Puzzled about the non-teaching of Monte Carlo method(s) for error analysis

[Vanadium 50]

has a Cauchy distribution

You mean a Breit-Wigner, right? Or maybe a Lorentzian?

 

[hutchphd]

The problem hurts my head. I think I will leave this problem to others.

I am certainly glad I was never "educated" in this stuff. And I used it a lot with great practical success, largely oblivious I guess.
All I require is that the deviations (rms errors) be small compared means, and they be "uncorrelated". Then the leading order terms for the rms deviations are as described independent of the form of the distributions. Of course if the Taylor expansion for the functional dependence blows up there is a problem, but in the real world this seldom happens.
These techniques are extraordinarily useful and robust. In my experience the only places requiring some care are low probability events (the wings of the distribution) where wrong assumptions will bite you. Do not be afraid. Say the magic words: "central limit theorem".

 

[Vanadium 50]

First, the thing we really want is the probability that the true value is inside the error bars to be 68%. That's not well-defined. (And to also require that the probability that the true value is also inside twice the error bars to be 90% is even less well-defined). So our whole statistical technique, analytic or Monte Carlo, isn't built on a mathematically rigorous foundation. But it's the best we have. And "not mathematically rigorous" is not the same as useless - there's real meaning in comparing error bars of 1%, 5% and 10%, even if none of these are exactly what we hoped to know.

What do we say if the voltage measurement is positive and the current measurement is negative?

Here's where you have to think like a physicist, not a mathematician. If my knowledge of the current is so poor I can't tell which way it is flowing, or even if it is flowing at all, I shouldn't be using it to calculate the resistance.

 

[FactChecker]

In my experience the only places requiring some care are low probability events (the wings of the distribution) where wrong assumptions will bite you. Do not be afraid. Say the magic words: "central limit theorem".

You are correct, of course. I was thinking of the 10% voltage error as being a huge error without realizing that a negative voltage would be 10 standard deviations below the mean. Stranger things have happened, but not since Moses.

 

[Stephen Tashi]

If the goal of the simulation example was to compare the simulation technique with a theoretical pencil-and-paper technique then we'd need to see the pencil-and-paper technique worked out to make the comparison. But to see the pencil-and-paper technique worked out, we'd need to define what problem is being analyzed in the first place!

 

[Dr Transport]

The bottom line is you need to know what the hell you are doing...The Monte Carlo methods give you numbers and very little insight. If you don't understand the theory, you will not know what you are doing.

My PhD advisor said exactly the same thing to me 25 years ago.

 

[FactChecker]

Insight and theory are great when they are correct. But things often get very complicated and confusing. If nothing else, Monte Carlo simulations can often be used to verify theoretical results or to point out errors.

Here is an example showing the difficulty of analyzing a fairly simple queueing problem: https://www.physicsforums.com/threads/waiting-time-in-a-queue-using-poisson-arrival.902175
And the only way I would feel confident of the analytical results is if there was a simulation that supported them. The only way to do real work in queueing theory is with MC simulation. IMHO, the only queueing problems that can be solved analytically are trivial ones.

Here is a problem involving a dice game where the analytical solution is messy and the MC simulation is simple. https://www.physicsforums.com/threads/probability-in-a-dice-game.989492/
Again, I would not be confident of the analytical solution at all if there was not an MC simulation result to support it.

 

[mpresic3]

I had a professor who taught Classical Electrodynamicsi at the graduate level, not from Jackson,but with his own notes involving exclusively, differential forms. I knew a professor that was proposing to teach graduate Classical Mechanics, not from Goldstein, but using category theory, alone. Both suggested these areas were sadly lacking in graduate physics education. After attending the Electrodynamics course, I had (two) graduate courses in Jackson. (I went to graduate school twice).

It may seem I am favor of conventional treatment of the physics curriculum. I think there is danger in uniformity, and it is good for some students to have different tools in their toolbox. However, it is hard to come up with areas in the tight physics curriculum that could be left out. Certainly, including MC methods at the expense of other important topics is going to be objected too by others.

It seems like, when the poster becomes the instructor head of the course, he or she can then teach whatever he or she wants. There is quite a bit academic freedom in the USA anyway. The professor who taught differential forms did not get much pushback. The professor who proposed category theory (as far as I know) did not get his course, because no student was interested in taking the course.

Also, the training of a physicist contains more than just physics courses. Physicists can run into MC techniques in computer science courses, or statistics courses. A good argument could be made that statistics and probability should be required. Maybe some would say, substitute probability, for Complex Analysis. However, a look at most graduate physics program, seems to regard complex analysis over probability. As I wrote, you can always find somebody to find something they feel should be part of the education, that is overlooked.