Review Logic: If our premises are that we have a six-sided die that we’ll toss, and that only one side will show on that toss, and that just one side of the six has six spots, then the probability that a six spot will show given these premises is 1/6. Probability, then, is a matter of logic. Usually non-deductive logic, and not necessarily inductive logic. Whew! We finally have that out of the way.
Frequentism The largest rival to logical probability is frequentism. This is the belief that the probability of some event is the limiting relative frequency of that event. Take an event, like a die roll, toss is a number of times which approaches infinity. Count the number of times it comes up “6 spot”, then divide by the number of throws. That limiting fraction becomes the probability. But only after we get to infinity. While this is a perfectly fine mathematical definition, it has nothing to do with reality.
What is the probability that “Hillary Clinton wins the 2012 USA presidential election”? This is a contingent event, and given that information, via logical probability, it is greater than a 0% chance and less than a 100% chance. Given other assumed information, like “She will be the Democrat nominee” then we can say, logically, the probability is 1/2 (implicitly, this assumes she has only one opponent). Incidentally, it is not pertinent that these assumptions are false in fact: assuming they are true, we can calculate a logical probability.
In any case, frequentism can never give a probability for any unique event, or any series of events that cannot, in theory, be infinite. And since those two classes encapsulate all real-world events of interest to humans, frequentism cannot deliver useful probabilities. Even if you wanted to embed the “Clinton wins” event into a set of events that can approach infinity, you are stuck with how. All women running for president of the USA? All women running for leader in democratic nations? All women whose husbands have been presidents? All women who take over any large organization? Each of these will arrive at difference answers—a theme that will often come back to haunt frequentist procedures.
Further, given the premise “Briggs was abducted by a green UFO”, the conclusion “Briggs was abducted by a UFO” has logical probability 1. But it can never have a relative frequency because, of course, it’s premise is false in fact. I think.
Subjective Bayesian The other name for logical probability is “objective Bayesian”, so you can imagine there is a lot of overlap with its subjective, willful brother. This is true: the math is, for all intents, identical; and in practice, nearly all subjectivists act like objectivists, so these objections are somewhat trivial.
However, a subjectivist is allowed to take, “If our premises are that we have a six-sided die that we’ll toss, and that only one side will show on that toss, and that just one side of the six has six spots” and announce the probability as any number he likes. He can say 0.01, or 0.987, or anything between 0 and 1—even including 0 and 1! However, as mentioned, hardly anybody does such a thing.
Next time Demystifying randomness. And finally some examples!