Hal, you have to bet at scalar odds. You’ve got to use a scalar quantity to weight the force of your subjective anticipations, and their associated utilities. Giving just the probability, just the betting odds, just the degree of subjective anticipation, does throw away information. More than one set of possible worlds, more than one set of admissible hypotheses, more than one sequence of observable evidence, can yield the final summarized judgment that a certain probability is 1⁄6.
The amount of previously observed evidence can determine how easy it is for additional evidence to shift our beliefs, which in turn determines the expected utility of looking for more information. I think this is what you’re looking for.
But when you have to actually bet, you still bet at 1:5 odds. If that sounds strange to you, look up “ambiguity aversion”—considered a bias—as demonstrated at e.g. http://en.wikipedia.org/wiki/Ellsberg_paradox
PS: Personally I’d bet a lot lower than 1⁄6 on ancient Mars life. And Tom, you’re right that 0 is a safer estimate than 10, but so is 9, and I was assuming the tree was known to be an apple tree in bloom.
Hal, you have to bet at scalar odds. You’ve got to use a scalar quantity to weight the force of your subjective anticipations, and their associated utilities. Giving just the probability, just the betting odds, just the degree of subjective anticipation, does throw away information. More than one set of possible worlds, more than one set of admissible hypotheses, more than one sequence of observable evidence, can yield the final summarized judgment that a certain probability is 1⁄6.
The amount of previously observed evidence can determine how easy it is for additional evidence to shift our beliefs, which in turn determines the expected utility of looking for more information. I think this is what you’re looking for.
But when you have to actually bet, you still bet at 1:5 odds. If that sounds strange to you, look up “ambiguity aversion”—considered a bias—as demonstrated at e.g. http://en.wikipedia.org/wiki/Ellsberg_paradox
PS: Personally I’d bet a lot lower than 1⁄6 on ancient Mars life. And Tom, you’re right that 0 is a safer estimate than 10, but so is 9, and I was assuming the tree was known to be an apple tree in bloom.
An assumption with no basis, I trust you realized on reflection.