# Probability Space & Aumann Agreement

The first part of this post de­scribes a way of in­ter­pret­ing the ba­sic math­e­mat­ics of Bayesi­anism. Eliezer already pre­sented one such view at http://​​less­wrong.com/​​lw/​​hk/​​pri­ors_as_math­e­mat­i­cal_ob­jects/​​, but I want to pre­sent an­other one that has been use­ful to me, and also show how this view is re­lated to the stan­dard for­mal­ism of prob­a­bil­ity the­ory and Bayesian up­dat­ing, namely the prob­a­bil­ity space.

The sec­ond part of this post will build upon the first, and try to ex­plain the math be­hind Au­mann’s agree­ment the­o­rem. Hal Fin­ney had sug­gested this ear­lier, and I’m tak­ing on the task now be­cause I re­cently went through the ex­er­cise of learn­ing it, and could use a check of my un­der­stand­ing. The last part will give some of my cur­rent thoughts on Au­mann agree­ment.

#### Prob­a­bil­ity Space

In http://​​en.wikipe­dia.org/​​wiki/​​Prob­a­bil­ity_space, you can see that a prob­a­bil­ity space con­sists of a triple:

• Ω – a non-empty set – usu­ally called sam­ple space, or set of states

• F – a set of sub­sets of Ω – usu­ally called sigma-alge­bra, or set of events

• P – a func­tion from F to [0,1] – usu­ally called prob­a­bil­ity measure

F and P are re­quired to have cer­tain ad­di­tional prop­er­ties, but I’ll ig­nore them for now. To start with, we’ll in­ter­pret Ω as a set of pos­si­ble world-his­to­ries. (To elimi­nate an­thropic rea­son­ing is­sues, let’s as­sume that each pos­si­ble world-his­tory con­tains the same num­ber of ob­servers, who have perfect mem­ory, and are la­beled with unique se­rial num­bers.) Each “event” A in F is for­mally a sub­set of Ω, and in­ter­preted as ei­ther an ac­tual event that oc­curs in ev­ery world-his­tory in A, or a hy­poth­e­sis which is true in the world-his­to­ries in A. (The de­tails of the events or hy­pothe­ses them­selves are ab­stracted away here.)

To un­der­stand the prob­a­bil­ity mea­sure P, it’s eas­ier to first in­tro­duce the prob­a­bil­ity mass func­tion p, which as­signs a prob­a­bil­ity to each el­e­ment of Ω, with the prob­a­bil­ities sum­ming to 1. Then P(A) is just the sum of the prob­a­bil­ities of the el­e­ments in A. (For sim­plic­ity, I’m as­sum­ing the dis­crete case, where Ω is at most countable.) In other words, the prob­a­bil­ity of an ob­ser­va­tion is the sum of the prob­a­bil­ities of the world-his­to­ries that it doesn’t rule out.

A pay­off of this view of the prob­a­bil­ity space is a sim­ple un­der­stand­ing of what Bayesian up­dat­ing is. Once an ob­server sees an event D, he can rule out all pos­si­ble world-his­to­ries that are not in D. So, he can get a pos­te­rior prob­a­bil­ity mea­sure by set­ting the prob­a­bil­ity masses of all world-his­to­ries not in D to 0, and renor­mal­iz­ing the ones in D so that they sum up to 1 while keep­ing the same rel­a­tive ra­tios. You can eas­ily ver­ify that this is equiv­a­lent to Bayes’ rule: P(H|D) = P(D H)/​P(D).

To sum up, the math­e­mat­i­cal ob­jects be­hind Bayesi­anism can be seen as

• Ω – a set of pos­si­ble world-histories

• F – in­for­ma­tion about which events oc­cur in which pos­si­ble world-histories

• P – a set of weights on the world-his­to­ries that sum up to 1

#### Au­mann’s Agree­ment Theorem

Au­mann’s agree­ment the­o­rem says that if two Bayesi­ans share the same prob­a­bil­ity space but pos­si­bly differ­ent in­for­ma­tion par­ti­tions, and have com­mon knowl­edge of their in­for­ma­tion par­ti­tions and pos­te­rior prob­a­bil­ities of some event A, then their pos­te­rior prob­a­bil­ities of that event must be equal. So what are in­for­ma­tion par­ti­tions, and what does “com­mon knowl­edge” mean?

The in­for­ma­tion par­ti­tion I of an ob­server-mo­ment M di­vides Ω into a num­ber of sub­sets that are non-over­lap­ping, and to­gether cover all of Ω. Two pos­si­ble world-his­to­ries w1 and w2 are placed into the same sub­set if the ob­server-mo­ments in w1 and w2 have the ex­act same in­for­ma­tion. In other words, if w1 and w2 are in the same el­e­ment of I, and w1 is the ac­tual world-his­tory, then M can’t rule out ei­ther w1 or w2. I(w) is used to de­note the el­e­ment of I that con­tains w.

Com­mon knowl­edge is defined as fol­lows: If w is the ac­tual world-his­tory and two agents have in­for­ma­tion par­ti­tions I and J, an event E is com­mon knowl­edge if E in­cludes the mem­ber of the meet I∧J that con­tains w. The op­er­a­tion ∧ (meet) means to take the two sets I and J, form their union, then re­peat­edly merge any of its el­e­ments (which you re­call are sub­sets of Ω) that over­lap un­til it be­comes a par­ti­tion again (i.e., no two el­e­ments over­lap).

It may not be clear at first what this meet op­er­a­tion has to do with com­mon knowl­edge. Sup­pose the ac­tual world-his­tory is w. Then agent 1 knows I(w), so he knows that agent 2 must know one of the el­e­ments of J that over­laps with I(w). And he can rea­son that agent 2 must know that agent 1 knows one of the el­e­ments of I that over­laps with one of these el­e­ments of J. If he car­ries out this in­fer­ence to in­finity, he’ll find that both agents know that the ac­tual world-his­tory is in (I∧J)(w), and both know the other know, and both know the other know the other know, and so on. In other words it is com­mon knowl­edge that the ac­tual world-his­tory is in (I∧J)(w). Since event E oc­curs in ev­ery world-his­tory in (I∧J)(w), it’s com­mon knowl­edge that E oc­curs in the ac­tual world-his­tory.

