# Probability vs. degree of belief in facts of nature (“Plausibility”)

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?

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This is the old argument against Bayesianism--- it's really philosophy. The point is that probability is a model for knowledge, and Bayesianism uses probability as a universal model for knowledge consistently. This is a good thing. –  Ron Maimon Jul 24 '12 at 15:36

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

Degrees of belief are also modelled by fuzzy logic. In tihs 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://www.mat.univie.ac.at/~neum/ms/fuzzy.pdf

My critique of using subjective probability in physics is given in Section 10.7 of my book http://lanl.arxiv.org/pdf/0810.1019v2.pdf

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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)

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