### The Principle of Indifference by Sorin Bangu

Probability Assignments and the Principle of Indifference.

A Reassessment of the Eliminative Strategies

A central and controversial component of the ‘classical’ conception of probability, the Principle of Indifference (hereafter PI) claims that equipossibility entails equiprobability. A more complete version can be formulated as follows:

Given a null state of background information, equal regions of the space of possible outcomes should be assigned equal probabilities (Howson and Urbach 2006, p. 266).

As is well known, the principle plays an important role not only in physics (in the foundations of statistical mechanics) but also in our everyday probabilistic inferences (e.g., in predicting the outcomes of coin-tossing). Yet many philosophers and scientists also hold that PI is subject to serious objections. These objections are of two kinds. The first kind has to do with the (logical) inconsistencies introduced by the use of the principle. It has long been claimed that the principle must be rejected because it leads to paradoxes – especially when employed in infinitary (continuous) contexts.[1] The second kind of criticism, advanced by Reichenbach (1949, §§ 68-71), maintains that the principle is dispensable in probabilistic inferences. Consequently, he proposed to pursue what I call here an eliminative strategy. In a nutshell, the guiding idea of this type of strategy is to show that reliance on the a priori principle is not necessary in order to infer the correct observed frequencies of outcomes.

My goal in this paper is to analyze in detail two attempts to implement the eliminative strategy.[2] While I agree that eliminativism is a very promising kind of strategy available when dealing with PI[3], I’ll maintain that both attempts proposed so far are fraught with problems. The paper is divided in two sections, each aiming to highlight the difficulties faced by those attempts. In the first section I show that Reichenbach’s implementation of the eliminative strategy fails; more precisely, I’ll show that one of the premises of his eliminative argument just assumes PI. In order to prove this, I’ll examine the same example taken by Reichenbach (following Poincaré) as paradigmatic in showing the effectiveness of PI, namely its employment in predicting the correct probabilities in the game of roulette. Since there is presumably nothing special about roulette (in the sense that the eliminative strategy can be adapted to other cases in which PI seems effective), it is natural to expect that the failure in this case will have similar consequences for other potential eliminative attempts.

In the second section I turn to another attempt to eliminate PI, Gillies’ recent heuristic approach (Gillies (2000)). Despite the fact that Gillies allows PI a certain role in our probabilistic inferences – namely, in conjuring probabilistic hypotheses – I’ll construe his view as another attempt to dispense with the role of PI in probabilistic reasoning. My reason for doing this is Gillies’ emphasis on the incapacity of the principle to justify those hypotheses. Although I agree that this line of thought is quite promising, I’ll close by raising doubts with regard to its cogency. More precisely, I show that the alternative method of justification / rejection of probabilistic hypotheses endorsed by Gillies (in essence, the method of statistical relevance tests) is subject to the same kind of difficulties as the method of a priori justification involving the application of the principle.

[1] van Fraassen, for instance, notes that “the great failure of symmetry thinking” is revealed in those situations “where indifference disintegrated into paradox.” (1989, p. 293).

[2] With one exception, none of the analyses of PI listed below pays attention to these strategies; instead, they focus exclusively on the relation between PI and the (Bertrand type) paradoxes. See Mikkelson (2004), Bartha and Johns (2000), Gillies (2000), Castell (1998), Marinoff (1994), Schlessinger (1993). The notable exception is Strevens (1998) and I’ll take issue with some of his points in the first section.

[3] Another line of objections to PI is of course based on its role in deriving the Bertrand type paradoxes. While I’ll be saying something about this role in section 2, the focus of this paper is not on the paradoxes.

A Reassessment of the Eliminative Strategies

A central and controversial component of the ‘classical’ conception of probability, the Principle of Indifference (hereafter PI) claims that equipossibility entails equiprobability. A more complete version can be formulated as follows:

Given a null state of background information, equal regions of the space of possible outcomes should be assigned equal probabilities (Howson and Urbach 2006, p. 266).

As is well known, the principle plays an important role not only in physics (in the foundations of statistical mechanics) but also in our everyday probabilistic inferences (e.g., in predicting the outcomes of coin-tossing). Yet many philosophers and scientists also hold that PI is subject to serious objections. These objections are of two kinds. The first kind has to do with the (logical) inconsistencies introduced by the use of the principle. It has long been claimed that the principle must be rejected because it leads to paradoxes – especially when employed in infinitary (continuous) contexts.[1] The second kind of criticism, advanced by Reichenbach (1949, §§ 68-71), maintains that the principle is dispensable in probabilistic inferences. Consequently, he proposed to pursue what I call here an eliminative strategy. In a nutshell, the guiding idea of this type of strategy is to show that reliance on the a priori principle is not necessary in order to infer the correct observed frequencies of outcomes.

My goal in this paper is to analyze in detail two attempts to implement the eliminative strategy.[2] While I agree that eliminativism is a very promising kind of strategy available when dealing with PI[3], I’ll maintain that both attempts proposed so far are fraught with problems. The paper is divided in two sections, each aiming to highlight the difficulties faced by those attempts. In the first section I show that Reichenbach’s implementation of the eliminative strategy fails; more precisely, I’ll show that one of the premises of his eliminative argument just assumes PI. In order to prove this, I’ll examine the same example taken by Reichenbach (following Poincaré) as paradigmatic in showing the effectiveness of PI, namely its employment in predicting the correct probabilities in the game of roulette. Since there is presumably nothing special about roulette (in the sense that the eliminative strategy can be adapted to other cases in which PI seems effective), it is natural to expect that the failure in this case will have similar consequences for other potential eliminative attempts.

In the second section I turn to another attempt to eliminate PI, Gillies’ recent heuristic approach (Gillies (2000)). Despite the fact that Gillies allows PI a certain role in our probabilistic inferences – namely, in conjuring probabilistic hypotheses – I’ll construe his view as another attempt to dispense with the role of PI in probabilistic reasoning. My reason for doing this is Gillies’ emphasis on the incapacity of the principle to justify those hypotheses. Although I agree that this line of thought is quite promising, I’ll close by raising doubts with regard to its cogency. More precisely, I show that the alternative method of justification / rejection of probabilistic hypotheses endorsed by Gillies (in essence, the method of statistical relevance tests) is subject to the same kind of difficulties as the method of a priori justification involving the application of the principle.

[1] van Fraassen, for instance, notes that “the great failure of symmetry thinking” is revealed in those situations “where indifference disintegrated into paradox.” (1989, p. 293).

[2] With one exception, none of the analyses of PI listed below pays attention to these strategies; instead, they focus exclusively on the relation between PI and the (Bertrand type) paradoxes. See Mikkelson (2004), Bartha and Johns (2000), Gillies (2000), Castell (1998), Marinoff (1994), Schlessinger (1993). The notable exception is Strevens (1998) and I’ll take issue with some of his points in the first section.

[3] Another line of objections to PI is of course based on its role in deriving the Bertrand type paradoxes. While I’ll be saying something about this role in section 2, the focus of this paper is not on the paradoxes.

Labels: Bertrand paradox., classical probability, eliminative strategy, Gillies, Popper, Principle of indifference, Reichenbach