Monday, December 31, 2018

What is probability?

Probability is a slippery and widely misunderstood concept. Sometimes it is defined as a frequency, propensity, or degree of belief.

Statistician Andrew Gelman endorses this:
Probability is a mathematical concept. To define it based on any imperfect real-world counterpart (such as betting or long-run frequency) makes about as much sense as defining a line in Euclidean space as the edge of a perfectly straight piece of metal, or as the space occupied by a very thin thread that is pulled taut. Ultimately, a line is a line, and probabilities are mathematical objects that follow Kolmogorov’s laws. Real-world models are important for the application of probability, and it makes a lot of sense to me that such an important concept has many different real-world analogies, none of which are perfect.
Mathematically, I agree with him. Those axioms do not even have anything to do with what non-mathematicians think of as randomness.

The curious thing is that he believes that quantum mechanics violates the laws of probability. This is because he was a physics major before becoming a statistician, and his professor gave him a faulty explanation of the double slit experiment.

The faulty explanation sometimes goes under the name "quantum logic". It says that the double slit experiment proves that the law of the excluded middle is false. You expect that if you fire a particle at a double slit, it goes thru one slit or the other. But you see an interference pattern, and not just the logical sum of particles going thru one slit or the other.

The much more reasonable explanation is the the particle is not a classical particle, but a wave. Waves show interference patterns, and nothing is surprising.

So which would you rather believe, that light has wave properties, or that the mathematical laws of logic and probability are broken?

Obviously, light has wave properties. The alternative is lunacy. It is like having a theory that predicts 2+2 and measuring 4, and then concluding that 2+2 is not 4. No, the better conclusion is that something is wrong with your theory or your measurement.

The theorems about conditional probability and the law of the excluded middle are mathematically valid, and as true for quantum mechanics as anything else.

Gelman says that the probability of the particle hitting a patch on the target screen should be the average of the conditional probabilities, where the condition is passing thru one slit or the other. But such a formula would not give an interference pattern.

His error is thinking that a particle goes thru one slit or the other. Quantum mechanics says that it does not.

When he previously posted about this, others tried to explain the physics to him. But he was not interested in the physics of quantum mechanics. He is interested in applying statistics to social sciences. If conventional probability theory is not good enough for physics, then he wonders whether there should be a generalized probability theory to cover both physics and social sciences.

The physics problem is that unrealized experiments have no results. Maybe there is an analogous principle in the social sciences, I don't know.

For how different people view probably, see this:
Michael Lewis's book "The Undoing Project" is concerned with the (mathematical) psychologists Daniel Kahneman and Amos Tversky. (Kahneman won the 2002 Nobel Prize; Tversky died in 1996.) On page 157, this question is quoted:

The mean IQ of the population of eighth graders in a city is known to be 100. You have selected a random sample of 50 children for a study of educational achievement. The first child tested has an IQ of 150. What do you expect the mean IQ to be for the whole sample.

Tversky and Kahneman stated: "The correct answer is 101. A surprisingly large number of people believe that the expected IQ for the sample is still 100" in Psychological Bulletin, vol. 76, 105--110 (1971).
So he got a Nobel Prize (actually a Bank of Sweden prize in economics) for saying people are irrational for reasoning differently from him.

I do not think I would give 101 as the answer. It is much more likely that the test was mis-normed, or that the children were unusual, or that the sampling was not random. NN Taleb mocks this thought experiment with a more extreme example:
Fat Tony is the foil to Dr. John. Dr. John is nerdy, meticulous, careful and academic; Fat Tony is confident, loud, careless and shrewd. Both of them make errors, but of different types. Dr. John can make gigantic errors that affect other people by ignoring reality in favor of assumptions. Fat Tony makes smaller errors that affect only himself, but more seriously (they kill him). ...

The most famous contrast between the two is the question of what to think about a fair coin that has tossed heads 99 times in a row. Dr. John insists that because the coin is fair, the answer has to be 50%.
Fat Tony deduces that the coin is not really fair, and says that heads is much more likely.

