Following Part II, here are some examples to show the differences between objectivist, subjectivist, and frequentist probabilities derived from fixed premises and set conclusions.
Not all probability is objectively strictly quantifiable. Premise: “Some X are F and x is X”. Conclusion: “x is F”. The objectivist can only say “The probability (given the premises) the conclusion is true is greater than 0 but less than or equal to 1.” This is because the logical “some” implies “at least some and perhaps all.”
The subjectivist is free to say, for example, “The probability (given the premises and my beliefs) the conclusion is true is 42.8%.” But he does so only by some mysterious introspection which, in effect, adds to or subtracts from the fixed premises. Of course, most subjectivists in practice would agree with the objectivist.
The frequentist, as with all his probabilities, must embed this fixed premise in an metaphysical infinite sequence of “identical but randomly different” premises, which is not exactly a coherent description. But again, like subjectivists, the frequentist would often in practice agree with the objectivist. If he disagrees, he is acting like the subjectivist because he is adding to or subtracting from the fixed premises in order to supply details of the infinite sequence. (There isn’t anything necessarily wrong with metaphysical infinite sequences: they are used all the time in analysis, for example. But in analysis, these sequences are precisely explained.)
In other words, the subjectivist and frequentist in disagreeing with the objectivist depart from the fixed premises, which is to say they change the evidence. But they speak as if they use the same evidence. To emphasize: they are not using the same premises. This is the primary reason for confusion and bad blood.
The problem is further highlighted by changing the question (keeping the same fixed premises and conclusion). “How much would you bet that x is F?” In this case, the problem unavoidably becomes subjective. Additional premises not supplied in the “fixed” list must be invented (though there is an infinitesimal bit of wiggle room because we know the probability is strictly greater than 0). These additional premises pertain to each individual’s idea of money or “utility” and the situation of the bet itself (who made it, why, the personal relationship between bettor and bookie, and on an on).
Subjectivists argue that because each individual can eventually come to a monetary amount (or utility), there exists a “true” probability for “x is F”. This is so, but only because the subjectivist supplies new premises to the official list. In other words, he changes the problem. It’s also the case that the subjectivist won’t be able to describe what these premises are; i.e., they will involve “gut feelings.” Frequentists in these cases also act like subjectivists.
Further examples are generated by changing “Some” in the premise with “Many”, “A few”, “Not that many”, “I’ve heard that a lot of”, “Most”, and others. This makes for a good exercise.
Donald Williams proposed the label Statistical Syllogism, an example of which is this. Premises: “There is a n-sided object, just one side of which is labeled ‘Q’; the object will be tossed and only one side can show.” Conclusion: “A ‘Q’ will show.” The objective deduction (given the premises) is the probability the conclusion is true is 1/n.
The subjectivist is free to change this number, but only if he changes the premises. The frequentist is on more familiar ground here because it appears “tosses” are ready-made for infinite sequences.
Notice that there is no need to add words about “random” tosses. “Random” only means unknown, and it is already unknown (given the premises) which side will show. That is, there is nothing in the premises which tells us how to deduce the outcome.
Suppose n = 2. The deduced probability is 1/2, given the fixed premises. The premises say nothing about the object being a coin, particularly being this coin in this real situation. If we had a real coin in a real situation, and if we were able to list additional premises which were probative of the conclusion, we might be able to deduce whether the coin would land one side or the other with certainty. People have done this (see the work of Persi Diaconis, for example.)
But if all we know is that we have a two-sided real coin and we are not physicists enough to add any additional premises, then the probability of the real toss is still 1/2. If you want to know about this real coin in this real situation and you won’t learn physics, you can experiment with the coin and add the observations as new premises. That’s what the science of statistics does.
Incidentally, we do not need to add the premise “unbiased” to our tosses. “Unbiased” makes the argument circular, because it has in it the notion that the probability of the conclusion just is 1/n. That would make the argument into this: “Given the probability the outcome is 1/n, i.e. ‘unbiased’, the probability of the outcome is 1/n.”
