It seems to me that if a probability is a quantity representing a degree of belief, and it's only meaningful in relation to another quantity representing another degree of belief, in the sense that we can only say than being X% sure is being more sure than being Y% sure, if X > Y, or equally sure, if X = Y, then such quantity only has an ordinal value, which is to say the quantity itself is meaningless. For it to be meaningful it has to have an interpretation that does not always refer us to another degree of belief. It also must not refer to a "degree of plausibility" since this is just another expression for "probability", and probability is what we are trying to define.
I'm not sure I understand where do you see a problem.
In the example of the degree of belief (between 0 and 1) that you have that the coin on my desk is showing one face or the other, don't you agree that the right numbers that represent your indifference are 0.5 and 0.5? The quantity itself is not meaningless.
You say indifference, but indifference with regards to what? In economics, an individual is said to be indifferent between two alternatives if those alternatives result in the same level of utility for him or her, utility being an abstract concept representing well-being. But you don't say why this person is indifferent to the coin showing one face or the other. Because if it's because he or she thinks both options are equally likely, then we again have a problem, since we don't know what "likely" means.
The argument is flawless, the problem is with the interpretation.
> p(A or B) = p(A) + p(B) - p(A and B)
How does one add degrees of belief and what sense do we make out of the result?
> p(H) = p(T) = 0.5
Sure, two equal quantities representing degrees of belief must mean the degrees of belief are of the same magnitude. But what about P(H) = 2P(T)? What does it mean for one degree of belief to be twice as large as the other?
> How does one add degrees of belief and what sense do we make out of the result?
That's how we postulate [1] that the numeric representations of degrees of belief are added. Doesn't that look like a property that you want a numeric representation of degrees of belief to have?
If you have some degree of belief about A, some degree of belief about B, and you believe that A and B are mutually exclusive, wouldn't you want the number representing the degree of belief of "any of them" p(A or B) to be the sum p(A)+p(B)?
>> p(H) = p(T) = 0.5
> Sure, two equal quantities representing degrees of belief must mean the degrees of belief are of the same magnitude. But what about P(H) = 2P(T)? What does it mean for one degree of belief to be twice as large as the other?
Consider p(H or T) = p(H) + p(T) = 2 p(H) = 2 p(T). Isn't it natural to quantify the degree of belief that I got any outcome with a number that is the sum of the numeric representations of the degrees of belief that I got each outcome?
Or say that, instead of flipping a coin, I toss two of them. They're lying flat on my desk right now. The number of heads up is 0, 1, or 2.
How would you describe your degree of belief about the statements "X=0: there are no heads", "X=1: there is one" and "X=2: there are two"?
Wouldn't you say that your degree of belief about "X=1" is of the same magnitude as your degree of belief about "X=0 or X=2"?
Wouldn't you say that your degree of belief about "X=0" is of the same magnitude as your degree of belief about "X=2"?
Wouldn't that make the numerical representation of your degree of belief about "X=1" twice as large as the numerical representations of your degrees of belief about each of "X=0" and "X=2"? (Where you assign numbers to degrees of belief using the representation we're discussing.)
p(X=1) = p(X=0) + p(X=2) = 2 p(X=0) = 2 p(X=2)
[1] in fact I think this is what we get from postulates which are a bit more general, but for the sake of the discussion we may stay in this level
Addition is required by the axioms of probability, not by the interpretation of it. When interpreting probability as a degree of belief this property is not only not useful, but is particularly troublesome, because adding and subtracting degrees of beliefs doesn't appear to make a lot of sense.
All in all, to me it's clear that these degrees of belief are a theoretical construct, not an empirical reality. I don't think people assess the truth value of a statement on a continuum from truth to false. This is not how the human psyche works. Personally, no, it's not natural for me to have a degree of belief (in the way that you have defined them) about a statement, and I have no idea how to interpret arithmetic operations involving these "things".
(Note: I enjoy the discussion but I won't be offended if I get no reply! You're free to consider probability just a construct and not accept its applicability to real-world reasoning.)
Ok, what can you say then regarding your degrees of belief about the statements "X=0: there are no heads", "X=1: there is one", "X=2: there are two" and “X={0,2}: there are none or two”?
Of course, if you don't think you have degrees of belief to start with there is no way you can make sense of assigning numbers to them. But I thought we had progressed to the point where you could accept that they existed and that they could be ordered.
> I don't think people assess the truth value of a statement on a continuum from truth to false.
So they cannot be ordered? Or there are no extremes?
> When interpreting probability as a degree of belief this property is not only not useful, but is particularly troublesome, because adding and subtracting degrees of beliefs doesn't appear to make a lot of sense.
Are the rules I proposed problematic in some specific way? They make a lot of sense as far as I can tell.
What I was saying before is that, as long as you only use the degree of belief as an ordinal value, you don't need to come up with an explanation for what the actual value represents (e.g. the percentage of neurons that agree or disagree with the statement, or whatever you think it means to think that a statement is X% true in terms of something that is quantifiable). But the problem is probability is not ordinal, it's a cardinal value, so you DO have to come up with an explanation. That was the argument.
Do the rules make sense? The rules don't have to make sense, they're axioms. They're assumed to be true whether they make sense or not. We are not discussing the axioms. We are discussing interpretations of probability. In my opinion a good interpretation of probability must provide a context in which these axioms (kind of) make sense. And that's one of the problems I have with the interpretation of probability as a degrees of belief, the rules don't make sense in the provided context, at least to me, because I don't know how to make sense of arithmetic operations involving degrees of belief. (But that doesn't mean that I think the rules themselves don't make sense.)
Finally, even if you think that the human mind doesn't work in the way degrees of belief are hypothesised to work, you may still find the concept useful as a means of giving an interpretation to probability. Personally, I don't think that the mind works like that, nor that they're useful as an interpretation of probability. This is basically my position.
Fine. I was trying to answer your question "what do people mean when they say they're 33% sure that tomorrow it will rain". They mean that they find twice as plausible that it will not rain. It's just that you don't understand it as they do.
Anyway, I think you're getting the direction of the argument wrong. It's not that you have probabilities and force an interpretation of them as degrees of belief.
You start with real numbers representing degrees of belief (with an ordinal meaning only, a larger number means more plausible) and some "common sense" properties they should have to be "rational":
- having identical information should result in the same degree of belief
- the degree of belief in "not A" should be a function of the degree of belief in A
- the degree of belief in "A and B" should be a function of the degree of belief in "A given B" and the degree of belief in B
The rules of probability _are_the_consequence_ (once the value of certainty is fixed to 1)
p(A) + p(not A) = 1
p(A and/or B) = p(A) + p(B) - p(A and B)
p(A and B) = p(A|B)p(B) = p(A)p(B|A)
and the use of probabilities to represent degrees of belief is not something you come up with. It is derived from the assumptions above (which don't involve probability at all).
In pages 5-9 he derives the rules of probability as a "reasonable expectations" extension to symbolic logic.
Of course you're right that this is a description of how rational thought should be and not necessarily a description of how people think. Actual beliefs can be inconsistent in the same way that one can believe things that go against the laws of logic.
It's always fun to discuss the foundations of probability. Thanks.
