**Next message:**Max More: "NEWS: Information Week on Ellison, anti-aging, ExI mention"**Previous message:**J. R. Molloy: "Re: Fuel Cell House, was Re: TECH: fuel cell car"**Next in thread:**Eliezer S. Yudkowsky: "Re: Opinions as Evidence: Should Rational Bayesian Agents Commonize Priors?"**Reply:**Eliezer S. Yudkowsky: "Re: Opinions as Evidence: Should Rational Bayesian Agents Commonize Priors?"**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]

I consider the situation of two uninformed Bayesian agents becoming aware of

differences in priors about a binary world-state. I derive their confidence

functions in the binary world-state upon the information of the other’s

priors and show the confidence functions coincide only if priors are drawn

from one particular distribution strongly informative on the world-state. I

non-rigorously prove the result generalizes to a world with any finite number

of possible states and to Bayesian agents with access to common public

information. I present arguments for the conjecture that the result further

generalizes to worlds with infinite states and to Bayesians with private

information. Hence, rational Bayesians with initially differing priors

should continue to disagree even when fully informed of each other’s beliefs.

Take a world with two possible states, Q and ~Q. Assume two Bayesian agents

with prior degrees of belief in Q, denoted as A and B, and degrees of belief

in ~Q of (1-A) and (1-B). For notational convenience in text transmission, I

denote the degree of belief of a Bayesian agent with initial degree of belief

X, given information Y, as (X)Y. Priors provide information in that they

depend on the state of the world; for any given prior P there is a

probability P|Q of that prior in world with Q and P|~Q in worlds with ~Q.

This defines a function f(P) = (P|Q)/(P|~Q) which eases notation of

confidence functions upon the information of a new agent with prior P.

On learning of B, A should now have a degree of belief in Q of A*B|Q and a

degree of belief in ~Q of (1-A)*(B|~Q). Normalizing A’s total degree of

belief to 1 and using f(B) to simplify notation, I derive:

(A)B = A*f(B)/( A*f( B)+(1-A)) (1)

And by symmetry

(B)A = B*f(B)/(B*f(A) + (1-B)) (2)

If A and B commonize their priors on learning of each other, (A)B = (B)A

Solving for f:

A*f(B)/(A*f(B) + 1-A) = B*f(A)/(B*f(A) + 1-B)

A*f(B)(B*f(A) + 1-B) = B*f(A)( A*f(B) + 1-A)

AB*f(B)*f(A) + A(1-B)f(B) = AB*f(B)*f(A) + B(1-A)f(A)

A(1-B)f(B) = B(1-A)f(A)

F(A)/f(B) = A(1-B)/B(1-A) (3)

This requires f(A) = cA/(1-A) (4), with some arbitrary constant c. Hence

priors must be highly dependent on the world-state. In particular,

completely uninformed individuals are almost never profoundly wrong: given

world-state Q, the chance of a prior with low A (relatively strong disbelief

in the actual world-state) goes to 0 as A goes to zero, and does so rather

rapidly. This disagrees markedly with actual experience, which shows most

completely uninformed people have very incorrect beliefs.

Let us suppose the agents with priors A and B have access to additional

public information E prior to learning each other’s priors. If they concur

at this stage, then ((A)E)B = ((B)E)A. By standard Bayesian inference,

((A)E)B = ((A)B)E so ((A)B)E = ((B)A)E. Hence, by Bayesian rules (A)B) =

(B)A. Hence two Bayesian agents with access to public information will

concur only under exactly the same restrictive conditions required to concur

in the absence of public information.

A world with three states Q, R, and S can be described with the binary

beliefs Q/~Q and R/~R, given that Q implies ~R. In order for two agent to

agree on degrees of belief to all three states, they must concur on both Q

and R. The requirements to concur on binary beliefs Q and R are as above.

By induction, two agents will concur on finite multi-state worlds only if the

probability of a degree of belief AN in each particular state N follows the

condition p(AN) = cAN(1-AN).

Even if the world consist of an infinite set of possible states, these can be

partitioned into two sets, each of which can be partitioned into two sets,

etc., leading to a sequence of binary possibilities Q1, Q2, Q3,… For two

agents to concur on the confidence function over all sets, it seems intuitive

they must concur on Q1, Q2, Q3 …, with each concurrence requiring the

distribution of priors on each Q follow condition (4).

Finally, if Bayesian agents with private information meet, the conclusion

above follows by replacing prior A with private-informed degree of belief A.

The requirement in (4) still holds, except that now private-informed beliefs,

rather than priors, must follow the distribution. I conjecture that if, in

some world, (4) holds for private-informed beliefs, gain or loss of private

information would in general cause (4) to no longer hold.

My personal experience is that priors and private information are only weakly

informative; i.e., even given world-state Q obtains, it isn’t particularly

difficult to find uninformed individuals with strong disbelief in Q. Given

this, the probability of a given degree of belief A in Q varies only mildly

with whether Q obtains. Hence the information derived from a given person

holding a degree of belief Q is small and a rational Bayesian should have

only a small change in belief on learning another’s opinions. Rational

Bayesians, then, generally should maintain differences of opinion due to

differences in priors. Under most circumstances, for two agents to commonize

priors requires a violation of Bayesian inference.

**Next message:**Max More: "NEWS: Information Week on Ellison, anti-aging, ExI mention"**Previous message:**J. R. Molloy: "Re: Fuel Cell House, was Re: TECH: fuel cell car"**Next in thread:**Eliezer S. Yudkowsky: "Re: Opinions as Evidence: Should Rational Bayesian Agents Commonize Priors?"**Reply:**Eliezer S. Yudkowsky: "Re: Opinions as Evidence: Should Rational Bayesian Agents Commonize Priors?"**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]

*
This archive was generated by hypermail 2b30
: Mon May 28 2001 - 10:00:03 MDT
*