Monte Carlo Integration

Transcription

1 Monte Carlo Integration SCX5005 Simulação de Sistemas Complexos II Marcelo S. Lauretto Reference: Robert CP, Casella G. Introducing Monte Carlo Methods with R. Springer, Chapter 3

2 Introduction I General idea: Monte Carlo Methods are numerical methods to solve mathematical problems through simulation of random variables. Origin: Stanislav Ulam, John von Neumann: considered the main creators in the late 1940s The Manhattan Project: atomic weapons Nickname inspired in the Monte Carlo casino Seminal paper: Metropolis & Ulam, 1949: The Monte Carlo method

3 Introduction II Two major numerical classes of problems in statistical inference: Optimization Integration It is not always possible to analitically compute the estimators associated with a given paradigm (maximum likelihood, Bayes, method of moments etc.) Therefore, numerical solutions are often needed

4 The laws of large numbers I

5 The laws of large numbers II

6 The laws of large numbers III

7 The laws of large numbers IV

8 The laws of large numbers V

9 The laws of large numbers VI

10 The Central Limit Theorem I Central Limit Theorem: If the random variables X 1,..., X n form a random sample of size n from a given distribution (discrete or continuous) with finite mean µ and finite variance σ 2, then for each fixed number x, lim P r n [ Xn µ σ/ n x ] = φ(x), where X = ( n i=1 X i) /n (the sample mean) and φ denotes the c.d.f. of the standard normal distribution, N(0, 1). Interpretation: If a large random sample is taken from any distribution with mean µ and variance σ 2, then the distribution of X will be approximately the normal distribution with mean µ and standard deviation σ/ n.

11 Classical Monte Carlo integration I Here, we consider the generic problem of evaluating the integral E f [h(x)] = h(x)f(x)dx, (1) where f is a probability density and χ denotes the set where the random variable X takes its values Principle of the Monte Carlo method for approximating (1): generate a sample (x 1..., x n ) from the density f and propose as an approximation the empirical average h n = 1 n h(x j ), n χ j=1 since h n converges almost surely to E f [h(x)] by the strong law of the large numbers

12 Classical Monte Carlo integration II Moreover, when h 2 (X) has a finite expectation under f, the speed of convergence of h n can be assessed since the asymptotic variance of the approximation is var(h n ) = 1 (h(x) E f [h(x)]) 2 f(x)dx, n χ which can also be estimated from the sample (x 1..., x n ) through v n = 1 n ( ) 2 h(xj n 2 ) h n j=1 Thus, due to the Central Limit Theorem, for large n, h n E f [h(x)] vn is approximately distributed as a N(0, 1) variable. v n : standard error of h n

13 Classical Monte Carlo integration III A method for determining when to stop generating new data: 1 Choose an acceptable upper bound ε 0.01 for the standard error v n 2 Generate at least 100 data values 3 Continue to generate additional data values (e.g. adding 100 data values in each step), stopping when you have generated a total of n values such that v n < ε 4 The estimate of E f [h(x)] is given by h n

14 Classical Monte Carlo integration IV Example: processing cost for a job batch: A company hires computing services from a cloud computing company for a special kind of processing job The time t (in hours) to perform each job has an exponential distribution with the rate parameter λ Notation: t Exp(λ) pdf: f(t λ) = λ exp ( λt) λ = number of jobs processed per hour average time processing = E[t] = 1/λ The price (in cents) per processed job is c(t) = a + b.t Problem: Assuming λ = 0.2, a = 10, b = 0.5: What is the expected total cost to process a batch of k = 10, 000 jobs? What is the 99th percentile for the total cost?

15 Classical Monte Carlo integration V Answer: The function to be integrated is E f [h(x)] = h(x)f(x)dx where x = (t 1,..., t k ), h(x) = k i=1 c(t i) = k.a + b. k i=1 t i and f(x) = ( k i=1 λ exp ( λt i) = λ k exp λ ) k i=1 t i Procedure: 1 for j = 1,..., n: (n =number of simulation replicates) 1.1 Generate x j = (t 1,..., t k ) where t i Exp(λ) 1.2 Compute h j = h(x j ) 2 Compute h n = 1/n n j=1 h j 3 Compute the 99th percentile: Sort the values h 1,..., h n, obtaining the ordering h (1) h (2)... h (n) ; take the value h ( 0.99n ), i.e. the 0.99n th largest value of h(x). χ

16 Classical Monte Carlo integration VI Confidence interval for h(x) h n (using the CLT): Compute the standard error v n for h n : v n = 1 n 2 n j=1 ( h(xj ) h n ) 2 CI(95%) for h n : [h n 1.96v n, h n v n ]