L3 Monte-Carlo integration

Transcription

1 RNG MC IS Examination RNG MC IS Examination Monte-Carlo and Empirical Methods for Statistical Inference, FMS091/MASM11 Monte-Carlo integration Generating pseudo-random numbers: Rejection sampling Conditional methods N (0, 1) numbers L3 Monte-Carlo integration Monte-Carlo integration Basic Monte-Carlo integration Law of large numbers & Central limit theorem Importance sampling Examination Computer exercise 1 The first project David Bolin - bolin@maths.lth.se MC Based Statistical Methods, L3 1/22 Inversion and transformation methods We want random numbers X R from a continuous distribution F. Takex = F 1 (u), whereu U (0, 1). If X has a density f, thenthedensityofz = h(x) is f(g(z)) g (z), whereg = h 1. If are X and Y independent random variables with densities f x and f y,thenthedensityofz = X + Y is f z (z) = f x (z t)f y (t) dt. Rejection sampling David Bolin - bolin@maths.lth.se MC Based Statistical Methods, L3 2/22 Algorithm: Given a density g(x) such that Mf(x) Kg(x) do 1. Draw (z 1,...,z d ) from g(x). 2. Draw z d+1 from U (0, Kg(z 1,...,z d )). 3. If z d+1 > Mf(z 1,...,z d ) goto 1 (reject). 4. (z 1,...,z d ) is now a sample from f(x). The efficiency of the algorithm is given by M/K. David Bolin - bolin@maths.lth.se MC Based Statistical Methods, L3 3/22 David Bolin - bolin@maths.lth.se MC Based Statistical Methods, L3 4/22

2 Example: Tail of a N (0, 1) We want samples from the tail of a N (0, 1) distribution, X > Proposal distribution: χ(5). Hasthesametailbehavior as the Normal, exp( x 2 /2), andhasmode 5 1 = Mf(x) = 16 exp( x 2 /2) x 4 exp( x 2 /2) = Kg(x). 3. M = 16 2π(1 Φ(2)) and K = Γ(2.5) gives an acceptance rate of M/K accept=false; while accept==false %sample x from chi(5) x = sqrt(chi2rnd(5)); %sample U from U(0,K*g) u = rand(1).*x. 4.*exp(-x. 2/2); %acceptance threshold Mf = 16*exp(-x. 2/2); Mf(x<2) = 0; %test if point is accepted accept = u<mf; end David Bolin - bolin@maths.lth.se MC Based Statistical Methods, L3 5/22 Conditional methods If we can decompose a multivariate density into conditional parts, the problem of sampling from a multivariate density can be reduce to sampling from several univariate densities. f(x 1,...,x n ) = f(x n x n 1,...,x 1 )f(x n 1 x n 2,...,x 1 ) f(x 2 x 1 )f(x 1 ) The problem is that it may be hard to find the conditional densities, e.g. f(x 1 ) = f(x 1,...,x n ) dx 2 x n Example: Tail of a N (0, 1) N (0, 1) numbers acceptance rate 24.66% David Bolin - bolin@maths.lth.se MC Based Statistical Methods, L3 6/22 There are several ways of generating N (0, 1) numbers. Use CLT, x = ( n i=1 u i n/2) 12/n The Box-Müller transform x 1 = 2lnu 2 cos(2πu 1 ) x 2 = 2lnu 2 sin(2πu 1 ) The polar coordinate method. Draw u 1, u 2 uniform on the circle u u2 2 1 using rejection sampling, and take 2ln(u 2 1 x 1 = u + u2 2 ) 1 u u2 2 2ln(u 2 1 x 2 = u + u2 2 ) 2 u u2 2 George Marsaglia s ziggurat algorithm. David Bolin - bolin@maths.lth.se MC Based Statistical Methods, L3 7/22 David Bolin - bolin@maths.lth.se MC Based Statistical Methods, L3 8/22

3 Monte Carlo integration Many quantities of interest in statistics can be written as expectations Expectations: E(X) = xf(x) dx Probabilities: P(X > 2) = 2 f(x) dx = I(x > 2)f(x) dx = E(I(x > 2)). Standard integrals 2π sin(2π/(x π) 2 ) dx 0 =2π sin(2π/(x π) 2 )I(0 x 2π)/(2π) dx ) = 2πE U(0,2π) (sin(2π/(x π) 2 ). Law of Large Numbers Theorem: Assume X 1,...,X n is a sequence of i.i.d. random variables with mean E(X i ) = τ and variance V(x i ) = σ 2 <. If T n = n 1 n i=1 X i,thent n converges in probability to τ, i.e. for any ε > 0 P( T n τ > ε) 0asn Project 1: Show convergence in the mean squared sense, i.e. show that under the assumptions of the theorem it also holds that E((T n τ) 2 ) = σ2 n 0asn Monte Carlo integration Algorithm: David Bolin - bolin@maths.lth.se MC Based Statistical Methods, L3 9/22 1. Draw N values x 1,...,x N from f. 2. Approximate τ = E(φ(X)) with t N = 1 N N φ(x i ) We have an approximation of the expectation. The next question is: How good is the approximation? The rate of the error is O ( n 1/2),independentofthe dimension d. Example of error rates for deterministic methods are O ( n 2/d) (trapezoidal) and O ( n 4/d) (Simpson). i=1 Central Limit Theorem David Bolin - bolin@maths.lth.se MC Based Statistical Methods, L3 10/22 Monte Carlo integration also provides a simple way of obtaining confidence intervals of the approximation. Theorem: Assume X 1,...,X n is a sequence of i.i.d. random variables with means E(X i ) = τ and variances V(x i ) = σ 2.If T n = n 1 n i=1 X i,then ( ) n(tn τ) P x Φ(x) as n σ Or, for large enough n, T n is approximately Normal with mean E(T n ) = τ and variance V(T n ) = σ 2 /n, wheresigmacan be estimated from X i.thisallowsustoconstructconfidence intervals for the approximation. David Bolin - bolin@maths.lth.se MC Based Statistical Methods, L3 11/22 David Bolin - bolin@maths.lth.se MC Based Statistical Methods, L3 12/22