Proof for the agree­ment the­o­rem then goes like this. Let E be the event that agent 1 as­signs a pos­te­rior prob­a­bil­ity (con­di­tioned on ev­ery­thing it knows) of q1 to event A and agent 2 as­signs a pos­te­rior prob­a­bil­ity of q2 to event A. If E is com­mon knowl­edge at w, then both agents know that P(A | I(v)) = q1 and P(A | J(v)) = q2 for ev­ery v in (I∧J)(w). But this im­plies P(A | (I∧J)(w)) = q1 and P(A | (I∧J)(w)) = q2 and there­fore q1 = q2. (To see this, sup­pose you cur­rently know only (I∧J)(w), and you know that no mat­ter what ad­di­tional in­for­ma­tion I(v) you ob­tain, your pos­te­rior prob­a­bil­ity will be the same q1, then your cur­rent prob­a­bil­ity must already be q1.)

Is Au­mann Agree­ment Over­rated?

Hav­ing ex­plained all of that, it seems to me that this the­o­rem is less rele­vant to a prac­ti­cal ra­tio­nal­ist than I thought be­fore I re­ally un­der­stood it. After look­ing at the math, it’s ap­par­ent that “com­mon knowl­edge” is a much stric­ter re­quire­ment than it sounds. The most ob­vi­ous way to achieve it is for the two agents to sim­ply tell each other I(w) and J(w), af­ter which they share a new, com­mon in­for­ma­tion par­ti­tion. But in that case, agree­ment it­self is ob­vi­ous and there is no need to learn or un­der­stand Au­mann’s the­o­rem.

There are some pa­pers that de­scribe ways to achieve agree­ment in other ways, such as iter­a­tive ex­change of pos­te­rior prob­a­bil­ities. But in such meth­ods, the agents aren’t just mov­ing closer to each other’s be­liefs. Rather, they go through con­voluted chains of de­duc­tion to in­fer what in­for­ma­tion the other agent must have ob­served, given his dec­la­ra­tions, and then up­date on that new in­for­ma­tion. (The pro­cess is similar to the one needed to solve the sec­ond rid­dle on this page.) The two agents es­sen­tially still have to com­mu­ni­cate I(w) and J(w) to each other, ex­cept they do so by ex­chang­ing pos­te­rior prob­a­bil­ities and mak­ing log­i­cal in­fer­ences from them.

Is this re­al­is­tic for hu­man ra­tio­nal­ist wannabes? It seems wildly im­plau­si­ble to me that two hu­mans can com­mu­ni­cate all of the in­for­ma­tion they have that is rele­vant to the truth of some state­ment just by re­peat­edly ex­chang­ing de­grees of be­lief about it, ex­cept in very sim­ple situ­a­tions. You need to know the other agent’s in­for­ma­tion par­ti­tion ex­actly in or­der to nar­row down which el­e­ment of the in­for­ma­tion par­ti­tion he is in from his prob­a­bil­ity dec­la­ra­tion, and he needs to know that you know so that he can de­duce what in­fer­ence you’re mak­ing, in or­der to con­tinue to the next step, and so on. One er­ror in this pro­cess and the whole thing falls apart. It seems much eas­ier to just tell each other what in­for­ma­tion the two of you have di­rectly.

Fi­nally, I now see that un­til the ex­change of in­for­ma­tion com­pletes and com­mon knowl­edge/​agree­ment is ac­tu­ally achieved, it’s ra­tio­nal for even hon­est truth-seek­ers who share com­mon pri­ors to dis­agree. There­fore, two such ra­tio­nal­ists may per­sis­tently dis­agree just be­cause the amount of in­for­ma­tion they would have to ex­change in or­der to reach agree­ment is too great to be prac­ti­cal. This is quite differ­ent from the un­der­stand­ing of Au­mann agree­ment I had be­fore I read the math.

• I think there’s an­other, more fun­da­men­tal rea­son why Au­mann agree­ment doesn’t mat­ter in prac­tice. It re­quires each party to as­sume the other is com­pletely ra­tio­nal and hon­est.

Act­ing as if the other party is ra­tio­nal is good for pro­mot­ing calm and rea­son­able dis­cus­sion. Se­ri­ously con­sid­er­ing the pos­si­bil­ity that the other party is ra­tio­nal is cer­tainly valuable. But as­sum­ing that the other party is in fact to­tally ra­tio­nal is just silly. We know we’re talk­ing to other flawed hu­man be­ings, and ei­ther or both of us might just be to­tally off base, even if we’re hang­ing around on a ra­tio­nal­ity dis­cus­sion board.

• I be­lieve Han­son’s pa­per on ‘Bayesian wannabes’ shows that even only par­tially ra­tio­nal agents must agree about a lot.

• Jaw-drop­pingly (for me), that pa­per ap­par­ently uses “Bayesi­ans” to re­fer to agents whose pri­mary goal in­volves seek­ing (and shar­ing) the truth.

IMO, “Bayesi­ans” should re­fer to agents that em­ploy Bayesian statis­tics, re­gard­less of what their goals are.

That Han­son ca­su­ally em­ploys this other defi­ni­tion with­out dis­cussing the is­sue or defend­ing his us­age says a lot about his at­ti­tude to the sub­ject.

• I as­sume this just means that their pri­mary epistemic goal is such, not that this is their util­ity func­tion.

• That’s why I used the word “in­volves”.

How­ever, surely there are pos­si­ble agents who are ma­jor fans of Bayesian statis­tics who don’t have the time or mo­tive to share their knowl­edge with other agents. In­deed, they may ac­tively spread dis­in­for­ma­tion to other agents in or­der to ma­nipu­late them. Those folk are not bound to agree with other agents when they meet them.

• Won’t the util­ity func­tion even­tu­ally up­date to match?

• Maybe I lack imag­i­na­tion—is it pos­si­ble for a strict Bayesian to do any­thing but seek and share the truth (as­sum­ing he is in­ter­act­ing with other Bayesi­ans)?

• Bayes rule is about how to up­date your es­ti­mates of the prob­a­bil­ity of hy­pothe­ses on the ba­sis of in­com­ing data. It has noth­ing to say about an agent’s goal, or how it be­haves. Agents can em­ploy Bayesian statis­tics to up­date their world view while pur­su­ing liter­ally any goal.

If you think the term “Bayesian” im­plies an agent whose goal nec­es­sar­ily in­volves spread­ing truth to other agents, I have to ask for your refer­ences for that idea.

• I am look­ing at the world around me, at the defi­ni­tion of Bayesian, and as­sum­ing the pro­cess has been go­ing on in an agent for long enough for it to be prop­erly called “a Bayesian agent”, and think to my­self—the agent space I end up in, has cer­tain prop­er­ties.

Of course, I’m us­ing the phrase “Bayesian agent” to mean some­thing slightly differ­ent than what the origi­nal poster in­tended.

