The principle of indifference states that in the absence of any relevant evidence, agents should distribute their credence (or "degrees of belief") equally among all the possible outcomes under consideration.
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.
As with coins, it is assumed that the initial conditions of throwing the dice are not known with enough precision to predict the outcome according to the laws of mechanics.
We draw a card from the deck; applying the principle of indifference, we assign each of the possible outcomes a probability of 1/52.
This example, more than the others, shows the difficulty of actually applying the principle of indifference in real situations.
In fact, some expert blackjack players can track aces through the deck; for them, the condition for applying the principle of indifference is not satisfied.
Applying the principle of indifference incorrectly can easily lead to nonsensical results, especially in the case of multivariate, continuous variables.
In general, the principle of indifference does not indicate which variable (e.g. in this case, length, surface area, or volume) is to have a uniform epistemic probability distribution.
Edwin T. Jaynes introduced the principle of transformation groups, which can yield an epistemic probability distribution for this problem.
If we have no reason to favour one trio of values over another, then our prior probabilities must be related by the rule for changing variables in continuous distributions.
, where K is a normalization constant, determined by the range of L, in this case equal to: To put this "to the test", we ask for the probability that the length is less than 4.
The fundamental hypothesis of statistical physics, that any two microstates of a system with the same total energy are equally probable at equilibrium, is in a sense an example of the principle of indifference.
However, when the microstates are described by continuous variables (such as positions and momenta), an additional physical basis is needed in order to explain under which parameterization the probability density will be uniform.
[4] The original writers on probability, primarily Jacob Bernoulli and Pierre Simon Laplace, considered the principle of indifference to be intuitively obvious and did not even bother to give it a name.
Attempts to put the notion on firmer philosophical ground have generally begun with the concept of equipossibility and progressed from it to equiprobability.