Monte Carlo Integration

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Monte Carlo Integration
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Goal Evaluate an integral
Why use random methods?
Computation by deterministic quadrature can
become expensive and inaccurate.

• grid points add up quickly in high dimensions

• bad choices of grid may misrepresent g(x)

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Hit-or-Miss Monte Carlo
a
b
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Sample uniformly from the rectangular region
a,bx0,c
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We can easily estimate p

• throw n uniform darts at the rectangle

• let X be the number of times you end up under
  the curve yg(x)

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Dont ever give an estimate without a confidence
interval!

• note that X is binomial(n,p)

• unravel the relationships to get an approximate
  confidence inerval for I

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(No Transcript)
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Example
(we know that the answer is e3-1 19.08554)

• an upper bound is e3

• 100,000 uniform draws over 0,3x0,e3

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Simulation Results
19.08554
0.31365 19.06217885 (18.88849413, 19.23586357)
0.31552 19.01216586 (18.83860388, 19.18572785)
0.31729 19.11882006 (18.94499712, 19.29264301)
0.31642 19.06639681 (18.8927018, 19.24009185)
0.31672 19.08447380 (18.9107346, 19.25821302)
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Sample Mean Monte Carlo
(Monte Carlo Integration)
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where Xunif(a,b)
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Example
(we know that the answer is e3-1 19.08554)
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• write this as

where Xunif(0,3)
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Simulation Results true
19.08554, n100,000
Simulation
1 19.10724
2 19.08260
3 18.97227
4 19.06814
5 19.13261
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Dont ever give an estimate without a confidence
interval!
This estimator is unbiased
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• furthermore,

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• an approximation

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• X1, X2, , Xn iid -gt g(X1), g(X2), ,
  g(Xn) iid

• Let Yig(Xi) for i1,2,,n

and we can once again invoke the CLT.
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For n large enough (ngt30),
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By the way
No one ever said that you have to use the uniform
distribution
Example
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Comparison of Hit-and-Miss and Sample Mean Monte
Carlo
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Comparison of Hit-and-Miss and Sample Mean Monte
Carlo
Sample mean Monte Carlo is generally preferred
over Hit-and-Miss Monte Carlo because

• the estimator from SMMC has lower variance

• SMMC does not require a non-negative integrand
  (or adjustments)

• HM MC requires that you be able to put g(x) in
  a box, so you need to figure out the max
  value of g(x) over a,b and you need to be
  integrating over a finite integral.