At 07:17 AM 6/7/99 -0700, you wrote:
>I thought that only the mode of the distribution on a prior has any
>effect on a decision. In the example David gives below, at first the
>prior of 0.5 is uncertain (has a wide distribution), whereas after 1M
>trials, the distribution on the prior of 0.5 is quite narrow -- but any
>decision to be made that depends on the prior is unchanged in the two
>cases.
>
>Isn't it provable that for expected utility decision making, uncertainty
>on the prior is irrelevant? Can someone point to that proof?
>
That expected utility decision making should depend only on the *mean* of
the prior on a probability p is a widely believed misconception. The
assertion is correct if utility depends only on a *single* realization of
an event with probability p. But as soon as you allow multiple events with
each event having the same probability p, then higher moments of the prior
can and do matter.
For example, Joe Halpern's decision whether to bet on the occurence of
between 450,000 and 550,000 heads in the next million tosses of a coin with
heads probability p certainly does depend on the entire prior distribution
of p, not just the mean of p. If the distribution of p is a spike at 0.5,
then the number N of heads in the next million tosses has approximately a
normal distribution with mean 500,000 and standard deviation 500, so
betting that N will fall in the interval 450,000 to 550,000 would be a good
idea regardless of the size of the bet or your utility function. On the
other hand, if p has a uniform(0,1) prior, then one can show that N will
have a discrete uniform distribution on the integers 0,1,2, ..., 1,000,000,
so the probability that N falls between 450,000 and 550,000 is around 1 in
10, something you should probably bet against.
For some references, Harrison (1977) shows that it is rational (consistent
with EU theory) to pay to reduce ambiguity (variability in the distribution
of p). Howard (1988) shows that EU decisions can depend on moments of p
beyond just the first. I have shown (1992) that ambiguity aversion in such
multiple-event situations is a direct consequence of risk aversion in one's
utility function.
Based on this, I think one can make a case that ignorance about whether an
event E will occur should depend on what one's second-order prior is on the
probability p of E, as Rolf Haenni suggested. David Poole's argument that
"I don't know" means the same thing regardless of the prior on p hinges, as
Uschi Sondhauss notes, on his assumption that the only question one would
want to ask is whether E will occur on a single trial. If we are
interested how many times E will occur in multiple independent trials, then
the prior on p makes a big difference and "I don't know" should refer, I
think, to the uniform prior on p.
Wang Pei writes:
>Yes, as long as current belief is concerned, the above three
>situations (0:0, 2:2, and 500000:500000) are the same: "I don't know
>(whether the next outcome will be H or T)."
>
>
>However, what distinguishes ignorance and known probability is how new
>evidence will revise current belief. After a new H (or T) is
>observed, the above three cases become different. I'd like to know
>how this difference can be captured by BN.
The answer is that one must include p with the appropriate prior in the BN.
New evidence will have little impact if the prior on p is tight, and
substantial impact if the prior on p is diffuse.
References
JM Harrison (1977), "Independence and calibration in decision analysis",
Management Science 24, 320-328.
GB Hazen (1992), "Decision versus policy: an expected utility resolution of
the Ellsberg paradox", in J Geweke, *Decision Making Under Risk and
Uncertainty: New Models and Empirical Findings*, Kluwer
RA Howard (1988), "Uncertainty about probability: a decision analysis
perspective", Risk Analysis 8, 91-98.
Gordon Hazen
Department of Industrial Engineering and Management Sciences
McCormick School of Engineering and Applied Science
Northwestern University
Evanston IL 60208-3119
Fax 847-491-8005
Phone 847-491-5673
www.iems.nwu.edu/~hazen/