# [Question] What do you do when you find out you have inconsistent probabilities?

I’ve re­cently been read­ing about a ra­tio­nal­ist blog­ger who con­verted to Catholi­cism. She may have as­signed sub­jec­tive prob­a­bil­ities like:

Then she may have in­tro­spected and come up with:

We can calcu­late:

Ab­bre­vi­at­ing Ob­jec­tive Mo­ral­ity by “OM”, and “God” by “G”, this state of af­fairs is in­con­sis­tent, be­cause we in­tu­itively see that:

To re­solve it, she could ei­ther in­crease her sub­jec­tive prob­a­bil­ity of there be­ing a God

or she could re­duce her prob­a­bil­ity of there be­ing some kind of ob­jec­tive morality

She could also re­con­sider P(God|Ob­jec­tive Mo­ral­ity) or P(Ob­jec­tive Mo­ral­ity|God).

Any­ways, I find my­self very con­fused by this state of af­fairs. Is this a solved ques­tion? Is there a purely prin­ci­pled way of re­solv­ing this which only takes into ac­count the 4 num­bers P(OM), P(G), P(OM|G) and P(G|OM)? Is there a stan­dard way of us­ing some kind of metaprob­a­bil­ities?

• I would make ex­plicit that her be­liefs about her sub­jec­tive prob­a­bil­ities are in­ac­cu­rate ob­ser­va­tions of her im­plied un­der­ly­ing log­i­cally om­ni­scient, con­sis­tent be­lief sys­tem. She can then as­sign each pos­si­ble un­der­ly­ing con­sis­tent be­lief sys­tem a prob­a­bil­ity, and up­date that as­sign­ment once she re­al­izes that some of the pos­si­ble sys­tems were not con­sis­tent. What this comes out to is that whether she should up­date her be­lief in God or Ob­jec­tive Mo­ral­ity comes down to which of her be­liefs she is less cer­tain about.

• The 4 given prob­a­bil­ities are ac­tu­ally perfectly con­sis­tent within the equa­tions you are us­ing. It is prov­able that what­ever 4 prob­a­bil­ities you use the equa­tions will be con­sis­tent.

There­fore the ques­tion be­comes “where did my maths go wrong?”

P(G|OM) = 0.055, not 0.55

I’m pretty con­fi­dent that the only way prob­a­bil­ities can ac­tu­ally be in­con­sis­tent is if it is over con­strained (e.g. in this case you define 5 rele­vant prob­a­bil­ities in­stead of 4). The whole point of hav­ing ax­ioms is to pre­vent in­con­sis­ten­cies pro­vided you stay in­side them.

P.S. Good job on notic­ing your con­fu­sion!

• 0.9 = P(Ob­jec­tive Mo­ral­ity) ≠ P(God) * P(Ob­jec­tive Mo­ral­ity | God) + P(No God) * P(Ob­jec­tive Mo­ral­ity | No God) = 0.05 * 0.99 + 0.95 * 0.02 = 0.0685. That’s in­con­sis­tent, right?

• Argh,you’re right,I didn’t check that one. P(OM) can­cels on the P(G) equa­tion so that one isn’t over con­strained.

How­ever for the equa­tion for P(OM) 4 vari­ables is over con­strained, 3 is enough.

• Some op­tions for ad­dress­ing this:

1) Be more spe­cific in your prob­a­bil­ities. What ex­pe­riences are in­cluded or ex­cluded from these pre­dic­tions? Often, this ex­er­cise will show you that you have un­rea­son­able es­ti­mates for one of these figures, which may or may not bring your be­liefs into con­sis­tency.

2) Rec­og­nize that these prob­a­bil­ity es­ti­mates are pretty wild guesses, and ac­cept that they’re prob­a­bly wrong. In­con­sis­tent be­liefs nec­es­sar­ily in­clude false­hoods, but that doesn’t mean you have enough in­for­ma­tion to im­prove them.

3) See if you can gather any ev­i­dence for some of the in­ter­me­di­ate prob­a­bil­ities you’re work­ing with. Th­ese may give hints to­ward which of them to ad­just.

• Note: Cleaned up the for­mat­ting a bit and changed in­line-la­tex into block-level LaTeX to make it read­able on phones.

• There may be a differ­ence be­tween “No God” and “Not God”. P(¬G) in­cludes ev­ery other pos­si­bil­ity − 2 gods, 3, 0, aliens cre­at­ing hu­mans, this is a simu­la­tion, ev­ery­thing we can think of and more. For this rea­son, some sug­gest odds over pri­ors (and us­ing Bayes rule ap­pro­pri­ately) be­cause the sum of prob­a­bil­ities we con­sider need not be one—we may de­ter­mine of the pos­si­bil­ities we are con­sid­er­ing that one is not likely to be true, in place of de­ter­min­ing what is true. (For ex­am­ple, if we are con­sid­er­ing the pos­si­bil­ity that deck of cards some­one else is us­ing for a poker game is or­di­nary, or has 4 ex­tra aces, we may ac­quire enough ev­i­dence, that the sec­ond pos­si­bil­ity has an or­der of mag­ni­tude more prob­a­bil­ity. There might not be 8 aces, but we may be very con­fi­dent that ei­ther the deck is not or­di­nary, or some­one is cheat­ing (pos­si­bly the per­son who is shuffling the deck).)

Also, if you ever use Bayes rule, and say “that can’t be right be­cause of X”, keep go­ing. Are there more givens you’re miss­ing?