Information about http://hanson.gmu.edu/prior.pdf
Uncommon Priors Require Origin Disputes …
Tags: belief, common knowledge, department of economics, george mason university, gul, hanson department, models, origins, partition, prior probability, probabilistic beliefs, probability distribution, rational agents, rationality, relevant event, robin hanson, standard model, variations,Pages: 7
Language: english
Created: Sat Jul 15 12:17:10 2006
Display cached document
Page 1
Page 2
Page 3
Page 4
Page 5
Page 6
Page 7
Uncommon Priors Require Origin Disputes
Robin Hanson
Department of Economics
George Mason University
July 2006, First Version June 2001
Abstract
In standard belief models, priors are always common knowledge. This prevents such
models from representing agents' probabilistic beliefs about the origins of their priors.
By embedding standard models in a larger standard model, however, pre-priors can
describe such beliefs. When an agent's prior and pre-prior are mutually consistent, he
must believe that his 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. Due to this, Bayesians who agree
enough about the origins of their priors must have the same priors.
Introduction
The most standard way to model a set of agents with beliefs that change over time is to
describe a set of possible states, a prior probability distribution for each agent, and an
information partition for each agent at each time. In some such standard models, all agents
have the same prior, while in other models priors differ. Some argue that common priors are
implied by common knowledge of rationality, since rational agents with the same information
should have the same beliefs (Aumann, 1998). Others do not find this argument compelling
(Morris, 1995; Gul, 1998).
Since it is logically possible either that agent priors are common, or that they are uncom-
mon, one might imagine that an agent could be uncertain about the priors of other agents.
Within a standard model, however, no agent can be unsure of any agent's prior; every prior
is always common knowledge. This is problematic, however, as it seems to preclude agents
from using probabilities to reason about the origins of their priors.
For example, if there were such a thing as a gene for optimism versus pessimism, you
might believe that you had an equal chance of inheriting your mother's optimism gene or your
rhanson@gmu.edu http://hanson.gmu.edu 703-993-2326 FAX: 703-993-2323 MS 1D3, Carow Hall, Fair-
fax VA 22030
1
father's pessimism gene. You might further believe that your sister had the same chances as
you, but via an independent draw, and following Mendel's rules of inheritance. You might
even believe that humankind would have evolved to be more pessimistic, had they evolved
in harsher environments. Beliefs of this sort seem central to scientific discussions about the
origin of human beliefs, such as occur in evolutionary psychology.
Beliefs about the origins of priors also seem relevant to the rationality of priors. For
example, if you learned that your strong conviction that fleas sing was the result of an
experiment, which physically adjusted people's brains to give them odd beliefs, you might
well think it irrational to retain that belief (Talbott, 1990). Similarly it might be irrational
to be more optimistic than your sister simply because of a random genetic lottery.
This paper presents a theoretical framework in which agents can hold probabilistic beliefs
about the origins of their priors, and uses this framework to consider how such beliefs might
constrain the rationality of priors. The basic approach is to embed a set of standard models
within a larger encompassing standard model. Each embedded model differs only in which
agents have which priors, while the larger encompassing model includes beliefs about which
possible prior combinations might be realized.
Just as beliefs in a standard model depends on ordinary priors, beliefs in the larger model
depend on pre-priors. We do not require that these pre-priors be common; pre-priors can
vary. But to keep priors and pre-priors as consistent as possible with each other, we impose a
pre-rationality condition. This condition in essence requires that each agent's ordinary prior
be obtained by updating his pre-prior on the fact that nature assigned the agents certain
particular priors.
This pre-rationality condition has strong implications regarding the rationality of un-
common priors. Consider, for example, two astronomers who disagree about whether the
universe is open (and infinite) or closed (and finite). Assume that they are both aware of
the same relevant cosmological data, and that they try to be Bayesians, and therefore want
to attribute their difference of opinion to differing priors about the size of the universe.
This paper shows that neither astronomer can believe that, regardless of the size of the
universe, nature was equally likely to have switched their priors. Each astronomer must
instead believe that his prior would only have favored a smaller universe in situations where
a smaller universe was actually more likely. Furthermore, he must believe that the other
astronomer's prior would not track the actual size of the universe in this way; other priors
can only track universe size indirectly, by tracking his prior. Thus each person must believe
that prior origination processes make his prior more correlated with reality than others'
priors.
As a result, these astronomers cannot believe that their differing priors arose due to the
expression of differing genes inherited from their parents in the usual way. After all, the usual
rules of genetic inheritance treat the two astronomers symmetrically, and do not produce
individual genetic variations that are correlated with the size of the universe.
This paper thereby shows that agents who agree enough about the origins of their priors
must have the same prior. This is a new argument for common priors, and one that depends
only on consistency relations between the beliefs of a single agent.
