Subjective probability

Subjective probability - the degree of personal belief of the agent (subject) in the possibility of the occurrence of some event.

The concept of subjective probability is one of the interpretations of the concept of probability along with frequency probability and logical probability ^[1] . It has found application in decision theory . Not all subjective opinions are useful. There is a requirement that the agent be rational ^[1] , that is, the product of his faith must obey certain rules. In particular, the sum of the probabilities of possible alternatives should not exceed unity.

Content

History

The concept of subjective probability was first formulated by Frank Plampton Ramsey in 1926 ^[1] . Paying tribute to the frequency interpretation of probability in statistics and physics, Ramsey notes that this does not impede the existence of a subjective interpretation. And he adds: “The main reason for the difference between the opinion of statisticians who more accept the frequency theory of probability and the opinion of logicians tending to a subjective theory is that the two schools really have a discussion about different things, and that the word“ probability ”is used by logicians in one sense, and by statisticians in another ” ^[2] . A significant contribution to the development of the theory of subjective probability was also made by Bruno de Finetti ^[3] , Leonard Savage ^[4] , Irwin Goode ^[5] and many others ^[1] .

Betting and the Dutch Book

Subjective probabilities are traditionally analyzed in terms of betting. Consider the following example. Let an individual have the opportunity to receive a sum of money S in case event E occurs. However, if it does not happen, then he will have to pay the same amount to the bank. This does not suit the individual, and he seeks changes in the terms of the contract. Under the new contract, he will not receive the entire amount S , but only its part pS if event E occurs. But then he will have to give him only this smaller amount if the event does not happen. He assigns p to himself. If an individual estimates the probability of occurrence of event E as high, then the value of p will also be high. If the probability of occurrence of an event is estimated by him as low, then the value of p will also be low.

According to personalist theory, the value of p is equal to the value of the subjective probability of the event E.

Despite its apparent arbitrariness, subjective probability must meet the requirements of the axiomatics of probability theory. Therefore, an assessment of its values should not violate the rules of calculating probabilities. To avoid this, special methods of filtering information are used. One of them has the conditional name "Dutch Book".

The Dutch book is a series of bets, each of which is quite acceptable for the agent, but in the aggregate they guarantee his loss if the agent involuntarily violated the rules for calculating probabilities. It is mathematically proved that if subjective probabilities violate the rules of calculating probabilities, then the Dutch book will feel it ^[1] .

Subjective probability and logical probability

The obvious usefulness of the concept of subjective probability in decision theory can only be explained by the fact that it grows from roots of an objective nature. On this occasion, Karnap writes the following ^[6] .

“I think that there should be no contradiction between the objectivist point of view and the subjectivist or personalist point of view. Both have a legitimate place in the context of our work, which consists in constructing many rules for determining probability values in relation to available evidence. At every step of this construction, a choice is made; which is not completely free, but limited to a certain framework. Basically, this is simply the difference in emphasis between the subjectivist tendency to emphasize the existence of freedom of choice and the objectivist tendency to strengthen the role of restrictions. ”

Notes

↑ ¹ ² ³ ⁴ ⁵ Hajek Alan. (2007). Interpretation of probability. In The Stanford Encyclopedia of Philosophy, ed. Edward N. Zalta, http://plato.stanford.edu/archives/fall2007/-entries/probability-interpret (link not available) .
↑ Ramsey Frank Plumton, Truth and probability. In RB Braithwaite, editor, The Foundation of Mathematics and Other Logical Essays, pages 156-198. Routledge and Kegan Paul, London, 1931. Read before the Cambridge Moral Sciences Club, 1926.
↑ Bruno de Finetti. La prevision: ses lois logiques, ses sources subjectives. Annales de l'Institut Henri Poincare, 7: 1-68, 1937.
↑ Savage Leonard J. The Foundations of Statistics. New York, Wiley, 1954.
↑ Good IJ The Estimation of Probabilities: An Essay on Modern Bayesian Mrthods. MIT Press, Cambridge, Mass., 1965.
↑ Zabell SL (2004). Carnap and the Logic of Inductive Inference. In Dov M. Gabbay, John Woods & Akihiro Kanamori (eds.), Handbook of the History of Logic. Elsevier 265-309.

[list1-1] ¹ ² ³ ⁴ ⁵ Hajek Alan. (2007). Interpretation of probability. In The Stanford Encyclopedia of Philosophy, ed. Edward N. Zalta, http://plato.stanford.edu/archives/fall2007/-entries/probability-interpret (link not available) .

[list2-2] Ramsey Frank Plumton, Truth and probability. In RB Braithwaite, editor, The Foundation of Mathematics and Other Logical Essays, pages 156-198. Routledge and Kegan Paul, London, 1931. Read before the Cambridge Moral Sciences Club, 1926.

[list3-3] Bruno de Finetti. La prevision: ses lois logiques, ses sources subjectives. Annales de l'Institut Henri Poincare, 7: 1-68, 1937.

[list4-4] Savage Leonard J. The Foundations of Statistics. New York, Wiley, 1954.

[list5-5] Good IJ The Estimation of Probabilities: An Essay on Modern Bayesian Mrthods. MIT Press, Cambridge, Mass., 1965.

[list6-6] Zabell SL (2004). Carnap and the Logic of Inductive Inference. In Dov M. Gabbay, John Woods & Akihiro Kanamori (eds.), Handbook of the History of Logic. Elsevier 265-309.