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Uncommon Priors Require Origin Disputes


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In a standard Bayesian belief model, the prior is always common knowledge. This prevents such a model from representing agents’ probabilistic beliefs about the origins of their priors. By embedding such a standard model in a larger standard model, however, we can describe such beliefs using "pre-priors". When an agent’s prior and pre-prior are mutually consistent in a particular way, she must believe that her prior would only have been different in situations where relevant event chances were different, but that variations in other agents’ priors are otherwise completely unrelated to which events are how likely. Thus Bayesians who agree enough about the origins of their priors must have the same priors. (More:

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Uncommon Priors Require Origin Disputes

  1. 1. Uncommon Priors Require Origin Disputes Robin Hanson George Mason University Rutgers Probability Seminar October 15, 2018
  2. 2. Standard Econ Bayesian Model 1. Grand State Space - resolves all uncertainty 2. One Prior - complete, same for all times/cases 3. Update – diff. beliefs only via diff. info: EP[X|I] 4. Expected Utility – picks max EP[U|I], U consta. 5. Common Prior – all agents use same prior (Gilboa, Postlewaite, & Schmeidler 2011, Synthese) An argument: differing prior act-equivalent to differing utility
  3. 3. We Can’t Agree to Disagree Aumann in 1976 • Any finite information • Re possible worlds • Common knowledge • Of exact E1[x], E2[x] • Would say next • For Bayesians • With common priors • If seek truth, not lie Since generalized to ® Infinite spaces ® Impossible worlds ® Common Belief ® A f(•, •), or who max ® Last ±(E1[x] - E1[E2[x]]) ® At core, or Wannabe ® Symmetric prior origins
  4. 4. Two Faces of Bayesian Priors • Prior = counterfactual probability would assign to states given least possible info – Prior gives any belief/expectation: EP[X|I] • Can have beliefs on prior origins/causes – Can this help constrain rational priors? • Problem: all priors always common kn. • Solution: embed standard model in extended model
  5. 5. Origins of Priors • Seems irrational to accept some priors – Imagine random brain changes for weird priors • In standard theories, your prior is not special – Species-common DNA • Selected to predict ancestral environment – Individual DNA variations (e.g. personality) • Random by Mendel’s rules of inheritance • Sibling differences independent of everything else! – Culture: random + adapted to local society • But you must think differing prior special! • Can’t express these ideas in standard models
  6. 6. Example of Standard Model X=1 p=1/4 q=1/3 X=2 p=1/4 q=1/3 X=3 p=1/4 q=1/6 X=4 p=1/4 q=1/6 A not A B not B P knows if A or not Q knows if B or not EP[X | A] = 2 EP[X | not A] = 3 EQ[X | B] = 1.5 EQ[X | not B] = 3.5
  7. 7. Example of Extended Model X=1 p=1/4 q=1/3 X=2 p=1/4 q=1/3 X=3 p=1/4 q=1/6 X=4 p=1/4 q=1/6 A not A B not B X=1 p=1/4 q=1/4 X=2 p=1/4 q=1/4 X=3 p=1/4 q=1/4 X=4 p=1/4 q=1/4 A not A r=1/6 r=1/6 r=1/12 r=1/12 r=1/8 r=1/8 r=1/8 r=1/8
  8. 8. Standard Bayesian Model Agent 1 Info Set Agent 2 Info Set Common Kn. Set A Prior
  9. 9. An Extended Model Multiple Standard Models With Different Priors
  10. 10. Standard Bayesian Model knowledgecommonis),,,(ModelIn )),(|()(Belief )(),(),(Info )()(Prior }21{Agent(finite)State 21 21 pp EEpEp ,...,, ,...,p,ppp,p ,...,N,iω t iiit Tt tt N tttt i Ni          
  11. 11. Extending the State Space )}(,:),({)),(( ~ (1))()| ~ (~ }:),({},:),({ ~ ~ ),(~state-Pre ,priorsPossible    t i t i ii N N i pppp EppEp ppppEpE p pp      P PP As event
  12. 12. An Extended Model .,ynecessarilnotbut,, ~ torelative knowledgecommonis),,,,,(ModelIn (2))| ~ (~)| ~ ( allow),(),~(prior-Pre ),~( ~ )~( )(),,...,,(),(info-Pre 21 21 StTt pq pEppEq qq,...,q,qqqωq STt tt ii jiNi t i t i St tt N tttt i        P  
  13. 13. Combining (1) & (2) 𝑝𝑖( 𝐸|𝑝) ≡ 𝑝𝑖(𝐸) (1) 𝑞𝑖( 𝐸|𝑝) = 𝑝𝑖( 𝐸|𝑝) (2) 𝑞𝑖( 𝐸|𝑝1, 𝑝2, ⋯ , 𝑝𝑖 ⋯ , 𝑝 𝑁) = 𝑝𝑖(𝐸)
  14. 14. My Differing Prior Was Made Special ).()| ~ (viaondependsE ~ any,In .given,anyoftindependenisE ~ any,In .ifgivenofis,In )|() ~ ,,...,,...,,| ~ ((2)&(1) 21 EppEqpq ppq P(A|C)P(A|BC)CBAP BEpBppppEq iiiii iiji iNii     2Theorem 1Theorem tindependen My prior and any ordinary event E are informative about each other. Given my prior, no other prior is informative about any E, nor is E informative about any other prior.
  15. 15. Corollaries ).()(then),| ~ ()| ~ (If EPEPPpEqPpEq iiii 1Corollary ).()(then ),,| ~ (),| ~ (If EPEP PpPpEqPpPpEq jiijii  2Corollary My prior only changes if events are more or less likely. If an event is just as likely in situations where my prior is switched with someone else, then those two priors assign the same chance to that event. Only common priors satisfy these and symmetric prior origins.
  16. 16. A Tale of Two Astronomers • Disagree if universe open/closed • To justify via priors, must believe: “Nature could not have been just as likely to have switched priors, both if open and if closed” “If I had different prior, would be in situation of different chances” “Given my prior, fact that he has a different prior contains no info” All false if they are brothers!