Probability vs. degree of belief in facts of nature ("Plausibility")

Question

I just came across a line in a paper:

"Assume the probability that a Lagrangian parameter lies between $a$ and $a + da$ is $dP(a) = $ [...]."

This reminded me again of my single biggest qualm I have with the "Bayesian school" - that they assign probabilities to facts of nature, say $P(m_\mathrm{Higgs} = 126\,\mathrm{GeV}$). I understand that you can assign a probability to the outcome of an experiment when you perform it many times. I would also say that you don't actually have to do the experiments, since you can reason about ensembles. You can consider the probability that there is an earthquake tomorrow without really having multiple earths. I would not say however that you can assign a probability to a fact of nature, like a natural constant. It is either such, or such, and there is no dynamics changing it.

Bayesians think of probability as "degree of belief", and in that case of course you can assign probabilities to arbitrary hypotheses. $P (\mathrm{SUSY\;exists})\approx 60\%$ and the such.

I myself like to keep the two cases distinct. I say "probability" when I'm talking about dice, or quantum mechanical decays and the such, and "plausibility" when I'm talking about degrees of belief. The probability that SUSY exists is either 1 or 0. The plausibility is a very sociological thing and depends on who you ask, and what experimental and theoretical inputs you consider. The nice thing is that the maths for plausibility is basically the same as for probability, so no new formulas to learn ;-).

I've been using that distinction in conversations for a while, and nobody seemed to find it strange, so I think this concept is pretty widespread. However, I can't find any literature on it (beyond the usual frequentist/Bayesian debate). My question is, are there any references / essays / lectures on the two different kinds of "$P\,$"? What's the degree-of-belief-$P$ (my "plausibility") usually called? What are the implications of keeping the two distinct?

Answer 1

In some of the Bayesian literature (see reference below) the distinction you describe is made quite clearly and distinctly. However, perhaps unfortunately, they've made the opposite choice from you regarding terminology: what you call "probability" is called "frequency" (as in the frequency of successful trials in a repeated experiment), and what you call "plausibility" is called "probability" (although it's also sometimes called plausibility). So you want to re-name the epistemically-interpreted part of probability theory to something else, whereas the Bayesian school decided to rename the part of it that's interpreted in terms of randomness to something else.

Historically, this does make sense - the Bayesian way of using probability theory would more accurately be called "Laplacian", and pre-dates what Bayesians call the "frequentist" interpretation by around a century or so. So the "plausibility" meaning of the word probability is the oldest one.

In terms of the practical advantages of making the distinction, by far the best reference (IMHO) is Edwin Jaynes' (1978) Where do we stand on maximum entropy?. At over 100 typewritten pages it's a long read, but it's quite entertaining, and if you're at all interested in the philosophy of Bayesian probability theory and its applications in physics then it's absolutely compulsory. Making the distinction between (plausibility-)probabilities and frequencies allows one to apply Bayesian inference to the outcomes of repeated experiments, and it's really quite amazing how much mileage Jaynes is able to get out of this simple trick.

If you're interested in understanding why Bayesians think of assigning probabilities (or plausibilities) to facts of nature, then it's really a good idea to read R. T. Cox's (1946) Probability, Frequency and Reasonable Expectation, where you'll also find the "frequency"/"plausibility" terminology. Cox shows that if one wants to extend propositional logic to the case where one doesn't know whether a statement is true or false then (under certain very mild assumptions) there is only one consistent way to do it - and that way is probability theory. Cox's argument is unpacked in a longer and perhaps more readable style in the first few chapters of Jaynes (2003) book, Probability Theory: the Logic of Science, the relevant parts of which are available online.

Answer 2

Probability is a special case of belief/plausibility. Belief and plausibility provide lower and upper bounds on probability, respectively. Belief/plausibility are based on assignments to intervals of outcomes (focal elements) while probability is based on assignments to individual outcomes (probability density function). Belief/plausibility can better address epistemic (state of knowledge) uncertainty than probability which addresses aleatory (stochastic or random) uncertainty. Belief/plausibility is a subjective approach, as is a Bayesian approach. There is a mis-conception that a Bayesian approach is always a good approach; however, a poor prior with little information to update to the posterior can be way off. In this case belief/plausibility is a much better approach. Necessity/possibility is another special case of belief/plausibility, applicable when the focal elements are nested. Search for papers by Dr. Jon Helton for more detailed information.

Answer 3

Not sure this goes in the same direction, but I like to think about these questions in terms of a problem called "Laplace's Rule of Succession":

Given that the sun has risen consecutively for the past $n$ days, what is the probability that the sun will rise tomorrow?

Nothing in said about the internal machinery of planetary motion. This is an update of our current knowledge based on past evidence. "Sun rising tomorrow" is an event we can experiment only once. Notwithstanding, we may have some degree of belief as to it's outcome based on other experiments (the past n days where the sun did rise)

Answer 4

Belief/plausibility (and its special cases probability and necessity/possibility) address ambiguity; uncertainty as to the outcome of a future event. Fuzzy sets address vagueness; uncertainty as to how to classify a known outcome. Fuzzy sets can be quantitative (fuzzy numbers) or qualitative (linguistics, or words). For example, for the variable "Outlook on Life" the outcomes can be the linguistics "Optimist" "Accepting" "Pessimist". Linguistic variables are appropriate when there is no appropriate numeric scale. The belief/plausibility measure (or its special cases) can be applied to variables whose outcomes are fuzzy sets. For example, for linguistic variables the variables are combined in an approximate reasoning rule base; focal elements are assigned to subsets of the linguistic outcomes for each variable and using the mathematics for "convolving" focal elements the uncertainty for a top level variable can be evaluated. For example, consider the top level variable "Happiness" modeled as a rule-based combination of "Quality of Life" and "Outlook on Life", and "Quality of Life" modeled as a rule-based combination of "Health" and "Wealth". For a specific individual, focal elements are assigned to each basic variable and the "Happiness" for the individual is generated including uncertainty using belief/plausibility. In practice, a group of subject matter experts (SMEs) identify the top level variable, the constituent lower level variables, the fuzzy outcomes for each variable, and the rule base for how the variables combine. This forms the "linguistic" equation (the equivalent of an algebraic equation for combining numeric variables). Then, for a specific instance, based on their state of knowledge the SMEs assign focal elements and the uncertainty in each variable for that instance can be evaluated. For a complex issue, different SMEs assign focal elements to different variables, and more then one SME addresses a given variable. In the past, I developed software that implements this process and applied it to numerous linguistic problems dealing with national security and counter terrorism. The following are screen shots from the software for the "Happiness" example.

Answer 5

Plausibility is most consistently modelled by Dempster-Shafer theory. https://en.wikipedia.org/wiki/Dempster-Shafer.

Degrees of belief are also modelled by fuzzy logic. In this context, you may be interested in my paper
A. Neumaier, Fuzzy modeling in terms of surprise, Fuzzy Sets and Systems 135 (2003), 21-38.
http://arnold-neumaier.at/ms/fuzzy.pdf

My critique of using subjective probability in physics is given in Section 10.7 of my book https://arxiv.org/abs/0810.1019
See also Ignorance in statistical mechanics