. . March 14th, 2011 Biostatistics 615/815 - Lecture 16 Hyun Min Kang March 14th, 2011 Hyun Min Kang Importance sampling Monte-carlo methods Biostatistics 615/815 Lecture 16: . . . . . . . Summary . Rejection sampling . Importance sampling Integration Introduction . . . . . . . . . 1 / 18 . . . . . . . . . .
. . . . . . . 815 Update . . . . . . . . Hyun Min Kang Biostatistics 615/815 - Lecture 16 March 14th, 2011 . . . Importance sampling . . . . . . . . . Introduction Integration . 2 / 18 . Rejection sampling . Summary Annoucements . Grading . . . . . . . . . . • Midterm will be given by Thursday • All the other homeworks will be given by next Tuesday • Send a brief progress update on the project • Schedule meeting with instructor if needed
. Importance sampling March 14th, 2011 Biostatistics 615/815 - Lecture 16 Hyun Min Kang . Summary . Rejection sampling . 3 / 18 Integration Introduction . . . . . . . . . . . . . . . . . . . Recap : Pseudo-random numbers using rand() #include <iostream> #include <cstdlib> int main(int argc, char** argv) { int n = (argc > 1) ? atoi(argv[1]) : 1; int seed = (argc > 2 ) ? atoi(argv[2]) : 0; srand(seed); // set seed -- same seed, same pseudo-random numbers for(int i=0; i < n; ++i) { std::cout << (double)rand()/RAND_MAX << std::endl; // generate value between 0 and 1 } return 0; }
. . March 14th, 2011 Biostatistics 615/815 - Lecture 16 Hyun Min Kang pseudo-random numbers function in PHP Recap : Good vs. bad random numbers Summary . Rejection sampling . Importance sampling Integration Introduction . . . . . . . . . 4 / 18 . . . . . . . . . . • Images using true random numbers from random.org vs. rand() • Visible patterns suggest that rand() gives predictable sequence of
. . March 14th, 2011 Biostatistics 615/815 - Lecture 16 Hyun Min Kang . Recap : Generating uniform random numbers in C++ Summary . Rejection sampling 5 / 18 Importance sampling Integration Introduction . . . . . . . . . . . . . . . . . . . #include <iostream> #include <boost/random/uniform_int.hpp> #include <boost/random/uniform_real.hpp> #include <boost/random/variate_generator.hpp> #include <boost/random/mersenne_twister.hpp> int main(int argc, char** argv) { typedef boost::mt19937 prgType; // Mersenne-twister : a widely used prgType rng; // lightweight pseudo-random-number-generator boost::uniform_int<> six(1,6); // uniform distribution from 1 to 6 boost::variate_generator<prgType&, boost::uniform_int<> > die(rng,six); // die maps random numbers from rng to uniform distribution 1..6 int x = die(); // generate a random integer between 1 and 6 std::cout << "Rolled die : " << x << std::endl; boost::uniform_real<> uni_dist(0,1); boost::variate_generator<prgType&, boost::uniform_real<> > uni(rng,uni_dist); double y = uni(); // generate a random number between 0 and 1 std::cout << "Uniform real : " << y << std::endl; return 0; }
. . March 14th, 2011 Biostatistics 615/815 - Lecture 16 Hyun Min Kang . . . . . . . . Sampling from complex distributions . Today Summary . . . . . . . . . . Introduction Integration Importance sampling . Rejection sampling 6 / 18 . . . . . . . . . . • Monte-Carlo Methods • Importance Sampling
. . . . . . An example problem . . . . . . . . Calculating Hyun Min Kang Biostatistics 615/815 - Lecture 16 March 14th, 2011 . . . . . . . . . . . . . Introduction . Importance sampling Integration Rejection sampling Monte-Carlo Methods . Informal definition . . 7 / 18 Summary . . . . . . . . . . • Approximation by random sampling • Randomized algorithms to solve deterministic problems approximately. ∫ 1 I = f ( x ) dx 0 where f ( x ) is a complex function with 0 ≤ f ( x ) ≤ 1 The problem is equivalent to computing E [ f ( u )] where u ∼ N (0 , 1) .
