If you see the empty ice cream containers, the sprinkles littering the ground, a kicked can of Reddi-wip in the trash, you can surmise what has happened: Sundaes. To surmise is to form an opinion or make a guess about something.
If you surmise that something is true, you don't have much evidence or knowledge about it. Near synonyms are guess, conjecture, and suppose. You might say, "I can't even surmise what he would do in such a situation." Surmise came to English from the French surmettre "to accuse," which is formed from the prefix sur- "on, upon" plus mettre "to put" (from Latin mittere "to send").