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
• Any 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
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
Impossible worlds
Common Belief
A f(•, •), or who max
Last ±(E1[x] - E1[E2[x]])
At core, or Wannabe
Symmetric prior origins
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  ( 1t , t2 ,..., tN ),   (  t )tT
Belief
pit ( E )  pi ( E |  it ( )),
E
In Model (, p,  ), p is common knowledge
Extending the State Space
Possible priors
pi  P, p  P N
~
N
~
Pre - state   ( , p )      P
As event
~
E  {( , p ) :   E}, p  { ( , p) : p  p}
~
~
pi ( E | p )  pi ( E )
(1)
~t
t




 i (( , p ))  {( , p ) : p  p,    i ( )}
An Extended Model
Pre - info
it ( ),  t  ( 1t , 2t ,..., Nt ),   (  t )tS
~t ~
~
 ( )   i ( ), t  T  S
t
i
~ ), q  ( q ,q ,...,q ), allow q  q
Pre - prior qi (ω
1 2
N
i
j
~
~
~
qi ( E | p )  pi ( E | p )
(2)
In Model (, P, , q, ), p is common knowledge
~t
t
relative to  , t  T , but not necessaril y  , t  S .
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 independen t of B given C if P(A|BC)  P(A|C).
~
Theorem 1 In qi , any E is independen t of any p j i , given pi .
~
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Theorem 2 In qi , any E depends on pi via qi ( E | pi )  pi ( E ).
Corollaries
~
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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),
then P( 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!