Question: It seems that frequentism and Bayesianism may not really be different as far as the the ultimate basis for what a probability is (relative frequency within a reference class) - it's just that frequentists do not allow for hypothetical reference classes (so I doubt these people really exist) or demand that these probabilities be present as physical properties of the world.
So, is there a process by which we develop subjective probabilities that is not, at its core, reliant on the construction of some hypothetical reference class about which we can reason in terms of frequencies? If so, why should we trust it?
Answers with references would be much appreciated.
Note: As has been pointed out by @amoeba in several posts, frequentist/Bayesian inference and frequentist/Bayesian probability are different concepts. I am solely focused on the latter.
Idea: So, it would seem that degrees of belief do (or should) track the relative frequency of some appropriate reference class. For example:
The probability it will rain tomorrow given that it is sunny today, with temp X and humidity Y, and with a cold front moving in at speed Z is 70%.
What does this 70% refer to? The above would suggest that if we had 1000's of sunny days that had exactly the same values for X,Y,Z, we expect 70% of them to be followed by rain the next day.
Ok, that one still has an air of plausibility for being repeatable. What about "Russia will declare war on X in the next 5 years"? Of course, this question will be conditional on a whole host of information. Let's call that information $\mathcal{I}$, so we are really asking about $P(\mathrm{Russia\; goes\; to \;war...}|\mathcal{I})$. We ask some experts to come up with a probability and they say 0.5%. What does this really mean? What reference class could we possibly be referring to?
In a sense, we can argue that all probabilistic statements are single case, but there must exist a reference class (likely implicit) in the heads of the experts from which they are drawing their estimates. For example, one expert may think, "Well, I know that when unemployment and inflation are high, people tend to be more open to violence (thinking of perhaps some reports and history). I don't know the exact figure, but I guess it would roughly triple the likelihood of going to war."
And we can imagine countless such inner and interpersonal dialogues like this.
Background: I was motivated to write this question after reading the excellent questions (and answers) to the following:
- Who are frequentists?
- Mathematical basis for frequentistm vs Bayesianism
- Bayesian vs Frequentist Probability
And from reading these two articles:
In particular, the following snippets from Standford Encyclopedia's discussion of subjective probability were quite interesting:
Probabilistic coherence plays much the same role for degrees of belief that consistency plays for ordinary, all-or-nothing beliefs. What an extreme subjectivist, even one who demands regularity, lacks is an analogue of truth, some yardstick for distinguishing the ‘veridical’ probability assignments from the rest (such as the 0.999 one above), some way in which probability assignments are answerable to the world. It seems, then, that the subjectivist needs something more.
And various subjectivists offer more. Having isolated the “logic” of partial belief as conformity to the probability calculus, Ramsey goes on to discuss what makes a degree of belief in a proposition reasonable. After canvassing several possible answers, he settles upon one that focuses on habits of opinion formation — “e.g. the habit of proceeding from the opinion that a toadstool is yellow to the opinion that it is unwholesome” (50). He then asks, for a person with this habit, what probability it would be best for him to have that a given yellow toadstool is unwholesome, and he answers that “it will in general be equal to the proportion of yellow toadstools which are in fact unwholesome” (50). This resonates with more recent proposals (e.g., van Fraassen 1984, Shimony 1988) for evaluating degrees of belief according to how closely they match the corresponding relative frequencies — in the jargon, how well calibrated they are. Since relative frequencies obey the axioms of probability (up to finite additivity), it is thought that rational credences, which strive to track them, should do so also.[7] [Emphasis mine]
This article was last updated in 2011.