This dichotomy in thinking goes back to Plato and Aristotle. Plato would make purely abstract theories, and doggedly insist on them. Aristotle was the empiricist. If theory differed from practice, Plato would side with the theory, and Aristotle with the practice.

Update: Gelman has commenters who make the usual mistakes about Bell and interpretations of quantum mechanics. Some say that Bell proved that locality implies an inequality inconsistent with QM, so the experiments prove nonlocality. Actually Bell assumed local hidden variables, so the experiments certainly do not prove nonlocality. They only show that local hidden variable models don't work.


  1. blah blah blah, it's not a particle, it's a WAVE...this solves the dilemma, yadda yadda yadda. Calling something you don't understand something else you can't understand does not solve the problem. We know there are particles, we know they do actually act classically in some cases... and yet not in others. This does not logically mean they are not particles, it means the picture is incomplete and that you are forcing a square peg into an ill-defined round hole and calling it a perfect match. Well... it isn't.

    pray tell, what is your wave composed of? Math? ALL energy has form, mass, and some sort of structure as it must or it isn't energy, it never can become a pure numerical abstraction like a amplitude without a cause. You have just cheerfully proposed an unassigned amplitude to the void that can somehow interact with matter. That's it. ALL WAVES are a product of something else moving in aggregate. Water has 'wave like' properties, Air has 'wave like' properties, hell, solid matter like metal or the ground (as in seismic) has wave like properties, YET, They are not composed of 'wave like' abstraction floating in the void, They are all an expressed pattern of aggregate behavior of something else moving through a pattern of particles.

    The double slit experiment is rife with gross assumptions about what is going on and how things are physically considered and depicted. The particles (or waves if you wish to insist) are treated as one kind of nearly intangible thing, and the slit itself and the sensor are treated as continuous solid matter, but at the scale they are interacting, which requires some kind of interaction of mass, there is no continuous solid matter, it's just loosely bound moving particles at that level.

    If I fire one single photon (and I'm only assuming a single photon can actually be's actually very unclear on this point as there is admission that when a single photon is purportedly detected, many other things happen in a cascade, not just one, which I suppose is like throwing an apple at an apple tree, you might have initially thrown one apple, but several more might get shaken loose or go flying, depending on how hard you threw it and where it collides with the structure of the tree, did it just graze one apple or several, did it just hit the trunk? In any case, if I throw one apple at the apple tree and two apples go flying out, I do not say my apple has multiplied. Which goes to the root of the problem with the double slit detections, you get the wave like pattern only when there is more than one slit, which indicates something else is going on in how the structure of the slit itself is interacting with the energy, it has to be, else why is it emitting? It can't be that photons with a set level of energy is now miraculously multiplying, that abandons all conservation of mass and energy. If I throw one photon and yet get the energy required for two impacts... unless you can explain where the additional mass came from, and it wasn't pulled from math or from the void or ether, then you have an accounting problem that needs to be solved, not a paradox to be foisted up as an explanation, like the ridiculous wave/particle malarky. At best, It's one OR (not inclusive) the other, or you have the wrong model being used entirely.

  2. When you strike a bell with a hammer, energy in the form of sound, and heat (and most likely some of the metal particles)radiate out in all directions, yet in a pattern which is influenced by the crystalline structure of the metal on an atomic level, there IS discrete moving structure to atoms, it isn't a Rutherford glommed monkey bread ball with orbiting satellite electrons, or clouds of probability or whatever. If we are uncertain of the very structure of the atoms, no wonder our understanding of what happens when they are pinged with photons is so ridiculous. I would be very curious of the geometry of detections by light sensors placed around the test slits, and see exactly what pattern emerges when more photons are detected outside as well behind the slits from the origin point of the wave pattern. Think rationally first, resort to causeless miracles only when you have exhausted curiosity and are ready to go to heaven.

    There are no paradoxes in actuality. A paradox is a logical error of valid syntax, you can easily construct a valid sentence that is not logically true ('zeus can do anything, he can create a rock so heavy he can't pick it up', 'The next sentence is true. The previous sentence is false', there is only incomplete understanding and/or faulty reasoning.