Taxonomy of Probability

# Subjective

 Ramsey The subjective interpretation identifies probability with degrees of belief. Problems: Its main problem is that the coherence constraint applies to a particular person at a particular time, and therefore allows anyone to change his mind, forming the most outlandish judgments, in the face of any evidence. There would be nothing irrational, on this approach, in holding that the probability of rain some time next year in New York is only one in six. de Finetti

'The subjective interpretation of probability, according to which the probability of a proposition is a measure of one's degree of belief in it, was developed by, e.g. Ramsey ("Truth and Probability," in his Foundations of Mathematics and other Essays, 1926); Definetti ("Foresight: Its Logical Laws, Its Subjective Sources," 1937, translated by H. Kyburg, Jr., in H. E. Smokler, Studies in Subjective Probability, 1964); and Savage (The Foundations of Stastics, 1954). Of course, subjective probability varies from person to person. Also, in order for this to be an interpretation of probability, so that the relevant axioms are satisfied, not all persons can count — only rational, or "coherent" persons should count. Some theorists have drawn a connection between rationality and probabilistic degrees of belief in terms of dispositions to set coherent bettings odds (those that do not allow a "Dutch book" — an arrangement that forces the agent to lose come what may), while others have described the connection in more general decision-theoretic terms.'
Audi (1999)

"One influential modern approach, pioneered by F. P. RAMSEY and Bruno de Finetti, is content to safeguard the links between frequency and probability by BERNOULLI'S THEOREM, and sees probability judgments as simply subjective expressions of confidence, subject not to empirical constraints, but only to a requirement of coherence. This prevents assignments such that if you bet on them you would lose whatever happens. The approach removes the sense that probabilities are there to be discovered, and it protects the close link between probability and practice. Its main problem is that the coherence constraint applies to a particular person at a particular time, and therefore allows anyone to change his mind, forming the most outlandish judgments, in the face of any evidence. There would be nothing irrational, on this approach, in holding that the probability of rain some time next year in New York is only one in six."
Flew and Priest (2002)

"The subjective theory identifies probability with the degree of belief of a particular individual. Here it is no longer assumed that all rational human beings with the same evidence will have the same degree of belief in a hypothesis or prediction. Difference of opinion are allowed."
Gillies (2000)

"The subjectivist theory analyses probability in terms of degrees of belief. A crude version would simply identify the statement that something is probable with the statement that the speaker is more inclined to believe it than to disbelieve it. Degrees of belief may be measured in terms of the bets the believer would be willing to place, and more refined versions of the theory say one is only entitles to use ‘probably’ if one's bets are ‘coherent’, in the sense that one does not bet on contradictory propositions in such a way that one is bound to lose whatever happens, which can be expressed by saying that one does not let oneself have a ‘Dutch book’ made against one. This, however, still bases probability on the attitudes of the believer. Because ‘coherence’ is required, subjectivism is sometimes described as the view that probability is the degree of the rational man's belief. However, when this means that calling something probable is saying that it is rational to believe it, it is not subjectivist, since it no longer analyses probability in terms of beliefs actually held. It then has no special name."
Lacey (1996)

"We can think of probability as a measure of degree of belief. This is often called the subjective interpretation of probability. On this interpretation, the probability of rain tomorrow for me is simply the degree of belief I give to its raining tomorrow. Here, degree of belief is not thought of as something measured by strength of feeling, but in terms of betting behaviour. For me to give 0.7 degree of belief to there being rain tomorrow is, roughly, for me to regard 70 cents as the fair price for a bet that returns \$1 if it rains tomorrow, and nothing if it does not.
If we interpret degree of belief in terms of betting behaviour it can be proved that people whose degrees of belief violate the axioms of the probability calculus can have a DUTCH BOOK made against them. For instance, if they give p v q a greater degree of belief than the sum of the degrees of belief they give to p and to q, minus the degree of belief they give to p & q, there will be a set of bets such that if they pay what they regard as a fair price for each bet, they must lose money. Thus, if one takes the possibility of having a Dutch book made against one to be a sufficient condition for having an irrational set of degrees of belief, degrees of belief must obey the calculus ‘provided’ they are rational.
We can also think of probability as a measure of the degree of support a body of evidence gives to a hypothesis, often written P(h/e). If I say that there ia a 0.7 probability of rain tomorrow, what I mean on this view is that relative to my evidence, or perhaos relative to all the available evidence, there is a 0.7 probability of rain tomorrow. The axioms of the calculus are then thought of as having an explicit or implicit relativity built into them. For instance, instead of ‘If p entails q, then P(p) ≤ P(q)’, we have ‘If p entails q, then P(p/e) ≤ P(q/e)’. We can connect this interpretation to the one in terms of degree of belief as follows: the support e gives to h is the degree of belief that someone ought to give to h if e is all they know. It is then possible to draw on this connection and the Dutch book argument mentioned above to show that the notion of degree of support obeys the calculus."
Mautner (2000)