We’ll start by skipping the famous Yogism about predicting. But living life successfully requires us to constantly predict the future – events that may or not happen in a time yet to come. It’s necessary to do that because we have to prepare for various futures by deciding what actions to take right now, or at least before the predicted event occurs. Most of our predictions are about whether a certain thing will happen or not, whether it will happen before a given time or not, or which one of, say, five possible things will come true. In short, very discrete things, things or events that can be characterized by a ‘yes’ or ‘no’. Then there are predictions that aren’t discrete and cover a range of values that are the anticipated outcome. Things like how many marbles are in a big jar, when will Billy get home from work, how much rain will the coming storm bring, what will our house sell for, a candidate’s poll number after the debate, the closing value of the Dow Industrials at week’s end, this year’s GDP, and so forth.
A little thought shows that all the CURRENT answers to such questions or unknown values are really what the techies call random variables. But for each such random variable or r.v., all of us have a hunch or belief about how well we can characterize our ignorance about the actual outcome or value realized in the future.
For discrete (random) events we often characterize the chance that something will come to pass as, say, ‘one out of three’ or state your odds such as ‘three to one it’ll rain tomorrow’. If we really don’t have a clue as to the outcome, we’ll use the old coin flip 50-50 analogy. For continuous r.v.s most people will state the smallest range of values that for them encompass all or almost all the possibilities. They’ll use phrases like ‘Tomorrow’s high temperature will be somewhere in the 45 to 55F range’ or ‘The fourth quarter GDP growth will be in the 1.5 to 2.0% range.’
However, when pressed or thought about some more, almost everyone is able to pick a value somewhere in their stated range which they believe will be at or near the most likely value that comes to pass. And they can also tell you how confident they are about choosing that most likely or best guess value by using words like ‘pretty sure’, ’50-50 it’ll be somewhere near there’, ‘it’s just a wild hunch’, ‘I’d be surprised if it’s not close to that’, and so on.
Well, it turns out that you can usefully summarize your belief in such a future value by just stating your intuition in four numbers – the low/high of the range (L and H), the best guess or most likely value within that range (M), and you confidence (C) in the best guess. C values will range between zero and one – low C values for little confidence and higher values expressing more confidence. These four values or the 4-tuple – [L, H, M, C] - expressing your belief defines a very useful and powerful measure of probability that can be combined with other such measures to judge, say, who’s prediction is best or what level of risk is represented by such a ‘state of knowledge’.