• Of course the agent space you end up in, has cer­tain prop­er­ties—but the is­sue is whether those prop­er­ties nec­es­sar­ily in­volve shar­ing the truth with oth­ers.

I figure you can pur­sue any goal us­ing Bayesian statis­tics—in­clud­ing goals that in­clude at­tempt­ing to de­ceive and mis­lead oth­ers.

For ex­am­ple, a Bayesian pub­lic re­la­tions officer for big to­bacco would not be bound to agree with other agents that she met.

• You’re speak­ing of Bayesian agents as a gen­eral term to re­fer to any­one who hap­pens to use Bayesian statis­tics for a spe­cific pur­pose—and in that con­text, I agree with you. In that con­text, your state­ments are cor­rect, by defi­ni­tion.

I am speak­ing of Bayesian agents us­ing the ideal­ized, Hol­ly­wood con­cept of agent. Maybe I should have been more spe­cific and referred to su­per-agents, equiv­a­lent to su­per-spies.

I claim that some­one who has lived and breathed the Bayes way will be sig­nifi­cantly differ­ent than some­one who has ap­plied it, even very con­sis­tently, within a limited do­main. For ex­am­ple, I can imag­ine a Bayesian su­per-agent work­ing for big to­bacco, but I see the prob­a­bil­ity of that event ac­tu­ally com­ing to pass as too small to be worth con­sid­er­ing.

• I don’t re­ally know what you mean. A “su­per agent”? Do you re­ally think Bayesian agents are “good”?

Since you haven’t re­ally said what you mean, what do you mean? What are these “su­per agents” of which you speak? Would you know one if you met one?

• Su­per-agent. You know, like James Bond, Mr. and Ms. Smith. Closer to the use, in con­text—Jeffreys­sai.

• Right… So: how about Lex Luthor or Gen­eral Zod?

• I’ve seen the pa­per, but it as­sumes the point in ques­tion in the defi­ni­tion of par­tially ra­tio­nal agents in the very first para­graph:

If these agents agree that their es­ti­mates are con­sis­tent with cer­tain easy-to-com­pute con­sis­tency con­straints, then… [con­clu­sion fol­lows].

But peo­ples’ es­ti­mates gen­er­ally aren’t con­sis­tent with his con­straints, so even for some­one who is suffi­ciently ra­tio­nal, it doesn’t make any sense what­so­ever to as­sume that ev­ery­one else is.

This doesn’t mean Robin’s pa­per is wrong. It just means that faced with a topic where we would “agree to dis­agree”, you can ei­ther up­date your be­lief about the topic, or up­date your be­lief about whether both of us are ra­tio­nal enough for the proof to ap­ply.

• [...] re­quires each party to as­sume the other is com­pletely ra­tio­nal and hon­est. [/​..] But as­sum­ing that the other party is in fact to­tally ra­tio­nal is just silly.

As­sum­ing hon­esty is pretty prob­le­mat­i­cal, too. In real-world dis­putes, par­ti­ci­pants are likely to dis­agree about what con­sti­tutes ev­i­dence (“the Bible says..”), aren’t ra­tio­nal, and sus­pect each oth­ers hon­esty.

• One ques­tion on your ob­jec­tions: how would you char­ac­ter­ize the state of two hu­man ra­tio­nal­ist wannabes who have failed to reach agree­ment? Would you say that their dis­agree­ment is com­mon knowl­edge, or in­stead are they un­cer­tain if they have a dis­agree­ment?

ISTM that peo­ple usu­ally find them­selves rather cer­tain that they are in dis­agree­ment and that this is com­mon knowl­edge. Au­mann’s the­o­rem seems to for­bid this even if we as­sume that the calcu­la­tions are in­tractable.

The ra­tio­nal way to char­ac­ter­ize the situ­a­tion, if in fact in­tractabil­ity is a prac­ti­cal ob­jec­tion, would be that each party says he is un­sure of what his opinion should be, be­cause the in­for­ma­tion is too com­plex for him to make a de­ci­sion. If cir­cum­stances force him to adopt a be­lief to act on, maybe it is ra­tio­nal for the two to choose differ­ent ac­tions, but they should ad­mit that they do not re­ally have good grounds to as­sume that their choice is bet­ter than the other per­son’s. Hence they re­ally are not cer­tain that they are in dis­agree­ment, in ac­cor­dance with the the­o­rem. Again this is in strik­ing con­trast to ac­tual hu­man be­hav­ior even among wannabes.

• One ques­tion on your ob­jec­tions: how would you char­ac­ter­ize the state of two hu­man ra­tio­nal­ist wannabes who have failed to reach agree­ment?

I would say that one pos­si­bil­ity is that their dis­agree­ment is com­mon knowl­edge, but they don’t know how to reach agree­ment. From what I’ve learned so far, dis­agree­ments be­tween ra­tio­nal­ist wannabes can arise from 3 sources:

• differ­ent priors

• differ­ent com­pu­ta­tional short­cuts/​ap­prox­i­ma­tions/​errors

• in­com­plete ex­change of information

Even if the two ra­tio­nal­ist wannabes agree that in prin­ci­ple they should have the same pri­ors and the same com­pu­ta­tions, and full ex­change of in­for­ma­tion, as of to­day they do not have gen­eral meth­ods to solve any of these prob­lems, can only try to work out their differ­ences on a case-by-case ba­sis, with high like­li­hood that they’ll have to give up at some point be­fore they reach agree­ment.

Again this is in strik­ing con­trast to ac­tual hu­man be­hav­ior even among wannabes.

Your sug­ges­tion of what ra­tio­nal­ist wannabes should do in­tu­itively makes a lot of sense to me. But per­haps one rea­son peo­ple don’t do it is be­cause they don’t know that it is what they should do? I don’t re­call a post here or on OB that ar­gued for this po­si­tion, for ex­am­ple.

• You mean “com­mon knowl­edge” in the tech­ni­cal sense de­scribed in the post?

If so, your ques­tions do not ap­pear to make sense.

• Why not? They both know they dis­agree, they both know they both know they dis­agree, etc… Per­haps Agent 1 doesn’t know 2′s par­ti­tion­ing, or vice versa. Or per­haps their par­ti­tion­ings are com­mon knowl­edge, but they lack the (com­pu­ta­tional abil­ity) to ac­tu­ally de­ter­mine the meet, for ex­am­ple, no?

• Wei was hy­poth­e­sis­ing dis­agree­ment due to an in­com­plete ex­change of in­for­ma­tion. In which case, the par­ties both know that they dis­agree, but don’t have the time/​en­ergy/​re­sources to sort each other’s opinions out. Then Au­mann’s idea doesn’t re­ally ap­ply.

