disagree. The expression is now widely used to mean "gives rise to [or brings about] the question." The expression is ambiguous. It has a meaning within philosophy (the technical meaning) and a meaning without philosophy (the everyday meaning).
I thought of a way for proponents of the everyday meaning to defend their usage. The word "beget" means "give rise to; bring about." If I beget the question, I give rise to (or bring about) the question. "Beg the question" can be understood as shorthand for "Beget the question." In case you're wondering, I got this idea from Rush Limbaugh, who uses the expression "beget the question" occasionally. Whether he's doing it for the reason I just gave or for some other reason, I don't know.