Subjective Versus Objective Bayes (Versus Frequentism): Part I

We first have to define what subjectivity and objectivity are and from these see what happens. For those unused to reading long stretches of prose, here is the conclusion, which will be proved in due time:

Like in The Highlander, “There can be only one”…correct interpretation of probability (uncertainty), but many approximations. Objective Bayes is the correct interpretation. Subjective Bayes fails in theory but often works in practice because subjective Bayesians act like objectivists. Frequentism fails except when it overlaps objective Bayesian methods (and when it does it works “in the long run” too). All others fail or are approximations too. Despite the labels, all three groups operate at least partly subjectively, partly objectively.

To assert without argument is to be subjective. There are two divisions to subjectivity, the internal and external. Internal subjective propositions are those which you believe (either not at all, somewhere in-between, or fully), whether via argument or observation or both. External subjective propositions are those which you seek to convince another to believe or disbelieve or to give weight to. It is subjective to say “I think I might be catching a cold¹,” “I am hot,” or “I dislike rice pudding”. If I am feeling or thinking these things, and I believe them and am not trying to fool my listener, then these propositions are true, given my internal state.

Externally, however, these kinds of propositions may be doubted. If you hear your neighbor say “My favorite color is red” and if you add to that the premises “My neighbor is sane and honest and sane and honest people faithfully report their beliefs” then it follows your neighbor’s favorite color is red. But other premises are possible which cause you to doubt your neighbor or to disbelieve him. Thus internal subjective statements of one person can be argued against externally by another. Think of a cop grilling a suspect. For example if I gave prior evidence of relishing rice pudding, you can put this to me. Or it may be freezing out which makes you question why I report being hot. Thus depending, or conditional, on the premises which are or aren’t accepted by the parties, the proposition in question can be true for one and false for another, or somewhere in between.

“Garfield was a better president than Tyler” is an internally subjective statement which has as much evidentiary support as the statement “Tyler was a better president than Garfield”, which is to say none except the implicit force of the speakers. That force is a premise, but difficult to clarify. To hear one and believe it and not the other is therefore to be subjective.

This next argument is also subjective, but only in part, and illustrates the sort of internal argument which happens when one hears a raw assertion like “Garfield’s better.” “Garfield was a better president than Tyler because Garfield’s vice president went on to be president himself and better presidents see their vice presidents go on to be presidents.” This is an (inverted) valid syllogism, and therefore the conclusion (“Garfield’s better”) is true given the premises. Not everybody will agree with the conclusion because not everybody will agree with the premises. The premises are merely asserted, and are therefore subjective. The argument as a whole is then also subjective, but interior to the argument there is no subjectivism; interior is it objective.

In order for an argument to avoid being subjective, and therefore be completely objective, it has to begin with premises which are true (and which has unambiguous terms) and ends with a conclusion which validly follows from the premises. Mathematical theorems are like this, though they are not usually presented in that fashion. Instead a theorem will begin with premises which are accepted but not proven in place as true. The premises are accepted as true because they have earlier been derived via other arguments, themselves with premises which are accepted but not proven (in place) as true. This chain eventually ends at axioms and rules of deduction which everybody believes are true. This is how we can all agree that propositions like “2 + 7 = 9” are true. The premises which led to this are suppressed, but always there.

Though with many propositions (like “2 + 7 = 9”), the premises are different for different folks, and hence so are the arguments. Quite a lot of people cannot prove these propositions true, but believe them anyway. But this is because they use a valid argument from authority, trusting in the experts who told them about propositions like this. This shows that there can be more than one path to a true conclusion.

The shorthand of only presenting a few premises when deducing new mathematical theorems is handy, but its use can be dangerous in other kinds of argument. Arguments can appear purely objective, like mathematical theorems are, but which are really not because the chain of premises do not lead back to unassailable axioms but instead to propositions which are dubious or do not share universal approval.

This is why propositions like “Garfield was a better president than Tyler” are often contentious. It is too difficult, especially seated at a bar stool or in a blog combox, to expose and agree to the exact and relevant chain of premises which lead unambiguously to proving the proposition certainly true or certainly false. It is even too difficult for sober scholars who devote their entire lives to studying the subject. What is the precise, unambiguous list of qualities and circumstances which defines presidential quality? It is also difficult or impossible to give a complete list of premises which lead me to say “I feel hot” or “I like this Mozart sonata better than that one.” Sometimes all I can do is just state it and believe it. Thus the hardest and simplest propositions are often irremediably subjective.

It isn’t always hopeless. Simple propositions are often proved via short chains of premise-arguments. It isn’t a long road to “One should do good and eschew evil” or “I am mortal.” And here lies the problem. Too often when seeking to understand a subject we try to peel off the more difficult problems first, like “What values do we give this prior?” when we should be starting with simple ones like we’ll see next time.

Part II.

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¹Note: the proposition “I think I might be catching a cold” differs from “I will catch a cold by tomorrow.” One is a report of health, the other is a forecast. The forecast is a “somewhere in-between” kind of belief, i.e. uncertain, i.e. probabilistic.