• Aaah, okay. Though pre­sum­ably at least one would know the prob­a­bil­ities that both as­signed (and said “I dis­agree”...) that is, it would gen­er­ally take a bit of a con­trived situ­a­tion for them to know they dis­agree, but nei­ther to know any­thing about the other’s prob­a­bil­ity other than that it’s differ­ent.

(What hap­pens if the suc­cess­fully ex­change prob­a­bil­ities, have un­bounded com­put­ing power, they have shared com­mon knowl­edge pri­ors… But they don’t know each other’s par­ti­tion­ing… Or would the lat­ter au­to­mat­i­cally be com­puted from the rest?)

• Just one round of com­par­ing prob­a­bil­ities is not nor­mally enough for the par­ties in­volved to reach agree­ment, though.

• Well, if they do know each other’s par­ti­tions and are com­pu­ta­tion­ally un­bounded, then they would reach agree­ment af­ter one step, wouldn’t they? (or did I mi­s­un­der­stand the the­o­rem?)

Or do you mean If they don’t know each other’s par­ti­tions, iter­a­tive ex­change of up­dated prob­a­bil­ities effec­tively trans­mits the needed in­for­ma­tion?

• I too found my un­der­stand­ing changed dra­mat­i­cally when I looked into Au­mann’s origi­nal pa­per. Ba­si­cally, the re­sult has a mis­lead­ing billing—and those cit­ing the re­sult rarely seemed to bother ex­plain­ing much about the ac­tual re­sult or its sig­nifi­cance.

I also found my­self won­der­ing why peo­ple re­mained puz­zled about the high ob­served lev­els of dis­agree­ment. It seems ob­vi­ous to me that peo­ple are poor ap­prox­i­ma­tions of truth-seek­ing agents—and in­stead pro­mote their own in­ter­ests. If you un­der­stand that, then the ex­is­tence of many real-world dis­agree­ments is ex­plained: peo­ple dis­agree in or­der to ma­nipu­late the opinions and ac­tions of oth­ers for their own benefit.

• Sure all by it­self this first pa­per doesn’t seem very rele­vant for real dis­agree­ments, but there is a whole liter­a­ture be­yond this first pa­per, which weak­ens the as­sump­tions re­quired for similar re­sults. Keep read­ing.

• I already scanned through some of the pa­pers that cite Au­mann, but didn’t find any­thing that made me change my mind. Do you have any spe­cific sug­ges­tions on what I should read?

• Seen Han­son’s own http://​​han­son.gmu.edu/​​de­ceive.pdf—and its refer­ences?

• Yes, I looked at that pa­per, and also Agree­ing To Disagree: A Sur­vey by Gi­a­como Bo­nanno and Klaus Nehring.

http://​​www.scot­taaron­son.com/​​pa­pers/​​agree-econ.pdf

He shows that you do not have to ex­change very much in­for­ma­tion to come to agree­ment. Now maybe this does not ad­dress the ques­tion of the po­ten­tial in­tractabil­ity of the de­duc­tions to reach agree­ment (the wannabe pa­pers may do this) but I think it shows that it is not nec­es­sary to ex­change all rele­vant in­for­ma­tion.

The bot­tom line for me is the fla­vor of the Au­mann the­o­rem: that there must be a rea­son why the other per­son is be­ing so stub­born as not to be con­vinced by your own tenac­ity. I think this in­sight is the key to the whole con­clu­sion and it is to­tally over­looked by most dis­agreers.

• I haven’t read the whole pa­per yet, but here’s one quote from it (page 5):

The de­pen­dence, alas, is ex­po­nen­tial in 1 /​ (δ^3 ε^6), so our simu­la­tion pro­ce­dure is still not prac­ti­cal. How­ever, we ex­pect that both the pro­ce­dure and its anal­y­sis can be con­sid­er­ably im­proved.

Scott is talk­ing about the com­pu­ta­tional com­plex­ity of his agree­ment pro­to­col here. Even if we can im­prove the com­plex­ity to some­thing that is con­sid­ered prac­ti­cal from a com­puter sci­ence per­spec­tive, that will still likely be im­prac­ti­cal for hu­man be­ings, most of whom can’t even mul­ti­ply 3 digit num­bers in their heads.

• To quote from the ab­stract of Scott Aaron­son’s pa­per:

“A cel­e­brated 1976 the­o­rem of Au­mann as­serts that hon­est, ra­tio­nal Bayesian agents with com­mon pri­ors will never agree to dis­agree”: if their opinions about any topic are com­mon knowl­edge, then those opinions must be equal.”

Even “hon­est, ra­tio­nal, Bayesian agents” seems too weak. Goal-di­rected agents who are forced to sig­nal their opinions to oth­ers can benefit from vol­un­tar­ily de­ceiv­ing them­selves in or­der to effec­tively de­ceive oth­ers. Their self-de­cep­tion makes their opinions more cred­ible—since they hon­estly be­lieve them.

If an agent hon­estly be­lieves what they are say­ing, it is difficult to ac­cuse them of dishon­esty—and such an agent’s un­der­stand­ing of Bayesian prob­a­bil­ity the­ory may be im­mac­u­late.

Such agents are not con­strained to agree by Au­mann’s dis­agree­ment the­o­rem.

• Goal-di­rected agents who are forced to sig­nal their opinions to oth­ers can benefit from vol­un­tar­ily de­ceiv­ing them­selves in or­der to effec­tively de­ceive oth­ers. Their self-de­cep­tion makes their opinions more cred­ible—since they hon­estly be­lieve them.

This seems to re­flect hu­man cog­ni­tive ar­chi­tec­ture more than a gen­eral fact about op­ti­mal agents or even most/​all goal-di­rected agents. That hu­mans are not op­ti­mal is noth­ing new around here, nor that the agree­ment the­o­rems have lit­tle rele­vance to real hu­man ar­gu­ments. (I can’t be the only one to read the pa­pers and think, ‘hell, I don’t trust my­self as far as even the weak­ened mod­els, much less Creation­ists and what­not’, and have lit­tle use for them.)

• The rea­son is of­ten that you re­gard your own per­cep­tions and con­clu­sion as trust­wor­thy and in ac­cor­dance with your own aims—whereas you don’t have a very good rea­son to be­lieve the other per­son is op­er­at­ing in your in­ter­ests (rather than self­ishly try­ing to ma­nipu­late you to serve their own in­ter­ests). They may rea­son in much the same way.

Prob­a­bly much the same cir­cuitry con­tinues to op­er­ate even in those very rare cases where two truth-seek­ers meet, and con­vince each other of their sincer­ity.

