What is the probability that “The Detroit Tigers win today’s game” (which has not yet been played)? The truth of the proposition (in quotes) is not known and is therefore uncertain. Enter probability.
Some will used words to express their answer (“Pretty likely”, “They don’t have a chance”, “No way they can lose”), others will provide rough quantification (“90%”, “3 to 1 against”), while still more will provide serious quantification (“$50 bucks says they win”). Finally, some will not answer at all (“I have no idea”, “I hate baseball”).
Which of these is the right answer? Assuming nobody is fibbing, they all are. (The frequentist response is given below.) Each reply is subjective because each is conditional of a set of premises supplied by each individual, premises which may or may not be articulated.
For example, “Pretty likely, given that they won their last three and the Astros (their opponent) are dead last.” Another thinks, “They don’t have a chance because I suspect Justin Verlander (Tigers’ starting pitcher) is injured.” But when you ask the man who was willing to bet $50 why, he might say, “I don’t know. It feels like the right amount.” Or he might say, “I always bet $50 on the team I think has the best chance” which again fails to provide a list of premises why he thinks the Tigers have the best chance.
This kind of situation is what people have in mind when they think of subjective probability. Answers can range from no probability at all (“I hate baseball”), to vague but real probabilities (“Pretty likely”), to actual quantifications (“3 to 1 against”, “$50 to win”). All depend on individual premises which we may or may not be able to elicit. This includes those situations where the person doesn’t want to or has no answer. For example, you might be asked, “底特律老虎隊奪冠的概率今天的比賽是什麼”? If you don’t speak Mandarin and haven’t any idea of the context the best answer is, “I have no idea what you’re talking about.” (Real speakers of Mandarin will say the same thing of this translation.) That is, there is no probability for you.
It is never an answer to say, “‘The event will either happen or it won’t’ therefore, the probability is 50%”. That number can never be deduced from a tautology. That is, it is always (as in always) true “The event will either happen or it won’t” for any event, which is what makes it a tautology, and that adding a tautology to a list of premises cannot change the truth or probability of a proposition. Any number of tautologies may be added, not just one. For example, “At today’s game it will either rain or it’ll stay dry, Verlander will either pitch well or he won’t, so the Tigers will lose or they will win.” There is no content in that phrase except that the Tigers will play (and Verlander will be the pitcher).
From these simple examples, we may conclude several things. (1) Probability is not always quantifiable; that is, not every probability is a precise number; (2) Probability is sometimes a range (another example: “They’ll either lose or come close to losing”); (3) Probability can be a fixed number; (4) Not all probabilities can be known; (5) The weight of evidence, how stable the probability of the proposition appears, depends on the list and strength of the premises.
What a person says may be at odds with what he believes, not only because of deception, but because sometimes words take slightly different meanings for different people or because not everybody is attentive to grammar. Our man might list as one of his premises, “Verlander will either pitch well or he won’t”, which is formally a tautology and therefore of no probative value, but subjectively he gives more weight to “pitch well” than to “not pitch well”, and so this tautology-in-form is actually informative. This is why there is confusion on the subject.
Consider two men. One gives the premise, “I don’t know much about the Tigers, but they won their last three.” Another says, “The Tigers’ batting is on fire; here are their stats. And Verlander is the best pitcher in baseball, and here is why” plus many more (a real fan). But suppose both men say the chance the Tigers will win is 80%. Adding or subtracting a premise from the second man will not change his stated probability by a great degree. But adding or subtracting a premise (particularly subtracting!) from the first man changes his by a lot. We would say the weight of evidence of the first is less than that of the second, even though both have the same probability. And this is because of the differences in the premises.
What is the objective probability the Tigers win? There isn’t one, at least, not yet. And the frequentist probability? Same answer: there isn’t one yet.
Now the difference between subjective and objective probability is this: when presented with a list of premises (of unambiguous words) a subjectivist can state any probability for the conclusion (proposition) he wishes, but the objectivist must take the premises “As Is” and from these deduce the probability. The subjectivist is free, while the objectivist is bound. This is why there is no objective probability the Tigers win, because there is no “official” list of premises for the proposition.
The lack of an official list of premises is also why the frequentist must remain mute, because in order to calculate any probability the frequentist must embed the proposition of interest in an infinite (as in infinite) sequence of events which are just like the event on hand, except that the other events are “randomly” different. This constrains the type and kind of premises which are allowable. (I discuss “random” in another post.)
For example, if the “official” list—which merely means those premises we accept for the sake of argument—are one: “The Tigers always win 80% of the time against the Astros”, the objectivist must say (given the plain English definitions of the words) the probability of a win is 80%. The subjectivist may say, if he likes, 4%. He won’t usually, but he is free to to do. The frequentist may be tempted to say 80%, but he has to first add the premise that “Tigers vs. Astros” events are unchanging (except “randomly” different) and will exist in perpetuity. Perpetuity means “in the long run.” But as Keynes reminded us, “In the long run we shall all be dead.” In other words, unless the frequentist “cheats” and adds to the official list of premises suppositions about infinite “trials”, he is stuck. Incidentally, the subjectivist who does say other than 80% is also usually cheating by adding or subtracting from the official premise list, or by (subjectively) changing the meaning of the words.
Now this is not nearly yet a complete proof which shows that frequentism or subjectivism are doomed; merely a taste of things to come. What is clear is that probability can seem subjective, but only because, as was showed in Part I, the list of agreed upon premises for a proposition can be difficult or impossible to discover. Next time: simpler examples. Maybe where “priors” come from.