2
Analysis
A standard1 Bayesian belief model M = (, p, ) regarding N agents consists of a finite
set of states, an assignment p = (p1 , p2 , . . . , pN ) of a prior probability distribution pi
over for each agent i, and an information history = (t)tT over times T , where
t = (t , t , . . . , t ) and each t is a partition2 of . At state and time t T ,
1 2 N i
agent i assigns to event E the subjective probability pit (E) = pi (E|t()), where pi (A|B)
i
pi (AB)/pi(B) and pi (E) E pi ().
In a standard model M, every prior pi (a map between states and probabilities) is always
common knowledge. To allow uncertainty about the priors of a standard model, let us
introduce an extended model M = (, P, , q, ), where P is a set of possible priors pi over
~
, and where q and are assignments of priors and information partitions over an extended
state space = × P N .
~
~
A pre-state = (, p) combines an ordinary state and a prior assignment
~
p = (p1 , p2 , . . . , pN ) P N . For each ordinary event E let us define a corresponding
event E {(, p) : E} in the extended space, and for each ordinary information
~ ~
set t (), let us define a corresponding information set
i
t ((, p)) {( , p ) : p = p , t ()}
~
i i
in the extended space. Let us also project an ordinary prior pi into a prior pi over the
~
extended space by defining
~ ~
pi (E|p) pi (E), (1)
where prior assignment p treated as an event denotes p = {(, p ) : p = p}. Agent prior pi
can also be treated as the event pi = {(, (p1 , p2 , . . . , pi , . . . , pN )) : pi = pi }, where i pi = p.
~
These definitions let us see the extended space as being partitioned into many ordinary
models M(p) = (p, p, ), which are identical to each other and to our original model M,
~ ~
except for having differing prior assignments p. Models M(p) describe the belief history of
N ordinary agents i at times t T . Within each model M(p), the beliefs of every ordinary
~
agent i at time t are described in terms of this extended space using the extended prior
pi and the extended information i
~ ~ t.
Even in this extended space, the priors p are always common knowledge for agents i.
This fact prevents any agent i from using beliefs derived from his prior pi and information
~
~ t to express beliefs about the origins of his and others' priors. To allow the expression of
i
such beliefs about origins, let us introduce the concept of a pre-agent i.
We introduce pre-agent i as a way to represent certain counter-factual beliefs of agent
i about the origins of priors. These beliefs are counter-factual in the sense that they are
counter to the fact that agent i must always know everyone's priors. While we want the
beliefs of agent i and pre-agent i to agree as much as possible, some of pre-agent i's beliefs
will describe what he considers to be reasonable beliefs about the processes that produced
the prior of agent i and the other agents. Such beliefs about prior origins might be based
3
on beliefs about genetic processes, cultural processes, or any other processes that the agent
believed to be relevant in producing the particular priors that the agents were given.
Formally, each pre-agent i has a pre-prior qi and pre-information t over the extended
i
~
space for times t S. The key to allowing uncertainty about agent priors is to introduce
new earlier times (s S such that s < t for all t T ) when agent i has no beliefs. At such
times s, pre-agent i can be uncertain about many aspects of the process that produced the
various agents' priors, including which exact priors p resulted. For this to be possible, the
pre-information s of pre-agent i should not make the event p common knowledge among
i
pre-agents at early times s.
Since pre-agent i represents the counterfactual beliefs of agent i, the beliefs of agent i
and pre-agent i should agree as much as possible. Therefore at the times when both agents
and pre-agents have beliefs, agent and pre-agent priors and information partitions should
agree. That is, for all i and all t T S, we should have t = t .
i
~
i
Furthermore, priors pi and pre-priors qi should satisfy a pre-rationality condition,
qi (E|p) = pi (E|p).
~ ~ ~ (2)
This condition says that agent priors are as if agent i had acquired his prior pi by conditioning
his pre-prior qi on the fact that nature assigned him and others certain priors. Note that
we do not assume pre-priors are common. Our assumptions do not directly constrain the
relations between agent beliefs, or the relations between pre-agent beliefs, but only the
relations between pre-agent and agent beliefs.
Equations 1 and 2 combine to give
~
qi(E|p1 , p2 , p3 , . . . , pi , . . . , pN ) = pi (E). (3)
The following results all follow trivially from equation 3.
Theorem 1 In pre-prior qi, given prior pi any event E is independent3 of other priors pj=i .
~
That is, once pre-agent i knows prior pi , no other prior pj can be informative about any
event E, nor can any event E be informative about any other prior pj .
~ ~
Theorem 2 In pre-prior qi, any E is directly dependent on prior pi via qi(E|pi ) = pi (E).
To pre-agent i, the prior pi and any event E are always informative about each other.
Corollary 1 If qi (E| pi = P ) = qi (E| pi = P ), then P (E) = P (E).
~ ~
My prior would only change when the events it forecasts become more or less likely.
Corollary 2 If qi (E| pi = P, pj = P ) = qi (E| pi = P , pj = P ), then P (E) = P (E).
~ ~
4
If event E were just as likely in situations where my prior had been exchanged with
someone else's prior, those priors must be the same regarding event E. And via Bayes' rule,
the same holds if exchanged priors are just as likely given E, and just as likely given not E.