• Uh oh, it looks like you guys are do­ing the Au­mann “meet” op­er­a­tion to up­date your be­liefs about Au­mann. Make sure to keep track of the lev­els of re­cur­sion...

• Should peo­ple re­ally adopt the “com­mon knowl­edge” ter­minol­ogy? Surely that ter­minol­ogy is highly mis­lead­ing and is re­spon­si­ble for many mi­s­un­der­stand­ings.

If peo­ple take com­mon English words and give them an es­o­teric tech­ni­cal mean­ing that differs dra­mat­i­cally from a literal read­ing, then shouldn’t they at least cap­i­tal­ise them?

• Sorry, I think I got a bit con­fused about the “meet” op­er­a­tion, mind clar­ify­ing?

is (I^J)(w) equal to the in­ter­sec­tion of I(w) and J(w) (which seems to be the im­plied way it works based on the over­all de­scrip­tion here) or some­thing else? (Since the defi­ni­tion of meet you gave in­volved unions rather than in­ter­sec­tions, and some sort of merg­ing op­er­a­tion)

Thanks.

EDIT: whoops. am stupid to­day. Meant to say in­ter­sec­tion, not disjunction

• Meet of two par­ti­tions (in the con­text of this post) is the finest com­mon coars­en­ing of those par­ti­tions.

Con­sider the coars­en­ing re­la­tion on the set of all par­ti­tions of the given set. Par­ti­tion A is a coars­en­ing of par­ti­tion B if A can be ob­tained by “lump­ing to­gether” some of the el­e­ments of B. Now, for this or­der, a “meet” of two par­ti­tions X and Y is a par­ti­tion Z such that

• Z is a coars­en­ing of X, and it is a coars­en­ing of Y

• Z is the finest such par­ti­tion, that is for any other Z’ that is a coars­en­ing of both X and Y, Z’ is also a coars­en­ing of Z.

• Meet of two par­ti­tions is the finest com­mon coars­en­ing of those par­ti­tions.

Un­der the us­ages fa­mil­iar to me, the com­mon coars­en­ing is the join, not the meet. That’s how “join” is used on the Wikipe­dia page for set par­ti­tions. Us­ing “meet” to mean “com­mon re­fine­ment” is the us­age that makes sense to me in the con­text of the proof in the OP. [ETA: I’ve been cor­rected on this point; see be­low.]

Of course, what you call “meet” or “join” de­pends on which way you de­cide to di­rect the par­tial or­der on par­ti­tions. Un­for­tu­nately, it looks like both pos­si­bil­ities are float­ing around as con­ven­tions.

• See for ex­am­ple on Wikipe­dia: Com­mon knowl­edge (logic)

It is not difficult to see that the com­mon knowl­edge ac­cessibil­ity func­tion [...] cor­re­sponds to the finest com­mon coars­en­ing of the par­ti­tions [...], which is the fini­tary char­ac­ter­i­za­tion of com­mon knowl­edge also given by Au­mann in the 1976 ar­ti­cle.

The idea is that the par­ti­tions define what each agent is able to dis­cern, so no re­fine­ment of what a given agent can dis­cern is pos­si­ble (un­less you perform ad­di­tional com­mu­ni­ca­tion). Au­mann’s agree­ment the­o­rem is about a con­di­tion for when the agents already agree, with­out any ad­di­tional dis­cus­sion be­tween them.

• Hmm. Then I am in a state of con­fu­sion much like Psy-Kosh’s. Th­ese op­pos­ing con­ven­tion aren’t helping, but, at any rate, I ev­i­dently need to study this more closely.

• It was con­fus­ing for me too, which is why I gave an im­per­a­tive defi­ni­tion: first form the union of I and J, then merge any over­lap­ping el­e­ments. Did that not help?

• Did that not help?

It should have. The fault is cer­tainly mine. I skimmed your defi­ni­tion too lightly be­cause you were defin­ing a tech­ni­cal term (“meet”) in a con­text (par­ti­tions) where I was already fa­mil­iar with the term, but I hadn’t sus­pected that it had any other us­ages than the one I knew.

• The term “meet” would cor­re­spond to con­sid­er­ing a coarser par­ti­tion as “less” than a finer par­ti­tion, which is nat­u­ral enough if you see par­ti­tions as rep­re­sent­ing “pre­ci­sion of knowl­edge”. The coarser par­ti­tion is able to dis­cern less. Great­est lower bound is usu­ally called “meet”.

• Great­est lower bound is usu­ally called “meet”.

It’s always called that, but the great­est lower bound and the least up­per bound switch places if you switch the di­rec­tion of the par­tial or­der. And there’s a lot of liter­a­ture on set par­ti­tions in which finer par­ti­tions are lower in the poset. (That’s the con­ven­tion used in the Wikipe­dia page on set par­ti­tions.)

The jus­tifi­ca­tion for tak­ing the meet to be a re­fine­ment is that re­fine­ments cor­re­spond to in­ter­sec­tions of par­ti­tion el­e­ments, and in­ter­sec­tions are meets in the poset of sets. So the ter­minol­ogy car­ries over from the poset of sets to the poset of set par­ti­tions in a way that ap­peals to the math­e­mat­i­cian’s aes­thetic.

But I can see the jus­tifi­ca­tion for the op­po­site con­ven­tion when you’re talk­ing about pre­ci­sion of knowl­edge.

• Ah, thanks. In that case… wouldn’t the meet of A and B of­ten end up be­ing the en­tire space?

For that mat­ter, why this coars­en­ing op­er­a­tion rather than the set of all the pos­si­ble pair­wise in­ter­sec­tions be­tween mem­bers of I and mem­bers of J?

ie, why coars­en­ing in­stead if “fine­ing” (what’s the ap­pro­pri­ate word there any­ways?)

When two ra­tio­nal­ists ex­change in­for­ma­tion, shouldn’t their con­clu­sions then some­times be finer rather than coarser since they have, well, each gained in­for­ma­tion they didn’t have pre­vi­ously?

• If I’ve got this right...

When two ra­tio­nal­ists ex­change all in­for­ma­tion, their new par­ti­tion is the ‘join’ of the two old par­ti­tions, where the join is the “coars­est com­mon fin­ing”. If you plot omega as the rec­t­an­gle with cor­ners at (-1,-1) and (1,1) and the ini­tial par­ti­tions are the x axis for agent A and the Y axis for agent B, then they share in­for­ma­tion and ‘join’ and then their com­mon par­ti­tion sep­a­rates all 4 quad­rants.

