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Subjective Versus Objective Bayes (Versus Frequentism): Part II

John Maynard Keynes, Chance Master
Read Part I.

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.

Read Part III.


13 thoughts on “Subjective Versus Objective Bayes (Versus Frequentism): Part II Leave a comment

  1. John Maynard Keynes changed PDQ after he met the ballerina Lydia Lopokova. Keynes was not only gay but very gay, so to speak. Keynes saw Lydia perform in 1921 and fell madly and quickly for her, in a manner not the least bit platonic. Lydia survived her husband by 31 years.

  2. What on earth is “The Tigers always win 80% of the time against the Astros” supposed to mean? How can anything happen both “always” AND “80% of the time”? The best I can make of it is that “Tigers vs. Astros” events are unchanging (except “randomly” different so that “always” in some mysterious way doesn’t actually have to mean 100% of the time) and in the long run the randomness leads to a Tigers win 80% of the time. So it seems that the objectivist, who you say considers this plain English interpretation of the words to be equivalent to saying that the probability of a win is 80%, is really a frequentist under the skin.

  3. Alan,

    Quickly: If you don’t love that one, then switch to “The Tigers will win 8 of their next 10 games against the Astros.” It matters not where the “official” list of premises arises.

    Also see Part I where we do not always know the official list (“Garfield is a better president than Tyler”).

  4. But to assign a probability of 4/5 to each game requires a whole bunch of other premises in order to distinguish that one from the syntactically equivalent “Mr Tigre will succeed in 8 of his next 10 attempts to beat the high jump record”.(I was tempted to try to express this in terms of ‘per ardua ad astra’ but with p=1/2 decided not to)

  5. Surely probabilities are parameters of a model used to predict the future and must be judged on that basis only. In physics they can be highly reliable whether quantum mechanical or classical and are used extensively in statistical mechanics. Note that in the latter neither word means what the layman might think that they mean. Casinos are highly successful because the physical models they use, whether in card games or roulette, are very accurate. In more complicated systems where predictability (knowledge of the proper odds) is low, such as horse racing, a parimutuel system must be used. For off track or other sporting events the bookie must be more cagey. For here even a dutch booking system is not fool proof if the betting is not spread uniformly across the field as implied by the odds.

  6. This little if any difference between a dutch book and a parimutuel system. Shaving money off the top; setting the odds according to the remainder; and not entering onto the game itself will only fail if the total shaved is less than operating cost.

    Horse gamblers (as a whole) are quite astute when it comes to assessing winners. It can be shown that the odds at the track (adjusted for the take) are very consistent with success rates. But even if they weren’t, the tracks wouldn’t care.

    But that raises an interesting question: just what exactly is probability? Even if it is assumed it is a measure of confidence in an outcome, down deep, it’s still based on the number of successes / number of tries of something (all though that something may only be circumstantial to the outcome in question).

    For example, Will the Lakers win the next game? Factors such as how often they’ve won at home and away are often considered . One may counter with things like a star athlete forced to sit out the game as an example of a non-counting basis but, even there, what makes the athlete a star if it that assessment isn’t based on how often he is associated with wins/losses or how his performance seems to increase the score (increase meaning toward success which in golf is a numerical decrease).

    The frequentists have taken this idea to an extreme where attempting to answer the Lakers question is nonsensical. OTOH, Mr. Briggs in a previous post said something to the effect that the frequencies are driven by the probability which, in my view, is an extreme in the opposite direction from the frequentist.

    What is probability? My answer is: it is a measure of belief but the belief is driven by past experience in counting successes. For those prone to juggle math then the measure gets expressed as a numerical value. For those inclined to leaving the counting (and the juggling) to the subconscious, it gets expressed in fuzzy terms like “more likely”.

  7. I am reluctant to accept without question the labeling and characterization of a position by one who does not hold it. Apparently, the objectivist (at least as represented by Briggs) does not claim the existence of a unique consistent assignment of probabilities to all propositions. Perhaps the freedom claimed by a subjectivist does not really include assigning arbitrary values to the probability of a specific event about which we have a body of relevant information to consider. It may only be to set some assumed underlying (prior?) probabilities within a range of possibilities that can be shown not to significantly affect the conditional probabilities that are deduced from an extended sequence of observations. (Nothing to do with statsig here, just the idea of a limit). Similarly the frequentist may not be constrained to imagine the repetition of a specific event over time, but rather the (Gibbs?) ensemble of possible scenarios consistent with some agreed on collection of past observations (cf Briggs’ “official” premises). Are these really all that different?

  8. Alan,

    The bookies can not change the adds on bets that have already been set. This is what distinguishes them from a parimutuel on track betting system. A bookie that tried what you suggest would soon be out of business and maybe tarred and feathered. The parimutuel would return all bets and take a loss on the overhead, which is also what they do if no one bets on the winning horse. This does not happen enough to be a concern for them, but the independent bookie must live by his wits.

  9. William Sears,

    Explain what happens if everyone bets on the winning horse.

    The tracks would lose money but not because of something inherent in the parimutuel system. In the U.S. anyway the minimum payoff is $2.10/$2 bet. However, that’s because of the requirement (real or perceived) that a winning bet must return a profit. There really (in theory) nothing stopping them from simply dividing up the pool returning less than the original bet. That may drive away customers. It’s surprising that many track bettors have no idea who they are betting against.

    Incidentally, if you were to look closely, you would see that the amount returned is not equal to the pool minus the take. That’s because the odds are truncated to the nearest nickel (in the U.S. anyway). It’s also why converting the published odds to a probability value then normalizing results in an effective take higher than advertised. It’s called breakage. The leftover is saved for those rainy negative pool days. They are allowed to pocket any remaining at the end of the year.

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