I would think that because bipolar transistors amplify the current not voltage. that the bias resistor would not be needed in this application.
I have never seen a pull up resistor used on a bipolar in such a simple application before.
Imagine that we literally put a capacitor between the base of the transistor and circuit ground. the charge would just drain through the transistor to the collector, maybe slowing the response to changing base current a little, not enough to notice unless it was a big giant electrolytic cap or something.
If we were dealing with a MOSFET, which operates on voltage, not current, then yes, we most definately would need a pull up or else the charge would just stay until it slowly discharged. but this is just a bipolar PNP.