“com­mon knowl­edge” is the set of ques­tions that they can both an­swer be­fore shar­ing in­for­ma­tion. This is the ‘meet’ which is the coars­est com­mon fin­ing. In the pre­vi­ous ex­am­ple, there is no in­for­ma­tion that they both share, so the meet be­comes the whole quad­rant.

If you ex­tend omega down to y = −2 and mod­ify the origi­nal par­ti­tions to both fence off this new piece on its own, then the join would be the origi­nal four squares plus this lower rec­t­an­gle, while the meet would be the square from (-1,1) to (1,1) plus this lower rec­t­an­gle (since they now have this as com­mon knowl­edge).

Does this help?

• wait, what? is it coars­est com­mon fin­ing or finest com­mon coars­en­ing that we’re in­ter­ested in here?

And isn’t com­mon knowl­edge the set of ques­tions that not only they can both an­swer, but that they both know that both can an­swer, and both know that both know, etc etc etc?

Ac­tu­ally, maybe I need to reread this a bit more, but now am more con­fused.

Ac­tu­ally, on reread­ing, I think I’m start­ing to get the idea about meet and com­mon knowl­edge (given that be­fore ex­chang­ing info, they do know each other’s par­ti­tion­ing, but not which par­tic­u­lar par­ti­tion the other has ob­served to be the cur­rent one).

Thanks!

• Nope; it’s the limit of I(J(I(J(I(J(I(J(...(w)...), where I(S) for a set S is the union of the el­e­ments of I that have nonempty in­ter­sec­tions with S, i.e. the union of I(x) over all x in S, and J(S) is defined the same way.

Alter­nately if in­stead of I and J you think about the sigma-alge­bras they gen­er­ate (let’s call them sigma(I) and sigma(J)), then sigma(I meet J) is the in­ter­sec­tion of sigma(I) and sigma(J). I pre­fer this some­what be­cause the ma­chin­ery for con­di­tional ex­pec­ta­tion is usu­ally defined in terms of sigma-alge­bras, not par­ti­tions.

• Then… I’m hav­ing trou­ble see­ing why I^J wouldn’t very of­ten con­verge on the en­tire space.

ie, sup­pose a su­per sim­plifi­ca­tion in which both agent 1 and agent 2 par­ti­tion the space into only two parts, agent 1 par­ti­tion­ing it into I = {A1, B1}, and agent 2 par­ti­tion­ing into J = {A2, B2}

Sup­pose I(w) = A1 and J(w) = A2

Then, un­less the two par­ti­tions are iden­ti­cal, wouldn’t (I^J)(w) = the en­tire space? or am I com­pletely mis­read­ing? And thanks for tak­ing the time to ex­plain.

• That sim­plifi­ca­tion is a situ­a­tion in which there is no com­mon knowl­edge. In world-state w, agent 1 knows A1 (mean­ing knows that the cor­rect world is in A1), and agent 2 knows A2. They both know A1 union A2, but that’s still not com­mon knowl­edge, be­cause agent 1 doesn’t know that agent 2 knows A1 union A2.

I(w) is what agent 1 knows, if w is cor­rect. If all you know is S, then the only thing you know agent 1 knows is I(S), and the only thing that you know agent 1 knows agent 2 knows is J(I(S)), and so forth. This is why the usual “ev­ery­one knows that ev­ery­one knows that … ” defi­ni­tion of com­mon knowl­edge trans­lates to I(J(I(J(I(J(...(w)...).

• Well, how is it not the in­ter­sec­tion then?

ie, Agent 1 knows A1 and knows that Agent 2 knows A2

If they trust each other’s ra­tio­nal­ity, then they both know that w must be in A1 and be in A2

So they both con­clude it must be in in­ter­sec­tion of A1 and A2, and they both know that they both know this, etc etc...

Or am I miss­ing the point?

• As far as I un­der­stand, agent 1 doesn’t know that agent 2 knows A2, and agent 2 doesn’t know that agent 1 knows A1. In­stead, agent 1 knows that agent 2′s state of knowl­edge is in J and agent 2 knows that agent 1′s state of knowl­edge is in I. I’m a bit con­fused now about how this matches up with the mean­ing of Au­mann’s The­o­rem. Why are I and J com­mon knowl­edge, and {P(A|I)=q} and {P(A|J)=q} com­mon knowl­edge, but I(w) and J(w) are not com­mon knowl­edge? Per­haps that’s what the the­o­rem re­quires, but cur­rently I’m find­ing it hard to see how I and J be­ing com­mon knowl­edge is rea­son­able.

Edit: I’m silly. I and J don’t need to be com­mon knowl­edge at all. It’s not agent 1 and agent 2 who perform the rea­son­ing about I meet J, it’s us. We know that the true com­mon knowl­edge is a set from I meet J, and that there­fore if it’s com­mon knowl­edge that agent 1′s pos­te­rior for the event A is q1 and agent 2′s pos­te­rior for A is q2, then q1=q2. And it’s not un­rea­son­able for these pos­te­ri­ors to be­come com­mon knowl­edge with­out I(w) and J(w) be­com­ing com­mon knowl­edge. The the­o­rem says that if you’re both perfect Bayesi­ans and you have the same pri­ors then you don’t have to com­mu­ni­cate your ev­i­dence.

But if I and J are not com­mon knowl­edge then I’m con­fused about why any event that is com­mon knowl­edge must be built from the meet of I and J.

• Then agent 1 knows that agent 2 knows one of the mem­bers of J that have non empty in­ter­sec­tion with I(w), and similar for for agent 2.

Pre­sum­ably they have to tell each other which of their own par­ti­tions w is in, right? ie, pre­sum­ably SOME sort of in­for­ma­tion shar­ing hap­pens about each other’s con­clu­sions.

And, once that hap­pens, seems like in­ter­sec­tion I(w) and J(w) would be their re­sul­tant com­mon knowl­edge.

I’m con­fused still though what the “meet” op­er­a­tion is.

Un­less… the idea is some­thing like this: they ex­change prob­a­bil­ities. Then agent 1 rea­sons “J(w) is a mem­ber of J such that it both In­ter­sects I(w) AND would as­sign that par­tic­u­lar prob­a­bil­ity. So then I can de­ter­mine the sub­set of I(w) that in­ter­sects with those” and de­ter­mine a prob­a­bil­ity from there.” And similar for agent 2. Then they ex­change prob­a­bil­ities again, and go through an equiv­a­lent rea­son­ing pro­cess to tighten the spaces a bit more… and the the­o­rem en­sures that they’d end up con­verg­ing on the same prob­a­bil­ities? (each time they state un­equal prob­a­bil­ities, they each learn more in­for­ma­tion and each one then comes up with a set that’s a strict sub­set of the one they were pre­vi­ously con­sid­er­ing, but each of their sets always con­tain the in­ter­sec­tion of I(w) and J(w))?

