The secret to guessing more accurately with maths

What do a 20th-century physicist, an 18th-century statistician and an ancient Greek philosopher have in common? They all knew how to extrapolate with incredible accuracy. Suppose I showed you a box and asked you to guess what is inside, without providing any more details. You might think this is completely impossible, but the nature of the container provides some information - the contents must be smaller than the box, for example, while a solid metal box can hold liquids and withstand temperatures that a cardboard box would struggle with. Is there a way to describe this process of guessing with limited information in a mathematically sensible way?