Is Bayesian probability individual, situational, or transcendental: a break with the usual subjective/objective bun fight.

Subjective versus objective is a two way split. The issue actually requires a three way spilt: individual, situational, transcendental. When two people agree that probability is situational, they commonly lack the vocabulary to express this. One stretches the meaning of the word subjective to cover situational. The other stretches the meaning of the word objective to cover situational. Then they quarrel even though they agree. Some quarrels are settled simply by extending the protagonists vocabularies so that they can express themselves clearly.

I develop two scenarios, leading to a three way split: individual, situational, transcendental. Bayesian probability is situational. My scenarios are fiction, but are always set in the ordinary, boring, real world. Werewolves don't exist, mental illness and delusion do. In England guns are outlawed and outlaws have guns.

Scenario one: The Gun

Mr A is in his workshop, carving wood. Mr B comes in, gun in hand, and threatens to shoot Mr A to death. Mr A. stabs Mr B with his chisel and Mr B dies. Was Mr B lawfully killed by Mr A acting in self defense?

The police investigate and turn up a tape recording of Mr B talking to Mr C, the man who sold Mr B the gun.

B: Can you get bullets? I can't find any bullets.
C: Why do you want bullets? The gun doesn't work.
B: What do you mean "the gun doesn't work"?
C: I'm not stupid enough to sell you a working gun. You're a nutter. You might shoot me with it. Look, you don't need bullets or a working gun. Trust me on this: no-one is going to call your bluff.
Soon the facts are established. The gun was unloaded. The gun was incapable of being fired. Mr A was not in danger of being shot to death. We usually describe this by saying that objectively Mr A was not in mortal peril, but he believed that he was, and it his subjective belief that he was going to be shot to death that is at issue in his excuse of self-defence. Here we are drawing the boundary between subjective and objective according to the actual facts.

Scenario two: The Werewolf.

Mr D is in his workshop, carving wood. Mr E comes to see him and they start to quarrel, about money, or a woman, or something. It grows late and as the full moon rises Mr D realises that Mr E is about to transform into a werewolf and tear him into pieces. Mr D looks at his chisel. It is carbon steel, not silver. Mr E is going to transform and become immune to base metals. Mr D doesn't want to die so he thinks harder. Mr E has not yet transformed; he is not yet immune. Mr D stabs him to death.

The two cases are similar in that neither A nor D actually faced the mortal peril that he believed that he was in. When faced with a legal case in which the defendant acted on beliefs that are false most jurisdictions have two different standards which could apply. The first asks whether the belief was genuine. The second asks whether the belief was reasonable. This second standard is often framed as a reasonable man test. Would a reasonable man, in the situation of the defendant, have had the belief that the defendant had. These two standards are sometimes labeled the subjective standard and the objective standard. The subjective standard is whether the defendant really believed it, perhaps because he is insane. The objective standard is whether a reasonable man would have believed it, give the same knowledge that the defendant had.

If the jurisdiction uses the objective standard in self defense cases Mr A succeeds in defending the prosecution against him. Mr C's testimony proves that a reasonable man would have feared being shot to death. Meanwhile Mr D is held to have behaved unreasonably. He killed Mr E unlawfully and the trial moves on to consider whether perhaps Mr D is not guilty by reason of insanity.

Now we have seen two different places at which we draw the boundary between subjective and objective. Sometimes according the actual facts. Other times according to a reasonable-man test. We must sort out the terminology. I propose a three way split: individual, situational, transcendental.

The individual permits personal eccentricity. You get to choose. If somebody else, in the same situation chooses differently, then you are both right. It is like being asked your favorite colour. You can only go wrong if, like the man in the Monty Python film, you answer with somebody else's favourite colour, different from your own, because you are copying their answer.

The situational has one right answer for each situation. You are right if you give the right answer for the situation that you are in. You go wrong if you give a different answer and there is no safety in copying the answer from somebody else. If their situation is different their answer is likely different and there is no merit in having the right answer but for the wrong situation.

The transcendental has one right answer. It is unclear how we find it. For a cubical die the probability of each side is 1/6 unless it is weighted, or the missing mass from the dimples/spots matters, or the angles are not exactly square, or it obeys Newtonian Mechanics and is therefore deterministic.

Bayesian probability is situational.

I illustrate this with a scenario with three characters, Superstish, Didact and Prankster. Didact is trying to teach Superstish about the importance of using the correct probabilities and not just making one up because it feels right.

They are alone in Didact's office. Didact shuffles a deck of cards and deals one face down. He looks at it himself, without showing it to Superstish before sealing it in an envelope. "There" says Didact, handing the envelope to Superstish, "Red or Black? Tell me what you know."