4 Bias Assume that we are interested in a function of the expectation. (e.g. h(e(x)) = E(X) 2 ). Anaturalestimatewouldbeh(E(X)) h(x). This estimate is no longer guaranteed to be unbiased, E(T n ) τ. Let τ = h(e(x)) and let T n = h(x). Theerrorcannowbe decomposed into variance and bias E((T n τ) 2 ) = V(T n ) + (E(T n ) τ) 2. Advantages of Monte Carlo integration More efficient than deterministic methods in high dimensions. If we can sample from a density, we don t need to know the normalizing constant to calculate integrals (rejections sampling). Handles strange integrands that causes problems for deterministic methods. 1 However the squared bias decreases as O ( n 2),while the variance decreases as O ( n 1). It can be shown that if X is asymptotically Normal with variance σ 2 /n then T n still is asymptotically Normal with variance σ 2 h (E(X)) 2 /n. David Bolin - bolin@maths.lth.se MC Based Statistical Methods, L3 13/ David Bolin - bolin@maths.lth.se MC Based Statistical Methods, L3 14/22 Problems with Monte Carlo integration Hard to sample from f(x). If we want to calculate E(φ(x)), whereφ(x) and f(x) are dissimilar, we might end of with a lot of samples where the integrand is small. Importance sampling can often solve these problems. Importance sampling The basis of importance sampling is to rewrite the integral as E f (φ(x)) = φ(x)f(x) dx = φ(x) f(x) g(x) dx g(x) ( ) φ(x)f(x) = E g = E g (ψ(x)) g(x) Construct the estimate T n as before and the variance is V(T n ) = 1 (ψ(x) τ) 2 g(x) dx. n Agoodchoiceforg(x) is one that makes ψ(x) close to constant. David Bolin - bolin@maths.lth.se MC Based Statistical Methods, L3 15/22 David Bolin - bolin@maths.lth.se MC Based Statistical Methods, L3 16/22

5 Example: Let X N(0, 1) and assume that we want to estimate P(X > 2) = f(x) dx = I(x > 2)f(x) dx = E(I(x > 2)) 2 MC integration: x = randn(1,n); phi = x>2; tn = mean(phi); Most of the samples of X are less than 2, whereweknow the integrand is zero! Alternatively we use that for large x, f(x) is similar to the χ(5) distribution. Importance sampling: y = sqrt(chi2rnd(5,1,n)); f = exp(-y. 2/2)/sqrt(2*pi); g = y. 4.*exp(-y. 2/2)/(gamma(2.5)*2 (1.5)); psi = (y>2).*f./g; tn2 = mean(psi); Importance sampling (cont.) David Bolin - bolin@maths.lth.se MC Based Statistical Methods, L3 17/22 An advantage with ordinary MC integration is that we don t need to know the normalizing constant of f(x), butnow both f(x) and g(x) are part of the expectation. Use MC integration to obtain the normalizing constant. E f (φ(x)) ( i φ(y i )Mf(y i ) g(y i ) ) / ( i Mf(x i ) g 2 (x i ) with y i samples from g and x i samples from g 2. ), Example (cont.) MC integration Importance sampling David Bolin - bolin@maths.lth.se MC Based Statistical Methods, L3 18/22 RNG MC IS Examination Computer exercise1 Project 1 Computer exercise 1; Random number generation & Monte Carlo-integration Answers to the following questions should be submitted no later than on the lecture Monday January 31. Note that the answers should be short (a couple of sentences only), and not constitute a complete report. 1. Compare the efficiency of the naïve method and the inversion method for generating from the geometric distribution, comments? 2. In task 2. above, which proposal distribution did you use, and what was the efficiency of the algorithm? 3. In task 2. above, what is the approximate 95% confidence interval for the probability P(X X2 3 2)? David Bolin - bolin@maths.lth.se MC Based Statistical Methods, L3 19/22 David Bolin - bolin@maths.lth.se MC Based Statistical Methods, L3 20/22

6 RNG MC IS Examination Computer exercise1 Project 1 Project 1; Simulation and Monte-Carlo integration The project: Asmallquestionontheory Importance Sampling Submission: AwrittenreportinPDF,PS,OpenOffice,oronPlain paper (No MS Word-files). An containing all your m-files. With a file proj1.m that runs your analysis. David Bolin - bolin@maths.lth.se MC Based Statistical Methods, L3 21/22 RNG MC IS Examination Computer exercise1 Project 1 Writing reports, some comments Introduce and/or explain notation; X =?. Describe/explain the model. The text should be readable without access to the Matlab code; write plain text instead of including the Matlab code in the report. Include your solutions in the text; Do not write calculations of? can be found in the matlab code, or similar. When referring to the lecture notes or the book, be specific (i.e. give Chapter/which lecture). Refer to your figures in the text. Explain colors etc. in the figure captions (A figure caption is almost never to long). Write clear motivations and reasonings, for any choices you have made. David Bolin - bolin@maths.lth.se MC Based Statistical Methods, L3 22/22