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					Uncommon Priors Require
    Origin Disputes
       Robin Hanson
   George Mason University
       ProLogic 2005
   Clarifying “Objective” Beliefs
• Regarding belief as truth estimate
   – vs. as expression of individuality, or what provokes
     intellectual progress
• Yes, distinguish possible topics of beliefs
   – what theory to apply now, vs. theory closest to
     ultimate truth, vs. mix of topics to focus research now
• Normative, not positive, claim
   – Clearly real people with similar info often disagree
   – But big scientists in big episodes may be wrong
• Need not know how to explicitly construct
   – Constraints: ≥0, Σ=0, update via Bayes’ rule, more?
   – If enough constraints, result must be unique
  The Puzzle of Disagreement
• Persistent disagreement ubiquitous
  – Speculative trading, wars, juries, …
  – Argue in science, politics, family, …
• Theory seems to say this irrational
• Possible explanations
  – We’re “just joshing”
  – Infeasible epistemic rationality
  – Fixable irrationality: all will change!
  – Non-epistemic rationality – truth not goal
  We Can’t Agree to Disagree
Aumann in 1976             Since generalized to
• Any information
• Re possible worlds          Impossible worlds
• Common knowledge            Common Belief
• Of exact E1[x], E2[x]       A f(•, •), or who max
• Would say next              Last ±(E1[x] - E1[E2[x]])
• For Bayesians               At core, or Wannabe
• With common priors          Symmetric prior origins
• If seek truth, not lie
  My Answer: We Self-Deceive
• We biased to think better driver, lover, …
   “I less biased, better data & analysis”
• Evolutionary origin: helps us to deceive
   – Mind “leaks” beliefs via face, voice, …
   – Leak less if conscious mind really believes
• Beliefs like clothes
   – Function in harsh weather, fashion in mild
• When see our self-deception, still disagree
   – So at some level we accept that we not seek truth
         Two Faces of Priors
• Prior help tell us what to believe
• We have beliefs about prior origins/causes
  – Can this help constrain rational priors?
               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
          Standard Bayesian Model


                           Agent 1 Info Set
A Prior

                           Agent 2 Info Set


                           Common Kn. Set
          An Extended Model



Multiple Standard
Models With
Different Priors
  Standard Bayesian Model

State ω   (finite)          Agent i  {1,2 ,...,N}

Prior pi ( ) ,               p  ( p1,p2 ,...,p )
                                                N


Info  it ( ),  t  ( 1, t2 ,..., tN ),   (  t )tT
                         t



Belief     pit ( E )  pi ( E |  it ( )),   E

In Model (, p,  ), p is commonknowledge
Extending the State Space

Possible priors     pi  P, p  P N
                          ~
            ~  (, p )      P N
Pre - state 
                       As event
~
E  {(, p ) :   E}, p  { (, p) : p  p}
               ~
           ~ ( E | p)  p ( E )
           pi            i                   (1)
~t
 i ((, p ))  {(, p) : p  p,    i ( )}
                                           t
          An Extended Model
Pre - info it ( ), t  ( 1t , 2t ,...,N ),   ( t )tS
                                            t


                     ~ ~
               ~ )   t ( ), t  T  S
             (
              i
               t
                       i

                ~
Pre - prior qi (ω), q  ( q1,q2 ,...,q ), allow qi  q j
                                      N

                        ~             ~
                                  ~ ( E | p)
                   qi ( E | p )  pi                     (2)

In Model (, P, , q, ), p is commonknowledge
              ~t
  relative to  , t  T , but not necessarily  , t  S .
                                               t
My Differing Prior Was Made Special
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.

                     ~                               ~
(1) & (2)      qi ( E | p1 , p2 ,..., pi ,..., pN , B )  pi ( E | B )

In P, A is independent of B given C if P(A|BC) P(A|C)
                                                     .
                      ~
Theorem 1 In qi , any E is independent of any p j i , given pi .
                      ~                        ~
Theorem 2 In qi , any E depends on pi via qi ( E | pi )  pi ( E ).
                          Corollaries
                     ~                   ~
 Corollary 1 If qi ( E | pi  P )  qi ( E | pi  P), then P( E )  P( E ).

  My prior only changes if events are more or less likely.
                     ~                             ~
 Corollary 2 If qi ( E | pi  P, p j  P )  qi ( E | pi  P, p j  P),
    thenP( E )  P( E ).
  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.
   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!

				
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posted:2/15/2012
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