This is the adverb twin of presumptive: you do something, or treat something, on the basis of a reasonable belief about what is likely, not on full proof. A person may be treated as presumptively innocent. A hospital may start medicine presumptively, meaning they act on what is likely while they wait to be sure. It is a formal, careful word — the kind you hear in law and medicine. It does not mean “in a rude, too-bold way.” That would be presumptuously, from a cousin word with a different job.