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"It is worth emphasizing that probabilities are assigned not measured." -- TJ Loredo "From Laplace to SN1987A" in "Max Entropy and Bayesian Methods" 1990.

Is there a logical way to distinguish what is assigned versus measured?

Or is this a scientific rather than a mathematical issue? If there is a logical distinction, it would be nice if it covered both frequentist and Bayesian concepts of probability.

(Maybe a starting point could be the distinction between the modes $X \to A$ versus $A \to X$ where, loosely, $A$ is better characterized than $X$)

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2 Answers 2

Classical probability theory is the study of theoretical random processes where everything is determined. There is nothing to measure. The problems only consist of being given some information that determines the rest, or at least what is asked for. So probability theory has no concept of a sample. The "sample space" is misnamed, as is the term "experiment." What is called an "experiment" is really a "trial" which is a basic concept in probability theory and the "sample space" is really the possible outcome space - meaning the possible outcomes of a trial.

Statistics on the other hand addresses what can be told about an unknown (or assumed) random process by making observations (measurements). It USES probability theory to generate it estimates but is an entirely different subject. "Sample" is a basic concept in statistics. (When I first studied statistics I called it "sadistics;" but it's not so bad once you catch on. But catching on can be arduous.)

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Good points. I agree that sample space is the space of possible outcomes, and that statistics is distinct from probability. But leaving aside statistics for the moment - how to mathematically distinguish what is an assignment vs what is measured? –  alancalvitti Dec 20 '12 at 18:32

There is a reason why probabilities are assigned and not measured : it's because probability is a mathematical concept, and in mathematics we do not measure things because there is no such thing as experience in mathematics (at least the theoretical part of it). In probability theory, when one defines a probability over a probability space, it is up to the mathematician to decide what probability is attributed to each event ; whether he sees fit that one probability works better than another is up to him.

Now what happens in real life is that there are standard intuition for probability definitions ; the most basic one would be to define the probability of an event as the number of possible cases where it happens over the number of possible cases in total. That makes sense if you want to count things ; in sampling theory, however, this does not fit the situation. In that case, the sampling plan needs to be adjusted so that the samples are chosen such that every sampling unit has a non-zero probability of being chosen, but there are many different kinds of sampling plans for many different reasons ; the most important one being when a sampling plan needs to take into account auxiliary information to improve accuracy of the sampling. That auxiliary information would've had to be measured, but the sampling plan is an assignment of the probabilities that each sample gets chosen.

I have not studied Bayesian statistics so I'm afraid I can't comment on that.

Hope that helps,

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Isn't the dimension of a vector space - which is also a mathematical concept - measured rather than assigned? –  alancalvitti Dec 19 '12 at 12:00
    
@alancalvitti : For me, measurement somehow refers to reality ; you measure speed of objects in physics, temperatures of solutions in chemistry, etc. In mathematics, we define things and we build theories upon those definitions. The dimension of a vector space is assigned by its definition ; we do not "measure it" just because we have to think about how many linearly independent vectors are required to generate the vector space. –  Patrick Da Silva Dec 21 '12 at 5:20
    
Ok forget vector spaces. Consider instead unit $L2$ balls in $\mathbb R^n$. Aren't the volumes and surface areas of the associated boundary (for each $n$) measurements? After all, the definition of the spheres doesn't automatically yield such information. The definition states only that $\|x\|<=1$. Yet these are mathematical concepts - it's not necessary to refer to real balls (though of course the definition is motivated by real world needs) –  alancalvitti Dec 21 '12 at 5:43
    
@alancalvatti : Again, you can define volume and surface in whatever way you want ; these things are defined (in full formalism) as integrals, and there are many ways to define integrals (Lebesgue, Riemann, etc.) and you can also choose which function you integrate to define the volume. You are not seeing the difference between what one measures in real life (i.e. the volume of a ball as we know it) and the mathematical definition of volume, which assigns to every subset of $\mathbb R^3$ the value of an integral which we call volume. –  Patrick Da Silva Dec 21 '12 at 23:05
    
Both notions coincide because mathematicians are smart, but what you do in real life, is you measure ; what you do in math, is you define and assign. There is no such thing as measurement in mathematics ; measuring belongs to other disciplines. –  Patrick Da Silva Dec 21 '12 at 23:05

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