I was browsing around my crackpottery achives, looking for something fun to write about. I noticed a link from one of them to an old subject of one of my posts, the inimitable Miles Mathis. And following that, I noticed an interesting comment from Mr. Mathis about Monte Carlo methods: “Every mathematician knows that ‘tools’ like Monte Carlo are used only when you’ve got nothing else to go on and you are flying by the seat of your pants.” I find Monte Carlo computations absolutely fascinating. So instead of wasting time making fun of more of Mathis rubbish, I decided to talk about Monte Carlo methods.
It’s a little hard to talk about Monte Carlo methods, because there’s a lot of disagreement about exactly what they are. I’m going to use the broadest definition: a Monte Carlo method is a way of generating a computational result using repeated computations and random sampling.
In other words, Monte Carlo methods are a way of using random sampling to solve problems.
I’ll start with a really simple example. Suppose you want to know the value of . (Pretend that you don’t know the analytical solution.) One thing that you could do is try to measure the circumference of a rod, and then divide it by its diameter. That would work, but it would be really hard to get much accuracy. You could, instead, get a great big square sheet of paper, and cover the whole thing in a single layer of grains of sand. Then, very carefully, you could remove the grains of sand that weren’t in the circle, compare it to the number of grains of sand that weren’t in the circle. By doing that, you could get a very, very accurate measurement of the area of the circle, and using that, you could get a much more accurate estimate of .
The problem with that is: it’s really hard to get a perfect single-grain layer of sand all over the paper. And it would be a lot of very, very tedious work to get all of the grains that weren’t in the circle. And it would be very tedious to count them. It’s too much trouble.
Instead, you could just take 1,000 grains of sand, and drop them randomly all over the circle and the square. Then you could count how many landed in the circle. Or ever easier, you could just go to a place where lots of drunk people play darts! Draw a square around the dartboard, and count how many holes there are in the square wall around it, versus how many in the dartboard!
You’re not going to get a super-precise value for – but you might be surprised just how good you can get!
That’s the basic idea of monte carlo simulation: you’ve got a problem that’s hard to compute, or one where you don’t know a closed-form solution to make it easy to compute. Getting the answer some other way is intractable, because it requires more work than you can reasonably do. But you’ve got an easy way to do a test – like the “is it in the circle or not” test. So you generate a ton of random numbers, and use those, together with the test, to do a sequence of trials. Then using the information from the trials, you can get a reasonable estimate of the value you wanted. The more trials you do, the better your estimate will be.
The more you understand the probability distribution of the space you’re sampling, the better your estimate will be. For example, in the example above, we assumed that the grains of sand/darts would be distributed equally all over the space. If you were using the dartboard in a bar, the odds are that the distribution wouldn’t be uniform – there’d be many more darts hitting the dartboard than hitting the wall (unless I was playing). If you assumed a uniform distribution, your estimate would be off!
That’s obviously a trivial example. But in reality, the Monte Carlo method is incredibly useful for a wide range of purposes. It was used during World War II by the Manhattan project to help design the first atom bomb! They needed to figure out how to create a critical mass that would sustain a nuclear chain reaction; to do that, they needed to be able to compute neutron diffusion through a mass of radioactive uranium. But that’s a very hard problem: there are so many degrees of freedom – so many places where things could proceed in several seemingly (or actually!) random directions. With the computers they had available to them at the time, there was absolutely no way that they could write a precise numerical simulation!
But, luckily for them, they had some amazing mathematicians working on the problem! One of them, Stanislav Ulam, had been sick, and while he was recovering, fooled around with some mathematical problems. One of them involved a solitaire game, called Canfield. Ulam wanted to figure out how often the game of Canfield was winnable. He couldn’t figure out how to do it analytically, but he realized that since the deals of cards are a uniform distribution, then if you were to take a computer, and make it play through 1000 games, the number of times that it won would be a pretty good estimate of how many times the game was winnable in general.
In that case, it’s obvious that a complete solution isn’t feasible: there are 52! possible deals – roughly ! But with just a couple of hundred trials, you can get a really good estimate.
Ulam figured that out for the card game. He explained it to Jon von Neumann, and von Neumann realized that the same basic method could be used for the Neutron diffraction process!
Since then, it’s been used as the basis for a widely applicable approach to numeric integration. It’s used for numerous physics simulations, where there is no tractable exact solution – for example, weather prediction. (We’ve been able to get reasonably accurate weather predictions up to six days in advance, using very sparse input data, by using Monte Carlo methods!) It’s an incredibly useful, valuable technique – and anyone who suggests that using Monte Carlo is in any way a half-assed solution is an utter jackass.
I’ll finish up with a beautiful example – my favorite example of combining analytical methods with Monte Carlo. It’s another way of computing an estimate of , but it gets a more accurate result with fewer trials than the sand/darts.
It’s based on a problem Buffon’s needle. Buffon’s needle is a problem first proposed by the Count of Buffon during the 1700s. He asked: suppose I drop a needle onto a panelled wood floor. What’s the probability that the needle will fall so that it crosses a one of the joints between different boards?
Using some very nice analytical work, you can show that if the panels have uniform width , and the needle has length , then the probability of a needle crossing a line is: . That gives us the nice property that if we let , then the probability of crossing a line is .
Using that, you can do a Monte Carlo computation: take a sheet of paper, and a couple of hundred matchsticks. Draw lines on the paper, separated by twice the length of a matchstick. Then scatter the matchsticks all over the paper. Divide the total number of matchsticks by the number that crossed a line. The result will be roughly .
For example – with 200 trials, I got 63 crossing a line. That gives me roughly 3.17 as an estimate of . That’s not half-bad for a five minute experimental estimate!