Is probability entirely subjective?

[Gábor Szabó]

Hi Guys,

It's quite possible, even likely, that you have already discussed
similar questions. In any case, it is a well known problem. The
specific context I started to think about the problem (as a thought
experiment) was this. I am sorry for the somewhat stupid example. I
hope no one is triggered or traumatized by the subject.

Suppose you are foolish teenage boys and you play a kind of mortal
danger game, where one of the players, you, should ride his bike into
a bend of the road on the opposite side where it is not possible to
see or hear oncoming traffic. Based on your previous experience, the
chance you survive this game without being hit by an approaching car
is 50 % at this particular hour, due to random fluctuation of traffic.
It is a thought experiment, there is no way you could get out of the
way once you had started your ride.

You go through the stretch of road without any problem. Were you in danger?

As a variant of the above, suppose that unbeknownst to you, a major
feeding road to your road had been cut due to an accident which
reduced traffic to 10 % of what was normally expected at this hour
(still fluctuating randomly). Does it change anything?

A further variant of this is that you intended to cheat by arranging
with a friend to watch traffic further up the road and call you if a
car were coming. In this way, you felt 100 % safe. Does it change
probabilities or only belief?

A still further variant is that unbeknownst to you and the watching
friend, there was a sideroad between you and the watching friend which
also fed traffic to the major road. Does it change anything?

I tend to be in the strongly subjectivist camp and I suppose it does
not make sense to speak about probabilties once an event happened or
did not happen and you know the result. On the other hand, I also tend
to subscribe to a many-world interpretation of reality where you are
dead in roughly half of possible worlds.

What do you think about this?

Kind regards,

Gabor

[Gábor Szabó]

I am not a lawyer, but I am quite sure the subjectivist argument would
not stand before a court. In the specific example, if I cajoled or
forced someone into playing this game, the judge is very unlikely to
accept the 'no danger' argument based on the fact that the guy
survived without a scratch (and condemning me only for the emotional
distress caused by going through this ordeal).

[v.sza...@gmail.com]

The reason I started to think about this problem was reading someone's criticism of an influential political analyst (Gábor Török) who gave a very low chance for Tisza victory a few months before the election. I pointed out that conditioning on the fact of Tisza victory, all probabilistic estimates lose sense. It would be somewhat strange to say that as Tisza won, now it seems they had more chance than previously supposed. If Mr Török gave a 90 % probabilistic estimate for Tisza victory, he would look better now, but in fact any estimate below 100 % would be wrong. Of course, in the case of an influential political analyst, publishing such an estimate itself changes the probability of a subsequent outcome. This is why I started to think about a different context.

[Timothy Underwood]

So 'probability' is a common language word, and as such has multiple referents that in practice usually run together, but there will be certain edge cases where some of them might not be present.

From the standpoint of how the word 'probability' is normally used, it is clear that the die had a 1/6 chance of coming up 1 is a thing that you can sensible say, even if the person actually rolled a 4. 

To the extent that the election is like throwing dice, saying that 'there was only a ten percent chance of this outcome, but Torok Gabor was unlucky' matches the way the word 'chance' is normally used.

Obviously though there is a sense in which the odds of a Tisza victory was never 10 percent, or 90 percent, or anything but 100 percent. In the world that we actually live in, Tisza was going to win. We just did not know several months ago if we were in a 100 percent Tisza world, or a 100 percent Fidesz world. After the election we now know that we were all along in a 100 percent Tisza world.

Of course this is the same thing with dice: Throwing a die is probably a sufficiently macroscopic action that even under a multiple worlds model, when a particular person starts to throw a particular die, it will probably come up the same way in the vast, vast majority of branches. A sufficiently powerful intelligence could simply watch the way the person was moving their arms, and know the instant the decision to release the dice was made what it would come up as.

On the other hand, I suspect that it would be impossible, or at least extremely difficult, for that intelligence to know what it would come up as before the person has even picked up the dice. So at some fundamental level, how the dice will come up is seen as unknowable--being smarter will not help me know what will come up the next time you throw a piece of dice, even if the outcome is actually deterministic. If I'd been smarter I could have known that Tisza would win, instead of simply taking the prediction market estimate and rounding down a bit.

Anyways, as you are probably aware, there is a great deal of discussion about whether probability and chance is part of reality (which it seems to be in the case of quantum level phenomenon, but that may or may not be the bottom level of reality), or a description of how we think about reality.

In so far as it is a description of how we think about reality, we can judge probability estimates after the fact based on whether they still seem like they were a good map or way of making decisions. Was the estimate well calibrated?

Under this view, if Torok Gabor had predicted a 100 percent chance of a Tisza victory, he would still seem stupid after the fact, because clearly he didn't actually know enough to be that confident--even though he was living in a world where that outcome would definitely happen. On the other hand, it does make a 20 percent estimate look like it was based on a worse model than a 90 percent estimate, but a single event isn't enough to tell us whether he got unlucky in his guess or if he had a bad model. This is why people are interested in tracking pundit predictions and seeing how accurate they are. It lets us distinguish those who are well calibrated in their probabilities from those whose probability estimates aren't correlated with the outcomes in any way.

I think saying, 'but the conditional probability of a Tisza victory given a Tisza victory is 1', while tautalogically true, doesn't tell us anything about how we should think about a question like 'what is the probability, given what we know, of Tisza getting back the EU funds'.

Tim 

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