Another statistical syllogism. Premise: “Two-thirds of the marbles in the bag are white and just one marble will be pulled from the bag.” Conclusion: “The marble pulled is white.” Notice we don’t need words on how marble was pulled, etc.
David Stove (a follower of Williams) liked examples like this one. Premise: “Exactly two-thirds of Martians wear hats and George is a Martian.” Conclusion: “George wears a hat.” The objectivist deduces 2/3 for the probability; the subjectiist will usually agree, but etc. The frequentist must remain mute forevermore because there is no infinite sequence of events. This is because there is no event to be embedded. And that is because there are no Martians, hat-wearing or not.
Premise: “If the bank would have made a loan to Jones and this list of Jones’s financial pertinents.” Conclusion: “Jones would have defaulted.”
This is a counterfactual, and a common one. The bank didn’t make the loan and is hoping that it was the right decision. Depending on the exact list of Jones’s financial pertinents, and probably given additional premises about how these pertinents are quantified, the objectivist could deduce a probability for the conclusion. So can the subjectivist. Here, both the objectivist and subjectivist are acting subjectively, unless that list of pertinents allows one to deduce—and not guess—what the relevant quantifications are.
The frequentist must again remain mute, for there is no embeddable sequence; there is no observation of any kind.
Premise: “Between 1/2 and 2/3 of X are F and x is X.” Conclusion: “x is F.” Objectivist deduction: the probability is the interval [1/2, 2/3], or possibly (1/2, 2/3) depending on what meaning is supplied for “between”. Subjectivists would usually agree, but etc. Frequentists weep, for the idea of probability as in interval is unknown to them. However, we already showed in Example 1 that probability can be an interval.
Another Stovian example. Premise: “Bob is a horse.” Conclusion: “Bob is a winged horse.” The objectivist probability of the conclusion is at least greater than the probability of the conclusion of this next argument. Premise: “Bob is a horse.” Conclusion: “It is not the case that Bob is a horse.” Now (quoting, Rationality of Induction, p. 166):
The latter has logical probability=0. So the former has logical probability > 0. But the schema for the former argument,
x is a horse
x is a winged horse,
has relative-truth-frequency=0. For the number of winged horses divided by the number of horses = 0/n, for some positive n.
Frequentism fails again.
There are many more examples which show frequentism fails but in which objectivism (i.e. logical probability) works (subjectivists in these cases usually agree with objectivists, but etc.). If there are any frequentists left, maybe some day we can go over these.
Statistics finally appeared (in Example 3), but we still haven’t got to the ideas of “priors” and all that. That will be next time. Told you it wasn’t easy! (If it was easy, we wouldn’t have some many arguments for so many years.)
Crucial Update: Example 7
Many people are understandably stuck on Example 2. This is for three reasons. The first is easy: because classical training emphasizes physical and not logical examples, it is natural to fall back to this training and to insist that all examples are physical in a sort of subjectivist frequentist manner.
Second. Change the example to this. Premises: “There is an n-state device, just one state of which is set at ‘Q’; the device will be used and only one state can attain.” Conclusion: “A ‘Q’ will attain.” The objective deduction (given the premises) is the probability the conclusion is true is 1/n. It is now very difficult (or impossible) to insist on adding premises about “equal sides” or “unbiasedness” or any other thing which makes the argument circular (see the comments).
Third. We are making logical not physical arguments. This distinction was easy to make in the counterfactual and Martian examples, but is often blurred in the cases of the statistical syllogism. Objective probability is a strict matter of logic. We must take the argument exactly as it is given, and add or subtract nothing from it. Just think: what if (as I am doing now) you were given just this premise and just this conclusion and asked to give a probability for the conclusion. What would you say? Why? On what grounds would you insist the probability cannot be known?
Lastly, if you’re still stuck, change “device” to this premise: “George the Martian will select from the numbers (integers) 1 through n and pick only one and Q is a number between 1 and n.” Same conclusion. It should now be starkly obvious that this is a logical and not physical argument. Note that we need no premise on how George will pick Q, just that he will pick it.
Incidentally, this example also works if we change “George the Martian” to “Briggs the statistician.”