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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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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?2012-12-20
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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?2012-12-19
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    @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.2012-12-21
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    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)2012-12-21
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    @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.2012-12-21
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    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.2012-12-21
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    I understand the distinction b/w math & real life, but I'm not convinced that measurement is extraneous to math. After all, Kolmogorov founded the (frequentist) theory of probability on *measure* theory - It's not called that by coincidence. It maps sets to real values. I was hoping for a deeper insight, such as what GC Rota wrote: that probability is the opposite: a functor *from* the Borel algebra on the reals, *to* the sample space. However, I've asked about that on MMA.SE and not everyone is convinced.2012-12-22
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    @alancalvitti : In measure theory the choice of word 'measure' might put you on the wrong course if you try to interpret it as a measurement ; the only thing the word 'measure' means in that context is that we attribute a weight to each subset of a set (in the $\sigma$-algebra) on which we put the measure. It is also not a measurement ; again a definition. You need to distinguish the two of them ; *measurement* immediately refers to something observable, because it implies physical reality ; math does not.2012-12-22
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    You used the word "measuring" in one of your answers (MMA.SE 7/25 - by the way I did not downvote it): "sampling real numbers in [0,1] and measuring them only up to 3 digits of precision". Does this measuring process necessarily refer to physical reality (eg, software pseudorandom generator)? Why can it not also refer to a mathematical quantization function?2012-12-22
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    @alancalvittt : Can you link? I want to see.2012-12-23
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    (meant to write MATH.SE not MMA.SE): http://math.stackexchange.com/questions/47532/sample-dont-confuse-measurements-with-actual-values2012-12-23
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    Here's another example: it's known that of the 32 rotational symmetry classes in which crystalline materials can be classified, 21 are non-centrosymmetric and 20 of these are piezoelectric (Band "Light and Matter"). What is the difference between measuring the symmetry group of a real crystal via x-ray crystallography versus measuring the properties of the mathematical representation of the crystal group?2012-12-23
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    Measurement has nothing to do with abstraction or corporeality. Measurement simply means that we obtain specific valuers as opposed to already knowing a function. It does not matter how they are obtained. In statistics we obtain them from hopefully random observations. In problems that we make up in probability theory or anywhere else, we make up the values arbitrarily or to simplify the solution or for personal taste or just to have values. The philosophers who dreamed up the distinction between abstract and so-called "real world" did the following 12500 generations a terrible disservice.2013-01-20