Re: probability and decision theory

From: Hal Finney <hal.domain.name.hidden>
Date: Mon, 16 Mar 1998 07:47:22 -0800

Mathematically, probability is defined as a measure applied to the set of
subsets of some base set. Each subset has a measure, which is a number
that must satisfy the property that measure of the union of two disjoint
subsets is the sum of the measure of the two subsets. If the measure of
the universal subset is 1, then this is considered a probability measure.

In applying this to the world, each subset corresponds to a possible
outcome of some event or experiment. From the definitions above, you
can show for example that the probability of the complement of an event
is 1 - the probability of the event, and other familiar results.

Wei Dai <weidai.domain.name.hidden> writes:
> On what basis do we choose a theory of probability? Does probability have
> any meaning outside the context of decision theory? It seems to me the
> answer is no. If this is true then the real question is how do we choose a
> decision theory. Is decision theory a positive theory? Is it supposed to
> make predictions about agents in the real world? At first I thought yes,
> we can accept or reject decision theory based on whether or not it makes
> correct predictions about real agents. But if probability theory is part
> of decision theory, then it would be circular to use probabilistic
> arguments on decision theory itself.
>
> Does anyone see a way out of this?

I think you can still look at evolution and see what kinds of
decision theory and probability theory will evolve. We know that we
are replicators and have developed our instinctive understanding of
probability over a long period of time. This should be able to be
analyzed and understood. You might be able to create a toy universe
of evolving replicators and see what kinds of probability theories are
most useful and successful.

The fact that we may apply probabilistic reasoning in understanding
evolution does not seem circular or paradoxical to me. Probability and
logic are tools which are applied successfully in a wide range of areas.
It may be that in some of our discussions we are extending them into areas
where they don't work so well (like your copying paradoxes). But that
does not mean that they are useless in general. Sticking to well defined
and explored areas, like evaluating the behavior of evolutionary systems,
we can confidently use probability theory, logic, and other tools to
develop our theories and understanding.

> P.S.
>
> Maybe the answer is that a decision theory has two roles, one as a
> normative theory about how one should act, and the other as a positive
> theory about how others do act. Perhaps the two roles could even be filled
> with two different theories. Once we accept a decision theory as a
> normative theory, the probability theory part of it would tell us which
> decision theory we should accept as a positive theory.
>
> If this is the right way to think about probability theory, then we should
> make clear in our discussions whether we're talking about it as a positive
> theory or a normative theory. But it is still not clear on what basis we
> should choose a normative probability (or decision) theory.

I don't really understand this. Assuming that we are like other people,
and if we have a theory that tells us how we should act, and we actually
follow the theory, then this same theory should tell how other people do
act. Maybe I'm missing the point here.

Hal
Received on Mon Mar 16 1998 - 07:54:23 PST