Stan's problem is very interesting. I believe it points to a more basic paradox concerning Laplace's "principle of indifference."
The principle of indifference states "given no reason to think otherwise, consider all alternatives equally likely." The rub is that the set of alternatives can be structured in different ways. To illustrate, suppose you're given an urn and told: 1. The urn contains 2 marbles 2. Each marble is either white or black 3. There are not 2 black marbles in the urn and then asked to estimate the probability that the urn has 2 white marbles. One could reason two different ways: View A There are two alternative states: i. the urn contains two white marbles ii. the urn contains one white and one black marble By this view, one estimates the probability of two white marbles is 1/2. View B There are three alternative states: i. The urn contains two white marbles ii. Marble 1 (by some arbitrary numbering) is white and Marble 2 is black iii. Marble 1 is black and Marble 2 is white By this view, one estimates the probability of two white marbles as 1/3. Either View A or View B might be considered simpler, depending on whom you ask. I suspect for Laplace, and for most nonstatisticians, View A is simpler. But if I knew that a statistician had filled the urn, View B might seem more obvious, as one would suppose a statistician would apply random sampling. I see parallels between this paradox and something philosophers call "Goodman's paradox," as well as parallels to Occams razor: "Choose from a set of otherwise equivalent models of a given phenomenon the simplest one" [but who decides when models are 'otherwise equivalent' and who decides which is 'simplest'?] and Kant's categorical imperative: "Act as if the maxim of your action were to become through your will a universal law of nature." [but the same action can define multiple maxims. For example, fighting a war in self-defense is both killing (immoral) and acting in self- defense (moral)]. -------------------------------------------------------------------------------- John Uebersax, PhD (858) 597-5571 La Jolla, California (858) 625-0155 (fax) email: [EMAIL PROTECTED] Statistics: http://ourworld.compuserve.com/homepages/jsuebersax/agree.htm Psychology: http://members.aol.com/spiritualpsych -------------------------------------------------------------------------------- . . ================================================================= Instructions for joining and leaving this list, remarks about the problem of INAPPROPRIATE MESSAGES, and archives are available at: . http://jse.stat.ncsu.edu/ . =================================================================
