Gerhard Schurz on Subjective Probability - Dictionary of Arguments
Def Objective probability/Schurz: the probability of an event type (e.g. Fx) is the relative frequency of its occurrence or the limit value of its relative frequency in the long run.
Notation p(-) resp. p(Fx)
Def Subjective probability /Schurz: the probability of a certain event or fact (e.g. Fa) is the rational degree of belief in the occurrence of an event by a given subject or all subjects of a rationality type,
Notation: w(-) or w(Fa).
Subjective probability: Pointe: Example coin toss: deviates from the objective probability! If more often number comes, one must assume that the coin is asymmetrical! This assumption is not expressible in the objective probability at all.
Probability theory/Schurz: problems:
(b) subjective probability: justification problems. On what grounds should rational degrees of belief satisfy Kolmogorov axioms?
What role should degrees of belief play in the goal of finding real truths?
Solution/Ramsey/de Finetti: Bet.
Bet/Betting Quotient/Ramsey/Schurz: thesis fair betting quotients of a person satisfy Kolmogorov Axioms A1 - A3 exactly if they are coherent, i.e. that there is no system where total loss is possible.
VsRamsey/Schurz: A bet is not yet a rational behavior in the sense of a search for truth! They are not truth-seeking, because the definition of the fair betting ratio refers only to the subjective degrees of belief, not to objective probability. The real frequency of success is not touched at all.
Ex Suppose a subjectivist enthusiastically accepts a bet, of 1 : 1, that he will roll a six. He is fair if he is also willing to accept the opposite bet, 1 : 1 that he will not roll a six.
Problem: he remains coherent and fair even if he has lost his entire fortune. He will only be surprised that no one will accept the counter bets he assumes to be fair. He cannot explain it as long as he is not allowed to consider the objective frequencies. This shows that the axioms A1 - A3 are at best a minimal condition. But this is too weak to exclude irrational behavior.
Principal Principle/PP/Statistics/Schurz: the subjective probabilities, if the objective probabilities are known, must be consistent with them.
Lewis: (1980)(1): singular PP: subjectivist. Here "objective" singular propensities are simply postulated.
SchurzVsPropensity/SchurzVsPopper: it remains unclear what property a singular propensity should correspond to in the first place.
Solution/de Finetti: one can also accept the objective notion of probability at the same time.
Conditionalization/Statistics/Schurz: on an arbitrary experience datum E(b1...bn) over other individuals b1,..bn is important to derive two further versions of PP:
1. PP for random samples, which is needed for the subjective justification of the statistical likelihood intuition.
2. the conditional PP, for the principle of the closest reference class and subject to the inductive statistical specialization inference.
1. Lewis, D. (1980). "A Subjectivist's Guide to Objective Chance". In: Jeffrey, R.C. (ed.)(1980), Studies in Inductive Logic and Probability, Vol 2, Berkeley: University of California Press._____________Explanation of symbols: Roman numerals indicate the source, arabic numerals indicate the page number. The corresponding books are indicated on the right hand side. ((s)…): Comment by the sender of the contribution. Translations: Dictionary of Arguments The note [Concept/Author], [Author1]Vs[Author2] or [Author]Vs[term] resp. "problem:"/"solution:", "old:"/"new:" and "thesis:" is an addition from the Dictionary of Arguments. If a German edition is specified, the page numbers refer to this edition.
Einführung in die Wissenschaftstheorie Darmstadt 2006