Superstish: I think its Black.
Didact: What? How do you know?
Superstish: I feel it in my bones, it is my hunch.
Didact: How sure are you?
Superstish: 75 to 25

Things are going according to Didactics plan. He is going to explain about Dutch books and how Superstish is going to lose his money if uses individual probabilities instead of situational probabilities. Just then Prankster turns up to see how Didact's plan is going. They fill him in on the details.

Didact: I'm trying to convince Superstish that, without seeing the card, his probably for the card being black should be 50%
Superstish: And I'm trying to convince Didact not to be so closed minded. Open yourself to the universe; trust your intuition. Its Black, I can feel it, I'm 75% sure its black.
Prankster collapses on the floor laughing hysterically. It is several minutes before Didact and Superstish can get any sense out of him.
Prankster: I fixed the deck. I took out the Diamonds and replaced them with Clubs from another pack. The probability that the card in the envelope is black is 75%.
Lets rewind this to just before Prankster starts to laugh. What is the Bayesian Analysis?

There are three situations. Didact has seen the card so his probability estimate for Black should be 100% or 0% accordingly. Superstish has seen a shuffle and a deal from a deck so his probability estimate should be 50%. Prankster fixed the deck so he should go with the odds appropriate to the fix: 75%

Notice that 75% is one of the right answers. Superstish insists that his hunch, 75%, is correct. Why do Bayesians insist that he is wrong, even when he is right? This seems unfair.

There are three situations and hence three right answers. The situations are different and those differences make the three right answers distinct. Bayesians insist that each of the three characters must come up with the single right answer appropriate to their actual circumstances.

The thing to do now is to work systematically through all the cross-over answers to see why, out of the six possibilities, one has psychological appeal, and the others are mostly invisible.

Start with Didact. He saw that the card is black so his answer is 100%. Why not cross over and take Pranksters answer of 75%? The answer appropriate to Superstish's situation is 50%. Why not take that? Both these options are unappealing for two reasons. The first reason is that uncertainty is the enemy. We want to be sure. There is no motivation for filling our mind with doubt. The other answers are less certain and thus less appealling. The second reason is that Didactic has seen that the card is black. Thinking it might be red is just silly.

Next think about Prankster. Why not take the 50:50 answer appropriate to Superstish's situation? Again we see two reasons. First there is no motivation. The less uncertain answer he already has is more emotionally satisfying. Second, he fixed the deck himself, it would be silly to say 50:50 when he knows that there are Spades, Hearts, Clubs, and more Clubs.

But Prankster could cross over to Didact's answer and back a hunch of 100%. That certainly has emotional appeal. We like to think that we know. The problem is that it is rather too certain. Prankster knows he left 13 Hearts in the deck. It might be nice for a while to claim absolute certainty, but that risks being just plain wrong.

Now for Superstish. If he claims that his intuitions give him certain knowledge, he will soon see that he is definitely wrong. We all shy away from being put on the spot and having to commit 100% when we don't really know. So crossing over to the correct answer for Didact's situation is not really on the agenda.

But look at the correct answer for Prankster's situation. It is much more appealing. One is not hopelessly adrift, without a clue, so it soothes the emotional pain of uncertainty. Further more, one is not giving too great a hostage to fortune. Perhaps playing a guessing game for a while will result in 3 Reds and 2 Blacks. One can say that the probability of Black is 75% and then shrug off the result as bad luck.

Of the six possible cross overs there is just one with emotional appeal. Why do Bayesians insist that Superstish is wrong even when he is right? They reject all six cross overs, and you have to admit that they have a point. The other five are definitely wrong, and since the sixth one only differs in psychological appeal, it has to be as wrong as the others.

We can wrap up with a comment on the principle of indifference. The principle of indifference instructs you that if you have no clues as to whether the card is red or black then the probability for your situation is ½.

If we think that probabilities are transcendental the principle of indifference offers us a free lunch. We get knowledge about the real world, merely by being ignorant of it. That is absurd.

If we think that probabilities are individual then the principle of indifference seems contrary. It insists on a single correct answer. If our eccentric, individual preferences are ever to be admitted as valid, the situation in which we have nothing else to go on is the one in which this is most promising.

If we think that probabilities are situational then the principle of indifference seems little better than a tautology. If you don't know, then you don't know.

Pretending to know will hurt you. Some-else might know, and if they offer you a bet you now have a clue and are not reduced to using the principle of indifference, although the game theory here could get really hard, really quickly. Meanwhile look at the entropy of a probability distribution with n options each with probability 1/n. The entropy is as big as it gets. The principle of indifference is telling you that if you haven't a clue you should try and get one, because not having a clue is a really bad situation to be in.