Principle of indifference

The principle of indifference (also called principle of insufficient reason) is a rule for assigning epistemic probabilities. Suppose that there are n > 1 mutually exclusive and collectively exhaustive possibilities. The principle of indifference states that if the n possibilities are indistinguishable except for their names, then each possibility should be assigned a probability equal to 1/n.

In Bayesian probability, this is the simplest non-informative prior. The principle of indifference is meaningless under the frequency interpretation of probability, in which probabilities are relative frequencies rather than degrees of belief in uncertain propositions, conditional upon state information.

The textbook examples for the application of the principle of indifference are coins, dice, and cards.

In a macroscopic system, at least, it must be assumed that the physical laws which govern the system are not known well enough to predict the outcome. As observed some centuries ago by John Arbuthnot (in the preface of Of the Laws of Chance, 1692),

Given enough time and resources, there is no fundamental reason to suppose that suitably precise measurements could not be made, which would enable the prediction of the outcome of coins, dice, and cards with high accuracy: Persi Diaconis's work with coin-flipping machines is a practical example of this.

A symmetric coin has two sides, arbitrarily labeled heads and tails. Assuming that the coin must land on one side or the other, the outcomes of a coin toss are mutually exclusive, exhaustive, and interchangeable. According to the principle of indifference, we assign each of the possible outcomes a probability of 1/2.

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