

Tricks with Monte Carlo Simulations - CountBayesie
http://www.countbayesie.com/blog/2015/3/3/6-amazing-trick-with-monte-carlo-simulations

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mturmon
This post has an opportunity to find the error bars on its estimate of π. The
estimate came out as 3.14616.

The number of in-the-circle counts is S ~ binomial(N,p) where N=1e5 and p=π/4,
which we'll take to be 0.75. Then

    
    
      var(S) = p(1-p)N ≤ N/4
    

and the standard deviation of S is the square root of this, or about ∆=137.

(If we weren't willing to use a guess for p in the variance above, we could
use the universal bound of N/4.)

The estimate of π is (S/N) __* 4, so the sdev on the estimate of π is (∆ /N)
__*4 = 0.0055.

So the estimate turned out to be a little less than one standard deviation
higher than π. Which is satisfying.

\--

The point being that, for just a little extra work, you get not only a Monte
Carlo estimate, but also a bound on its error. Sometimes the bound is just as
important as the estimate.

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pmalynin
I think the most interesting application of Monte Carlo that I've come in
contact with is for pruning the game trees for Go (the game). It was similar
to using minimax, except the tree was only evaluated at points chosen by Monte
Carlo and decisions were made from there. From what I know, this seems to be
one of the better methods for playing go ATM.

~~~
fryguy
This approach is known as Monte Carlo Tree Search, which usually follows a
weighted search by how good the move is until it finds a leaf in the explored
tree. Then it makes a new set of leaves for its moves and scores the path by
randomly making moves until the game is finished and propagating the result up
the path to the root. The wikipedia article has a better explanation:
[http://en.wikipedia.org/wiki/Monte_Carlo_tree_search](http://en.wikipedia.org/wiki/Monte_Carlo_tree_search)

