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ch3 - Monte Carlo Methods with R Monte Carlo Integration[1...

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Monte Carlo Methods with R : Monte Carlo Integration [1] Chapter 3: Monte Carlo Integration “Every time I think I know what’s going on, suddenly there’s another layer of complications. I just want this damn thing solved.” John Scalzi The Last Colony This Chapter This chapter introduces the major concepts of Monte Carlo methods The validity of Monte Carlo approximations relies on the Law of Large Numbers The versatility of the representation of an integral as an expectation
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Monte Carlo Methods with R : Monte Carlo Integration [2] Monte Carlo Integration Introduction We will be concerned with evaluating integrals of the form integraldisplay X h ( x ) f ( x ) d x, ⊲ f is a density We can produce an almost infinite number of random variables from f We apply probabilistic results Law of Large Numbers Central Limit Theorem The Alternative - Deterministic Numerical Integration R functions area and integrate OK in low (one) dimensions Usually needs some knowledge of the function
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Monte Carlo Methods with R : Monte Carlo Integration [3] Classical Monte Carlo Integration The Monte Carlo Method The generic problem: Evaluate E f [ h ( X )] = integraldisplay X h ( x ) f ( x ) d x, ⊲ X takes its values in X The Monte Carlo Method Generate a sample ( X 1 , . . . , X n ) from the density f Approximate the integral with h n = 1 n n summationdisplay j =1 h ( x j ) ,
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Monte Carlo Methods with R : Monte Carlo Integration [4] Classical Monte Carlo Integration Validating the Monte Carlo Method The Convergence h n = 1 n n summationdisplay j =1 h ( x j ) integraldisplay X h ( x ) f ( x ) d x = E f [ h ( X )] Is valid by the Strong Law of Large Numbers When h 2 ( X ) has a finite expectation under f , h n E f [ h ( X )] v n → N (0 , 1) Follows from the Central Limit Theorem ⊲ v n = 1 n 2 n j =1 [ h ( x j ) h n ] 2 .
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Monte Carlo Methods with R : Monte Carlo Integration [5] Classical Monte Carlo Integration A First Example Look at the function h ( x ) = [cos(50 x ) + sin(20 x )] 2 Monitoring Convergence R code
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Monte Carlo Methods with R : Monte Carlo Integration [6] Classical Monte Carlo Integration A Caution The confidence band produced in this figure is not a 95% con- fidence band in the classical sense They are Confidence Intervals were you to stop at a chosen number of iterations
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