Re: Bayesian Networks and Belief Functions
Jonathan Weiss (jjweiss@ix.netcom.com)
Thu, 10 Jun 1999 13:55:44 -0400
At 6/9/99 06:04 PM, Gordon Hazen wrote:
...
>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.
An even more extreme example: The coin is either two-headed or two-tailed,
with equal probability. Assuming once the coin is chosen, the same coin is
flipped 1,000,000 times, the outcome will be either 0 heads or 1,000,000
heads,
each with probability 0.5. The probability of N falling between 450,000 and
550,000 is zero, even though the mean of that distribution is 500,000. On the
other hand, if we are tossing a fair, unbiased coin, or if we choose a new
coin
from the two-heads or two-tails bag for each toss, we obtain a distribution
with standard deviation 500, and therefore overwhelming odds that N will fall
within the interval from 450,000 to 550,000.