. . . B B . Desirable properties of Monte-Carlo methods . . . . . . . . Consistency : Estimates converges to true answer as B increases Unbiasedness : E Minimal Variance Hyun Min Kang Biostatistics 615/815 - Lecture 16 March 14th, 2011 . . . . . . . . . . . . . Introduction Integration Importance sampling . Rejection sampling . Summary The crude Monte-Carlo method . Algorithm . . . 8 / 18 . . . . . . . . . . • Generate u 1 , u 2 , · · · , u B uniformly from U (0 , 1) . • Take their average to estimate θ θ = 1 ˆ ∑ f ( u i ) i =1
. . . . . B B . Desirable properties of Monte-Carlo methods . . . . . . . . Hyun Min Kang Biostatistics 615/815 - Lecture 16 March 14th, 2011 . 8 / 18 . . . . . . . . . . . Introduction Integration Importance sampling . Rejection sampling . . . Summary The crude Monte-Carlo method . Algorithm . . . . . . . . . . • Generate u 1 , u 2 , · · · , u B uniformly from U (0 , 1) . • Take their average to estimate θ θ = 1 ˆ ∑ f ( u i ) i =1 • Consistency : Estimates converges to true answer as B increases • Unbiasedness : E [ˆ θ ] = θ • Minimal Variance
. B . Variance . . . . . . . . B f u du BE f u . B Consistency . . . . . . . . lim B Hyun Min Kang Biostatistics 615/815 - Lecture 16 March 14th, 2011 . B 9 / 18 . . . . . . . . . . Introduction Integration Importance sampling . Rejection sampling . Summary Analysis of crude Monte-Carlo method . Bias . . . . . . . B B . . . . . . . . . . θ ] = 1 E [ f ( u i )] = 1 E [ˆ ∑ ∑ θ = θ i =1 i =1
. . . . Variance . . . . . . . . B B Consistency B . . . . . . . . lim B Hyun Min Kang Biostatistics 615/815 - Lecture 16 March 14th, 2011 B 9 / 18 . Importance sampling Rejection sampling . Summary Analysis of crude Monte-Carlo method . Integration Introduction . . . . . Bias . . . . . B . . . . . B . . . . . . . . . . . . θ ] = 1 E [ f ( u i )] = 1 E [ˆ ∑ ∑ θ = θ i =1 i =1 ∫ 1 1 σ 2 ( f ( u ) − θ ) 2 du = 0 BE [ f ( u ) 2 ] − θ 2 1 =
. . . Variance . . . . . . . . B B Consistency B . . . . . . . . lim Hyun Min Kang Biostatistics 615/815 - Lecture 16 March 14th, 2011 B . B Bias . . . . . . . . . Introduction Integration Importance sampling . Rejection sampling . Summary Analysis of crude Monte-Carlo method . 9 / 18 . . B . . . . . . . . . . . . . . . . θ ] = 1 E [ f ( u i )] = 1 E [ˆ ∑ ∑ θ = θ i =1 i =1 ∫ 1 1 σ 2 ( f ( u ) − θ ) 2 du = 0 BE [ f ( u ) 2 ] − θ 2 1 = ˆ θ = θ B →∞
. . . . . . . . . . . . . . 5 Repeat step 3 and 4 for B times . . h Hyun Min Kang Biostatistics 615/815 - Lecture 16 March 14th, 2011 . . . . . . . . . . . . . Introduction Integration . Importance sampling Rejection sampling Accept-reject (or hit-and-miss) Monte Carlo method . . Algorithm . 10 / 18 Summary . . . . . . . . . . . 1 Define a rectangle R between (0 , 0) and (1 , 1) 2 Set h = 0 (hit), m = 0 (miss). 3 Sample a random point ( x , y ) ∈ R . 4 If f ( x ) < y , then increase h . Otherwise, increase m 6 ˆ θ = h + m | R | .
. . . . . h . Variance . . . . . . . . B Hyun Min Kang Biostatistics 615/815 - Lecture 16 March 14th, 2011 . . . . . . . . . . . . . Introduction Integration Importance sampling . Rejection sampling . Summary Analysis of accept-reject Monte Carlo method . Bias . 11 / 18 . . . . . . . . . . Let u i , v i follow U (0 , 1) . [ ] E [ˆ θ ] = E = θ h + m
. . . . . h . Variance . . . . . . . . B Hyun Min Kang Biostatistics 615/815 - Lecture 16 March 14th, 2011 . . . . . . . . . . . . . Introduction Integration Importance sampling . Rejection sampling . Summary Analysis of accept-reject Monte Carlo method . Bias . 11 / 18 . . . . . . . . . . Let u i , v i follow U (0 , 1) . [ ] E [ˆ θ ] = E = θ h + m σ 2 = θ (1 − θ )
. Summary March 14th, 2011 Biostatistics 615/815 - Lecture 16 Hyun Min Kang method The crude Monte-Carlo method has less variance then accept-rejection B B B B . crude Which method is better? 12 / 18 . Rejection sampling . . . Importance sampling . . . . . . . Introduction Integration . . . . . . . . . . BE [ f ( u ) 2 ] + θ 2 θ (1 − θ ) − 1 σ 2 AR − σ 2 = θ − E [ f ( u )] 2 = ∫ 1 1 = f ( u )(1 − f ( u )) du ≥ 0 0
. . March 14th, 2011 Biostatistics 615/815 - Lecture 16 Hyun Min Kang B B B B . Revisiting The Crude Monte Carlo Summary 13 / 18 Rejection sampling . . . . . . . . Introduction Integration . . Importance sampling . . . . . . . . . . ∫ 1 = E [ f ( u )] = f ( u ) du θ 0 1 ˆ ∑ = f ( u i ) θ i =1 More generally, when x has pdf p ( x ) , if x i is random variable following p ( x ) , ∫ = E p [ f ( x )] = f ( x ) p ( x ) dx θ 1 ˆ ∑ θ = f ( x i ) i =1
. Rejection sampling March 14th, 2011 Biostatistics 615/815 - Lecture 16 Hyun Min Kang B B . Importance sampling Summary . 14 / 18 . Integration . . . Introduction . . . . . . Importance sampling . . . . . . . . . . Let x i be random variable following pdf p ( x ) . [ f ( x ) ] ∫ ∫ f ( x ) θ = f ( x ) dx = p ( x ) g ( x ) du = E g p ( x ) 1 f ( x i ) ˆ ∑ θ = p ( x i ) i =1
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