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Monte Carlo is an estimation procedure. The basic idea is as follows. You want to know the average value of some random variable. You cannot work out what its distribution is exactly, or you do not want to do integrals numerically, but you can take samples from that distribution. The random variable may, for instance, be some complicated function of variables with simple distributions, or the distribution may have a hard-to-compute normalizing factor ("partition function" in statistical mechanics). To estimate it, you simply take samples independently and average them. If you take enough samples, then the Law of Large Numbers says your average must be close to the true value. The Central Limit Theorem says that your average has a Gaussian distribution around the true value.

Here is one of the canonical examples. Say you want to measure the area of a shape with a complicated, irregular outline. The Monte Carlo approach is to draw a square around the shape and measure the square. Now you throw darts into the square, as uniformly as possible. The fraction of darts falling on the shape gives the ratio of the area of the shape to the area of the square. Now, in fact, you can cast almost any integral problem, or any averaging problem, into this form. So you need a good way to tell if you are inside the outline, and you need a good way to figure out how many darts you should throw. Last but not least, you need a good way to throw darts uniformly, i.e. a good random number generator.

Now, in fact, you do not strictly need to sample independently. You can have dependence, so long as you end up visiting each point just as many times as you would with independent samples. This is useful, since it gives a way to exploit properties of Markov chains in designing your sampling strategy, and even of speeding up the convergence of your estimates to the true averages. The classic instance of this is the Metropolis-Hastings algorithm, which gives you a way of sampling from a distribution where all you have to know is the ratio of the probability densities at any two points. This is extremely useful when, as in many problems in statistics and statistical mechanics, the density itself contains a complicated normalizing factor. It drops out of the ratio.



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