• Try a con­crete ex­am­ple: Two dice are thrown, and each agent learns one die’s value. In ad­di­tion, each learns whether the other die is in the range 1-3 vs 4-6. Now what can we say about the sum of the dice?

Sup­pose player 1 sees a 2 and learns that player 2′s die is in 1-3. Then he knows that player 2 knows that player 1′s die is in 1-3. It is com­mon knowl­edge that the sum is in 2-6.

You could graph it by draw­ing a 6x6 grid and cir­cling the in­for­ma­tion par­ti­tion of player 1 in one color, and player 2 in an­other color. You will find that the meet is a par­ti­tion of 4 el­e­ments, each a 3x3 grid in one of the cor­ners.

In gen­eral, any­thing which is com­mon knowl­edge will limit the meet—that is, the meet par­ti­tion the world is in will not ex­tend to in­clude world-states which con­tra­dict what is com­mon knowl­edge. If 2 peo­ple dis­agree about global warm­ing, it is prob­a­bly com­mon knowl­edge what the cur­rent CO2 level is and what the his­tor­i­cal record of that level is. They agree on this data and each knows that the other agrees, etc.

The thrust of the the­o­rem though is not what is com­mon knowl­edge be­fore, but what is com­mon knowl­edge af­ter. The claim is that it can­not be com­mon knowl­edge that the two par­ties dis­agree.

• What I don’t like about the ex­am­ple you provide is: what player 1 and player 2 know needs to be com­mon knowl­edge. For in­stance if player 1 doesn’t know whether player 2 knows whether die 1 is in 1-3, then it may not be com­mon knowl­edge at all that the sum is in 2-6, even if player 1 and player 2 are given the info you said they’re given.

This is what I was con­fused about in the grand­par­ent com­ment: do we re­ally need I and J to be com­mon knowl­edge? It seems so to me. But that seems to be an­other as­sump­tion limit­ing the ap­pli­ca­bil­ity of the re­sult.

• Not sure… what hap­pens when the ranges are differ­ent sizes, or oth­er­wise the type of in­for­ma­tion learn­able by each player is differ­ent in non sym­met­ric ways?

Any­ways, thanks, upon an­other read­ing of your com­ment, I think I’m start­ing to get it a bit.

• Differ­ent size ranges in Hal’s ex­am­ple? Noth­ing in par­tic­u­lar hap­pens. It’s ok for differ­ent ran­dom vari­ables to have differ­ent ranges.

Otoh, if the play­ers get differ­ent ranges about a sin­gle ran­dom vari­able, then they could have prob­lems. Sup­pose there is one d6. Player A learns whether it is in 1-2, 3-4, or 5-6. Player B learns whether it is in 1-3 or 4-6.
And sup­pose the ac­tual value is 1.
Then A knows it’s 1-2. So A knows B knows it’s 1-3. But A rea­sons that B rea­sons that if it were 3 then A would know it’s 3-4, so A knows B knows A knows it’s 1-4. But A rea­sons that B rea­sons that A rea­sons that if it were 4 then B would know it’s 4-6, so A knows B knows A knows B knows it’s 1-6. So there is no com­mon knowl­edge, i.e. I∧J=Ω. (Omit­ting the ar­gu­ment w, since if this is true then it’s true for all w.)

And if it were a d12, with ranges still size 2 and 3, then the par­ti­tions line up at one point, so the meet stops at {1-6, 7-12}.

• In­ter­est­ing that the prob­lems with Au­mann’s the­o­rem were pointed out ten years ago, but be­lief in it con­tinues to be preva­lent.

• Di­a­grams would be won­der­ful, any­one up to draw­ing them?

• I think that I un­der­stand this proof now. Does the fol­low­ing di­alogue cap­ture it?

AGENT 1: My ob­ser­va­tions es­tab­lish that our world is in the world-set S. How­ever, as far as I can tell, any world in S could be our world.

AGENT 2: My ob­ser­va­tions es­tab­lish that our world is in the world-set T. How­ever, as far as I can tell, any world in T could be our world.

TOGETHER: So now we both know that our world is in the world-set ST—though, as far as we can tell, any world in ST could be our world. There­fore, since we share the same pri­ors, we both ar­rive at the same value when we com­pute P(E | ST), the prob­a­bil­ity that a given event E oc­curred in our world.

ETA: janos’s com­ment in­di­cates that I’m miss­ing some­thing, but I don’t have the time this sec­ond to think it through. Sounds like the set that they ul­ti­mately con­di­tion on isn’t ST but rather a sub­set of it.

ETA2: Well, I couldn’t re­sist think­ing about it, even though I couldn’t spare the time :). The up­shot is that I don’t un­der­stand janos’s com­ment, and I agree with Psy-Kosh. As stated, for ex­am­ple, in this pa­per:

The meet [of par­ti­tions π1 and π2] has as blocks [i.e., el­e­ments] all nonempty in­ter­sec­tions of a block from π1 with a block from π2.

From this it fol­lows that the el­e­ment of I∧J con­tain­ing w is pre­cisely I(w) ∩ J(w). So, un­less I’m miss­ing some­thing, my di­alogue above com­pletely cap­tures the proof in the OP.

ETA3: It turns out that both pos­si­ble ways of ori­ent­ing the par­tial or­der re­la­tion are in com­mon use. Every­thing that I’ve seen dis­cussing the the­ory of set par­ti­tions puts re­fine­ments lower in the lat­tice. This was the con­ven­tion that I was us­ing above. But, as Vladimir Nesov points out, it’s nat­u­ral to use the op­po­site con­ven­tion when talk­ing about epistemic agents, and this is the us­age in Wei Dai’s post. The clash be­tween these con­ven­tions was a large part of the cause of my con­fu­sion. At any rate, un­der the con­ven­tion that Wei Dai is us­ing, the el­e­ment of I∧J con­tain­ing w is not in gen­eral I(w) ∩ J(w).

• Your di­a­log is one way to achieve agree­ment, and what I meant when I said “sim­ply tell each other I(w) and J(w)” how­ever it is not what Au­mann’s proof is about. The di­a­log shows that two Bayesi­ans with the same prior would always agree if they ex­change enough in­for­ma­tion.

