ch5 - Monte Carlo Methods with R Monte Carlo Optimization[1...

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Monte Carlo Methods with R : Monte Carlo Optimization [1] Chapter 5: Monte Carlo Optimization “He invented a game that allowed players to predict the outcome?” Susanna Gregory To Kill or Cure This Chapter Two uses of computer-generated random variables to solve optimization problems. The first use is to produce stochastic search techniques To reach the maximum (or minimum) of a function Avoid being trapped in local maxima (or minima) Are sufficiently attracted by the global maximum (or minimum). The second use of simulation is to approximate the function to be optimized.
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Monte Carlo Methods with R : Monte Carlo Optimization [2] Monte Carlo Optimization Introduction Optimization problems can mostly be seen as one of two kinds: Find the extrema of a function h ( θ ) over a domain Θ Find the solution(s) to an implicit equation g ( θ ) = 0 over a domain Θ. The problems are exchangeable The second one is a minimization problem for a function like h ( θ ) = g 2 ( θ ) while the first one is equivalent to solving ∂h ( θ ) /∂θ = 0 We only focus on the maximization problem
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Monte Carlo Methods with R : Monte Carlo Optimization [3] Monte Carlo Optimization Deterministic or Stochastic Similar to integration, optimization can be deterministic or stochastic Deterministic: performance dependent on properties of the function such as convexity, boundedness, and smoothness Stochastic (simulation) Properties of h play a lesser role in simulation-based approaches. Therefore, if h is complex or Θ is irregular, chose the stochastic approach.
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Monte Carlo Methods with R : Monte Carlo Optimization [4] Monte Carlo Optimization Numerical Optimization R has several embedded functions to solve optimization problems The simplest one is optimize (one dimensional) Example: Maximizing a Cauchy likelihood C ( θ, 1) When maximizing the likelihood of a Cauchy C ( θ, 1) sample, ( θ | x 1 , . . . , x n ) = n productdisplay i =1 1 1 + ( x i θ ) 2 , The sequence of maxima (MLEs) θ = 0 when n → ∞ . But the journey is not a smooth one...
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Monte Carlo Methods with R : Monte Carlo Optimization [5] Monte Carlo Optimization Cauchy Likelihood MLEs (left) at each sample size, n = 1 , 500 , and plot of final likelihood (right) . Why are the MLEs so wiggly? The likelihood is not as well-behaved as it seems
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Monte Carlo Methods with R : Monte Carlo Optimization [6] Monte Carlo Optimization Cauchy Likelihood-2 The likelihood ( θ | x 1 , . . . , x n ) = producttext n i =1 1 1+( x i θ ) 2 Is like a polynomial of degree 2 n The derivative has 2 n zeros Hard to see if n = 500 Here is n = 5 R code
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Monte Carlo Methods with R : Monte Carlo Optimization [7] Monte Carlo Optimization Newton-Raphson Similarly, nlm is a generic R function using the Newton–Raphson method Based on the recurrence relation θ i +1 = θ i bracketleftbigg 2 h ∂θ∂θ T ( θ i ) bracketrightbigg 1 ∂h ∂θ ( θ i ) Where the matrix of the second derivatives is called the Hessian
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