What is The Principle of Indifference?
The principle of indifference is a rule which tells us how to assign probabilities when we don’t have any special knowledge of a situation. It is also known as the principle of insufficient reason.
Let’s say there are a number of different alternative possibilities. The rule states that each possibility should be assigned an equal probability—assuming there’s no reason for choosing one above the other. Since the sum of probabilities for all possible outcomes always adds up to one, this means that in a situation with n possibilities each possibility is assigned a probability of 1/n.
The Partition Problem
The principle of indifference can be used to generate irreconcilable and illogical results because of the partition problem: the set of all possibilities can be partitioned in any number of ways, and if probabilities are assigned via the principle of indifference, we will get different answers each time.
Example of Conflicting Probabilities
André Kukla, in his book Extraterrestrials: A Philosophical Perspective, gives an example of how absurd the principle of indifference can really get. Let’s say we’ve discovered two new Earth-like planets (P and Q). Consider the question: is there life on other planets?
Let’s assume for a moment that you have no idea if life is possible on other planets, and life on an Earth like planet is equally possible. You’ve got four possibilities:
- There’s no life on P or Q.
- There’s life on P, but not Q.
- There’s life on Q, but not P.
- There’s life on P and Q.
Giving us the probability of life on at least one of these two planets as 3/4 = 75%. But before you send out that exploratory spaceship, pause to consider if that’s really a good solution, even though it’s a very simple partition. An even simpler partition is:
- Yes (there’s life).
- No (there’s no life).
This not only raises the odds a bit, but it also results in conflicting probabilities: the solution to the same question is both 3/4 and 1/2.
Example: Light Switch and Urn
Another example is the classic light switch and urn problem.
Suppose you have a jar of blue, white, and black marbles, of unknown proportions. One is picked at random, and if it is blue, the light is turned on. If it is black or white, the light stays off (or is turned off). What is the probability the light is on?
There isn’t one single answer. In fact, there are several possible answers.
- You might decide to assign a 1/2 probability to the light being on, because you’ve got no reason to assign any other odds. It’s either on (50%) or off (50%).
- You could assign the blue marble a 1/3 probability of being selected (after all, you know that there are three colors). From this it would follow that you have a 1/3 chance of the light being on, and 2/3 chance of the light being off.
This problem was partially addressed by Michael Huemer (cited in Doody 2015), who stated that the principle of indifference should be applied to the partition that is most “explanatorily basic.” This simple idea doesn’t always work, as the life on planets example clearly shows.
Probability or “Magic”?
Choosing the most basic solution makes the decision clear in some cases; in others, however, multiple partitions may be equally basic with no clear way of choosing between them. So the principle of indifference can, in some cases, be a useful rule of thumb, but the results it gives should not be taken as certainties by any means. In fact, some argue that it shouldn’t be used at all. For example, Salmon (1966) states that:
“Knowledge of Probabilities is concrete knowledge about occurrences; otherwise, it is useless for prediction and action.”
He goes on to call the principle of indifference nothing more than “epistemological magic”.
Doody, R. Principle of Indifference & Imprecise Credences. April 10, 2015. Retrieved from http://www.mit.edu/~rdoody/Phil%20Probability%20Handouts/POI&impreciseCr.pdf on April 14, 2018
Kukla, A. (2010). Extraterrestrials: A Philosophical Perspective. Rowman & Littlefield.
Maher, P. Lecture 7: Keynes on the Principle of Indifference. Scientific Thought II, Spring 2010. Retrieved from http://patrick.maher1.net/318/lectures/keynes2.pdf on April 14, 2018
Weisstein, E. “Principle of Insufficient Reason.” MathWorld–A Wolfram Web Resource. Retrieved from http://mathworld.wolfram.com/PrincipleofInsufficientReason.html on April 14, 2018
Stephanie Glen. "Principle of Indifference / Insufficient Reason: Definition & Examples" From StatisticsHowTo.com: Elementary Statistics for the rest of us! https://www.statisticshowto.com/principle-of-indifference/
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