Au­mann’s proof is not re­ally about how to reach agree­ment, but why dis­agree­ments can’t be “com­mon knowl­edge”. The proof fol­lows a com­pletely differ­ent struc­ture from your di­a­log.

From this it fol­lows that the el­e­ment of I∧J con­tain­ing w is pre­cisely I(w) ∩ J(w).

No, this is wrong. Please edit or delete it to avoid con­fus­ing oth­ers.

• From this it fol­lows that the el­e­ment of I∧J con­tain­ing w is pre­cisely I(w) ∩ J(w).

No, this is wrong. Please edit or delete it to avoid con­fus­ing oth­ers.

The im­pli­ca­tion that I as­serted is cor­rect. The con­fu­sion arises be­cause both pos­si­ble ways of ori­ent­ing the par­tial or­der on par­ti­tions are com­mon in the liter­a­ture. But I’ll note that in the com­ment.

• The prob­lem is not in con­ven­tions and the liter­a­ture, but in whether your in­ter­pre­ta­tion cap­tures the state­ment of the the­o­rem dis­cussed in the post. Am­bi­guity of the term is no ex­cuse. By the way, “meet” is Au­mann’s us­age as well, as can be seen from the first page of the origi­nal pa­per.

• Am­bi­guity of the term is no ex­cuse.

In­deed. I plead guilty to read­ing hastily. I saw the term “meet” be­ing used in a con­text where I already knew its defi­ni­tion (the only defi­ni­tion it had, so far as I knew), so I only briefly skimmed Wei Dai’s own defi­ni­tion. Ob­vi­ously I was too care­less.

How­ever, it re­ally bears em­pha­siz­ing how strange it is to put re­fine­ments higher in the par­tial or­der of par­ti­tions, at least from the per­spec­tive of the gen­eral the­ory of par­tial or­ders. Un­der the cat­e­gory the­o­retic defi­ni­tion of par­tial or­ders, PQ means that there is a map PQ. Now, to say that a par­ti­tion Q is a coars­en­ing of a par­ti­tion P is to say that Q is a quo­tient P/​~ of P. But such a quo­tient cor­re­sponds canon­i­cally to a map PQ send­ing each el­e­ment p of P to the equiv­alence class in Q con­tain­ing p. In­deed, Wei Dai is in­vok­ing just such maps when he writes “I(w)”. In this case, Ω is con­strued as the dis­crete par­ti­tion of it­self (where each el­e­ment is in its own equiv­alence class) and I is used (as an abuse of no­ta­tion) for the canon­i­cal map of par­ti­tions I: Ω → I. The up­shot is that one of these canon­i­cal par­ti­tion maps PQ ex­ists if and only if Q is a coars­en­ing of P. There­fore, that is what PQ should mean. In the con­text of the gen­eral the­ory of par­tial or­ders, coarser par­ti­tions should be greater than finer ones.

• Efforts to illu­mi­nate Au­mann’s dis­agree­ment re­sult do seem rather rare—thanks for your efforts here.

• sec­onded!

• It ap­pears to me that re­duc­ing this to an equa­tion is to­tally ir­rele­vant, in that it ob­scures the premises of the ar­gu­ment, and an ar­gu­ment is only as good as the re­li­a­bil­ity of the premises. More­over, the the­o­rem ap­pears faulty based on in­duc­tive logic, in that the premises can be true and the con­clu­sion false. I’m re­ally in­ter­ested in why this thought pro­cess is wrong.

• While I see your point, I wouldn’t say that the agree­ment is­sue is over rated at all.

There are many dis­agree­ments that don’t change at all over ar­bi­trar­ily many iter­a­tions, which sure don’t look right given AAT. Even if the be­liefs don’t con­verge ex­actly, I don’t think its too much to ask for some mo­tion to­wards con­ver­gence.

I think the more im­por­tant parts are the parts that talk about pre­dict­ing disagreements

• Could ro­bust statis­tics be rele­vant for ex­plain­ing fixed points where dis­agree­ments do not change at all?

Roughly speak­ing, the idea of ro­bust statis­tics is that the me­dian or similar con­cepts may be prefer­able in some cir­cum­stances to the mean—and un­like the mean, the me­dian rou­tinely does not change at all, even when an­other dat­a­point changes.

• I don’t think that re­ally helps. If you’re treat­ing some­ones be­liefs as an out­lier, then you’re not re­spect­ing that per­son as a ra­tio­nal­ist.

Even if you did take the me­dian of your metaprob­a­bil­ity dis­tri­bu­tion (which is not the odds you want to bet on, though you may want to pro­fess them for some rea­son), even­tu­ally you should change your mind (most both­er­some dis­agree­ments in­volve peo­ple con­fi­dently on op­po­site sides of the spec­trum so the di­rec­tion in which to up­date is ob­vi­ous).

It could be that in prac­tice most peo­ple up­date be­liefs ac­cord­ing to some more “ro­bust” method, but to the ex­tent that it freezes their be­liefs un­der new real ev­i­dence, its a sucky way of do­ing it and you don’t get a ‘get out of jail free’ card for do­ing it.

• The main prob­lem I have always had with this is that the refer­ence set is “ac­tual world his­tory” when in fact that is the ex­act thing that ob­servers are try­ing to de­ci­pher.

We all re­al­ize that there is in fact an “ac­tual world his­tory” how­ever if it was known then this wouldn’t be an is­sue. Us­ing it as a refer­ence set then, seems spu­ri­ous in all prac­ti­cal­ity.

The most ob­vi­ous way to achieve it is for the two agents to sim­ply tell each other I(w) and J(w), af­ter which they share a new, com­mon in­for­ma­tion par­ti­tion.

I think that sum­ma­tion is a good way to in­ter­pret the prob­lem I ad­dressed in as prac­ti­cal a man­ner as is cur­rently available; I would note how­ever that most peo­ple ar­bi­trar­ily weight ob­ser­va­tional in­fer­ence, so there is a skew­ing of the data.

The sad part about the whole thing is that both or all ob­servers ex­chang­ing in­for­ma­tion may be the same de­vi­a­tion away from w such that their com­bined prob­a­bil­ities of l(w)are fur­ther away from w than ei­ther in­di­vi­d­u­ally.

• Huh? The refer­ence set Ω is the set of pos­si­ble world his­to­ries, out of which one el­e­ment is the ac­tual world his­tory. I don’t see what’s wrong with this.

• I sup­pose my post was poorly worded. Yes, in this case omega is the refer­ence set for pos­si­ble world his­to­ries.

What I was refer­ring to was the baseline of w as an ac­cu­rate mea­sure. It is a nor­mal­iz­ing refer­ence, though not a set.