What Are Probabilities, Anyway?

In Prob­a­bil­ity Space & Au­mann Agree­ment, I wrote that prob­a­bil­ities can be thought of as weights that we as­sign to pos­si­ble world-his­to­ries. But what are these weights sup­posed to mean? Here I’ll give a few in­ter­pre­ta­tions that I’ve con­sid­ered and held at one point or an­other, and their prob­lems. (Note that in the pre­vi­ous post, I im­plic­itly used the first in­ter­pre­ta­tion in the fol­low­ing list, since that seems to be the main­stream view.)

  1. Only one pos­si­ble world is real, and prob­a­bil­ities rep­re­sent be­liefs about which one is real.

    • Which world gets to be real seems ar­bi­trary.

    • Most pos­si­ble wor­lds are life­less, so we’d have to be re­ally lucky to be al­ive.

    • We have no in­for­ma­tion about the pro­cess that de­ter­mines which world gets to be real, so how can we de­cide what the prob­a­bil­ity mass func­tion p should be?

  2. All pos­si­ble wor­lds are real, and prob­a­bil­ities rep­re­sent be­liefs about which one I’m in.

    • Be­fore I’ve ob­served any­thing, there seems to be no rea­son to be­lieve that I’m more likely to be in one world than an­other, but we can’t let all their weights be equal.

  3. Not all pos­si­ble wor­lds are equally real, and prob­a­bil­ities rep­re­sent “how real” each world is. (This is also some­times called the “mea­sure” or “re­al­ity fluid” view.)

    • Which wor­lds get to be “more real” seems ar­bi­trary.

    • Be­fore we ob­serve any­thing, we don’t have any in­for­ma­tion about the pro­cess that de­ter­mines the amount of “re­al­ity fluid” in each world, so how can we de­cide what the prob­a­bil­ity mass func­tion p should be?

  4. All pos­si­ble wor­lds are real, and prob­a­bil­ities rep­re­sent how much I care about each world. (To make sense of this, re­call that these prob­a­bil­ities are ul­ti­mately mul­ti­plied with util­ities to form ex­pected util­ities in stan­dard de­ci­sion the­o­ries.)

    • Which wor­lds I care more or less about seems ar­bi­trary. But per­haps this is less of a prob­lem be­cause I’m “al­lowed” to have ar­bi­trary val­ues.

    • Or, from an­other per­spec­tive, this drops an­other an­other hard prob­lem on top of the pile of prob­lems called “val­ues”, where it may never be solved.

As you can see, I think the main prob­lem with all of these in­ter­pre­ta­tions is ar­bi­trari­ness. The un­con­di­tioned prob­a­bil­ity mass func­tion is sup­posed to rep­re­sent my be­liefs be­fore I have ob­served any­thing in the world, so it must rep­re­sent a state of to­tal ig­no­rance. But there seems to be no way to spec­ify such a func­tion with­out in­tro­duc­ing some in­for­ma­tion, which any­one could in­fer by look­ing at the func­tion.

For ex­am­ple, sup­pose we use a uni­ver­sal dis­tri­bu­tion, where we be­lieve that the world-his­tory is the out­put of a uni­ver­sal Tur­ing ma­chine given a uniformly ran­dom in­put tape. But then the dis­tri­bu­tion con­tains the in­for­ma­tion of which UTM we used. Where did that in­for­ma­tion come from?

One could ar­gue that we do have some in­for­ma­tion even be­fore we ob­serve any­thing, be­cause we’re prod­ucts of evolu­tion, which would have built some use­ful in­for­ma­tion into our genes. But to the ex­tent that we can trust the prior speci­fied by our genes, it must be that evolu­tion ap­prox­i­mates a Bayesian up­dat­ing pro­cess, and our prior dis­tri­bu­tion ap­prox­i­mates the pos­te­rior dis­tri­bu­tion of such a pro­cess. The “prior of evolu­tion” still has to rep­re­sent a state of to­tal ig­no­rance.

Th­ese con­sid­er­a­tions lead me to lean to­ward the last in­ter­pre­ta­tion, which is the most tol­er­ant of ar­bi­trari­ness. This in­ter­pre­ta­tion also fits well with the idea that ex­pected util­ity max­i­miza­tion with Bayesian up­dat­ing is just an ap­prox­i­ma­tion of UDT that works in most situ­a­tions. I and oth­ers have already mo­ti­vated UDT by con­sid­er­ing situ­a­tions where Bayesian up­dat­ing doesn’t work, but it seems to me that even if we set those aside, there is still rea­son to con­sider a UDT-like in­ter­pre­ta­tion of prob­a­bil­ity where the weights on pos­si­ble wor­lds rep­re­sent how much we care about those wor­lds.