The standard for a warrant is probable cause, which is more stringent than reasonable belief.

Reasonable belief is what allows for police to take warrantless actions. Cop sees someone in a neighborhood walking around looking inside car windows and trying door handles. He now has reasonable belief enough to temporarily detain that person and ask what he's doing. No arrest or search may be conducted.

vs.

A court issued warrant requires probable cause. Cop let the suspect go in the first example (as he should with no probable cause for an arrest), and the next day someone in the neighborhood reports that their car was broken into and their laptop stolen. Cop checks local pawn shops and finds the laptop, the person that sold it to the pawn shop is the same person the cop stopped last night. NOW the cop has enough probable cause to seek a search warrant to look for other stolen items.

Point being, reasonable belief or reasonable suspicion isn't and shouldn't be enough to search or detain. You need probable cause, and that probable cause needs to be affirmed by a judge and a warrant issued.