Quote:
Originally Posted by lastcardcharlie
Suppose the probability of an event happening is an irrational number. Is it possible to outline briefly how Bayesians and frequentists interpret this statement?
Unless I misunderstand your question, they interpret it the same way they would if it were rational. You might enjoy reading an
old post of mine about probability interpretations. The subjective and logical interpretations are the two flavors of "Bayesianism." For convenience, here is the relevant part:
Quote:
First: how do we define "probability of an event"? We define it as a limit of a ratio of events we interested in to total number of outcomes of a trial when number of trials goes to infinity.
In Euclidean geometry, the word "point" is undefined. Rather than being explicitly defined, it is characterized by the axioms of geometry. A point is any object which satisfies those axioms. Similarly, the probability of an event is simply a number between 0 and 1 which satisfies the axioms of probability. There is no mathematical definition beyond that.
What you are wondering about is the
interpretation of probability. There are many competing philosophical interpretations of probability. The four major competitors are the frequency, propensity, subjective, and logical interpretations.
Here is a very loose description of the four interpretations. Imagine we flip a biased coin and we wonder about the probability of heads. According to the frequency interpretation, we can only talk about this probability if the flip is part of a long sequence of flips. In that case, the probability of heads is simply the limiting ratio of the number of heads to the number of flips. This is what you mentioned in your OP. If the coin is only flipped once and then subsequently destroyed, then according to the frequency interpretation, it does not make sense to talk about the probability of heads.
The propensity interpretation, however, does not require a long sequence of flips. Suppose we claim the probability of heads is 0.6. According to the propensity interpretation, this means the coin (or rather, the entire experimental setup, including the coin, the flipper, the air in the room, etc.) has a propensity for producing, in the long run, 6 heads for every 10 flips. It does not matter if the coin is destroyed after one flip. The probability of 0.6 refers to the potential frequency that would arise if we could flip it many times.
Both the frequency and propensity interpretations regard the probability of heads as describing some real physical property. With frequency, it is a property of the sequence. With propensity, it is a property of the experimental setup. The subjective and logical interpretations, on the other hand, do not regard probabilities as representing physical realities. Instead, they represent degrees of belief.
Suppose Joe says the probability of heads is 0.6. According to the subjective interpretation, this means that if Joe was offered a choice between these two bets:
(a) win $4 on heads, lose $6 on tails,
(b) lose $4 on heads, win $6 on tails,
then Joe would be indifferent as to which bet to take. The probability of 0.6 represents Joe's personal betting preferences regarding this event. Another person, say Jack, may have different preferences. Jack might, for instance, say the probability of heads is 0.3. In the subjective interpretation, neither is right or wrong. The statements they are making are not contradictory. They are simply subjective. This looks like it might be a totally useless interpretation, but there is one caveat which somewhat fixes this subjectivity. In the subjective interpretation, Joe's and Jack's probabilities must be "coherent," which means that they must obey the axioms of probability. These axioms ensure that if the coin is flipped many times, then Joe and Jack will no longer disagree about the probability of heads. Their subjective opinions will get closer and closer to each other, and in the limit of infinitely many flips, they will exactly agree on the probability of heads. In fact, their subjective opinions will match the frequency of heads to total coin flips. However, if the coin is flipped only once, then their opinions may differ dramatically, and (in the subjective interpretation) no one can say who is right or wrong, because in fact neither is right or wrong.
The logical interpretation is similar to the subjective. Probabilities represent degrees of belief, but they are not meant to be subjective. In the logical interpretation, probabilities arise because we have uncertainty about some proposition or event. This uncertainty exists because we have only partial information about the thing in question. In the logical interpretation, it is postulated that there exists some ideal form of reasoning that can be applied to this partial information which will yield a probability. In other words, if Joe and Jack have the same information about the coin flip, then they should arrive at the same degree of belief. In the logical interpretation, probabilities do not belong to the person stating the probability, but rather to the information which that person possesses.