Start with standard logic; think syllogisms. This is not a trick question: Is it true or false that C = “George wears a hat”?
There is no way to answer, because why? Because of course the first parts of the syllogism, the premises, are missing. I know what these premises are, but I’m going to hold off telling you about them and again ask you to just have a guess whether C is true or false.
Well, C could be true or—this really is necessary to say—it could be false. The statement T = “C is either true or false” is called a tautology, and by the rules of logic we know that tautologies are always true. Point is that there is no information afforded us in T (in the tautology) to guess whether C is true or false. Be sure you get this.
There is no information elsewhere, either, though you may be tempted to make some up. You might say: Y = “I know this Briggs, who always wears a hat (and he looks quite rakish, too), therefore he’ll probably but not certainly make George wear one.” But you can’t from Y deduce C. It is not a valid argument to say “Y therefore C,” which I hope is obvious. With me so far?
Here’s a premise: P1 = “All Martians wears hats & George is a Martian.” Now it is valid to say “P1 therefore C”, which is to say, C is true given P1. Another way to state it: We deduce C given P1. Etc.
Here’s another premise: P2 = “No Martian wears hats & George is a Martian.” Now, given P2, C is false. Etc. Right?
Gist is that it is the premises which say whether C is true or false. C is not true or false “by itself”. The only statements which are true “by themselves” are those like T or axioms (like tautologies, these are self-obvious truths).
Everybody who has ever studied logic knows and agrees with everything I’ve said so far. There is no dispute about any of it, no controversy. Further, the material has the added benefit of being provably true. This is why logicians (the ones who don’t go nuts, an occupational hazard) are in such a happy place.
What does any of this have to do with Bayesian probability? Well, everything. Just like logical statements (including mathematical theorems etc.), all (as is all) probability statements are conditional on their premises, or evidence, sometimes called their “data.” And this is because probability statements just are logic statements, only where the conclusions might not be true or false, but somewhere in between.
It is because people forget that all probability statements are conditional that they say probability is subjective. Yet it is no more subjective than logic is: once the premises and the conclusion are set, whether the conclusion can be deduced or falsified depends not a whit on anybody’s subjective feelings or their betting proclivities (a popular way to argue for subjectivity).
On the other hand, probability and logic can sure seem subjective. Go back to the beginning and ask whether C is true. We don’t know unless we pick a premise. Which premise do we pick? Well, there is no guidance. If I pick P1 and you P2 then we come to opposite answers. The kicker is that we are both right, given our choices, which are indeed subjective. By “subjective” I mean there is no objective criteria arguing for either P1 or P2; the choice is yours and it is almost certainly the case that you cannot articulate why you went with one and not the other. (The choice of C was also subjective in this sense. If you understand this parenthetical remark, you understand all.)
The interview question
Consider this “interview question”, which is based on an episode of Columbo. Q = “Given any car in Los Angeles, what is the chance a contact lens will be found in its trunk”?1 This is the kind of quirky (though somewhat asinine) question a Google human resources person might ask you. The asker doesn’t think there is a right answer; he just wants to see how you arrive at your wrong one.
But there is a right answer: that is, there are in fact, at this fixed moment, a certain fixed number of cars in Los Angeles, some of which will have contact lenses in their trunks, many of which won’t. Consider R = “there are N cars M of which have contact lenses.” Given R, the probability Q is true is thus M/N, a number we deduce. We do not deduce N or M, however, because we have no premises from which to do so.
It is at this point where probability seems subjective. We are free to—yes, subjectively—choose whatever premises we like to give us information on N and M. I wager that no two human beings would arrive at an identical set. Consider the questions you might ask yourself: How big is L.A.? How many people? How many cars per people? It’s February 4, 6 a.m., how many people have driven in or out of the city? How many people wear contacts? How many would have stored contacts in a trunk? How many people might have leaned over and lost one? How many people stored a dead body in which the deceased wore contacts? Etc., etc. If you are a student of Bayesian probability, you might even go through some betting calculus to help you hone in on the target.
All of these considerations you will weight, plus and minus, and because I and the human resource interviewer insist, you must come to an answer. The point is, whatever set of premises you supply, once they are set, the answer is not itself subjective, whatever the answer is will necessarily follow. It will be deduced; the answer is not subjective given Q and your premises.
It is likely your answer will be vague, because your premises will be. Therefore your N/M will have fuzziness to it. This is fine, because not all probability is quantifiable.
Consider C (George’s hat) and P3 = “Some Martians wears hats & George is a Martian.” Then the probability of C given P3 is somewhere between 0 and 1, nobody can say where. If I insist that you give me a precise number, you are right to slap me in the face. But since I’m probably bigger than you, you might try to state a number after all. But if you do so, you are adding to P3 (or changing it). You cannot deduce a single number from P3, but you might be able to if you augment P3 with further evidence, which itself will be subjectively applied.
Probability is not subjective.
1Given the episode I saw, with Robert Culp as the bad guy, it is 1.