One concern about the above analysis is that our beliefs about the processes that pro-
duce priors are based on information we have received during our lives as agents. So this
information would not have been available to our pre-agents, who we imagine held beliefs
before we ever had any beliefs.
Fortunately, we can generalize equation 3 to include any background information B,
qi (E|p1 , p2 , p3 , . . . , pi , . . . , pN , B) = pi (E|B).
~ ~
Thus theorems 1 and 2 and corollaries 1 and 2 all easily generalize to condition on B.
~ ~
For example, theorem 2 becomes qi (E|pi B) = pi (E|B). So if we set B to represent the
background information on which we base our beliefs about the basic processes that produce
priors, all of the above results will apply conditional on that background information.
Discussion
These constraints on beliefs about the origins of priors are strong and highly asymmetric.
Each agent must believe that his prior would "track truth" in the sense that his prior would
only assign a higher probability to an event in situations where that event actually was more
likely. Furthermore, he must believe that other agent's priors would only track truth to the
extent that their priors covaried with his prior; he believes any additional variation in the
priors of others must be completely unrelated to any other events of interest.
In contrast, standard scientific beliefs about the origins of individual human variations
do not offer much support for the belief that some people's initial beliefs tendencies track
truth much better than other people's tendencies.
For example, some have argued that many general attitudes, such as general optimism
or pessimism, are influenced by our genes (Olson, Vernon, Harris, & Jang, 2001). (Others
disagree with this claim.) Mendel's rules of genetic inheritance, however, are symmetric
and random between siblings. If optimism were coded in genes, you would not acquire an
optimism gene in situations where optimism was more appropriate, nor would your sister's
attitude gene track truth any worse than your attitude gene does.
Thus it seems to be a violation of pre-rationality to, conditional on accepting Mendel's
rules, allow one's prior to depend on individual variations in genetically-encoded attitudes.
Having your prior depend on species-average genetic attitudes may not violate pre-rationality,
but this would not justify differing priors within a species.
Similar problems apply to cultural processes that produce priors. If priors are transmitted
culturally via children copying visible adults, standard theories about individual variations
in such culturally-transmitted belief tendencies offer little support for the idea that some
children are better able to select the most truth-tracking cultural elements from among
the available cultural transmissions. Perhaps some children could better extract relevant
5
information from the available cultural transmissions, but any differences in information
must represented in differing information partitions, and not in differing priors.
Without some basis for believing that the process that produced your prior was sub-
stantially better at tracking truth than the process that produced other peoples' priors, you
appear to have no basis for believing that beliefs based on your prior, are more accurate
than beliefs based on other peoples' priors.
Conclusion
In standard models priors are common knowledge, which makes it hard to express prob-
abilistic beliefs that agents might have about the origins of their priors. This paper has
introduced a formal framework to allow standard agents to reason probabilistically about
the origins of their priors. Standard models corresponding to different priors are embedded
in a larger space where priors can vary, and pre-agents are imagined who can be uncertain
about which priors ordinary agents will get. Pre-priors, the priors of these pre-agents, then
describe counterfactual beliefs an agent has regarding the origins of ordinary priors.
Using this concept of a pre-prior, this note has presented a new argument for the ratio-
nality of common priors. The argument is that a simple plausible constraint relating rational
priors and pre-priors seems to conflict with our usual scientific stories about the origins of
individual variations in early belief tendencies. While people often disagree about the origins
of humankind, they also seem to accept the idea that nature treated individuals symmetri-
cally ex ante, and that most other topics of disagreement are irrelevant to estimating those
origin processes. It thus seems hard to attribute most human disagreements to these sort of
rationally differing priors.
Acknowledgments
For their comments, I thank Nick Bostrom, Tyler Cowen, Ronald Heiner, Sahotra Sarkar,
William Talbott, one very helpful anonymous referee, and ten other anonymous referees. I
thank the Center for Study of Public Choice, the Mercatus Center, and the International
Foundation for Research in Experimental Economics for financial support.
Notes
1
Belief hierarchies are also a standard form. The mapping between these forms is unique given common
priors.
2
It is common to also require that t weakly refine s when t > s.
i i
3
In distribution P , event A is independent of B when P (A|B) = P (A). In P , event A is independent of
B conditional on C when P (A|BC) = P (A|C). Independence is a symmetric relation.
6
References
Aumann, R. J. (1998). Common Priors: A Reply to Gul. Econometrica, 66 (4), 929938.
Gul, F. (1998). A Comment on Aumann's Bayesian View. Econometrica, 66 (4), 923927.
Morris, S. (1995). The Common Prior Assumption in Economic Theory. Economics and
Philosophy, 11, 227253.
Olson, J. M., Vernon, P. A., Harris, J. A., & Jang, K. (2001). The Heritability of Attitudes:
A Study of Twins. Journal of Personality and Social Psychology, 80 (6), 845860.
Talbott, W. (1990). The Reliability of the Cognitive Mechanism: A Mechanist Account of
Empirical Justification. Garland Publishing, New York.
7