MONTE CARLO METHODS. Hedibert Freitas Lopes

Transcription

1 MONTE CARLO METHODS Hedibert Freitas Lopes The University of Chicago Booth School of Business 5807 South Woodlawn Avenue, Chicago, IL

2 1 A bit of Outline

3 in the 40s and 50s Stan Ulam soon realized that computers could be used in this fashion to answer questions of neutron diffusion and mathematical physics; He contacted John Von Neumann and they developed many algorithms (importance sampling, rejection sampling, etc); In the 1940s Nick Metropolis and Klari Von Neumann designed new controls for the state-of-the-art computer (ENIAC); Metropolis and Ulam (1949) The. Journal of the American Statistical Association. Metropolis et al. (1953) Equations of state calculations by fast computing machines. Journal of Chemical Physics.

4 We introduce several () for integrating and/or sampling from nontrivial densities. Simple via importance sampling (IS) sampling Sampling importance resampling (SIR) Iterative sampling Metropolis-Hastings algorithms Simulated annealing Gibbs sampler Based on the book by Gamerman and Lopes (1996).

5 A few references (Geweke, 1989) (Gilks and Wild, 1992) SIR (Smith and Gelfand, 1992) Metropolis-Hastings algorithm (Hastings, 1970) Simulated annealing (Metropolis et al., 1953) Gibbs sampler (Gelfand and Smith, 1990)

6 Two main tasks 1 Compute high dimensional integrals: E π [h(θ)] = h(θ)π(θ)dθ 2 Obtain a sample {θ 1,..., θ n } from π(θ) when only a sample { θ 1,..., θ m } from q(θ) is available. q(θ) is known as the proposal/auxiliary density.

7 Bayes via appear frequently, but not exclusively, in modern Bayesian statistics. Posterior and predictive densities are hard to sample from: Posterior : π(θ) = f (x θ)p(θ) f (x) Predictive : f (x) = f (x θ)p(θ)dθ Other important integrals and/or functionals of the posterior and predictive densities are: Posterior modes: max θ π(θ); Posterior moments: E π [g(θ)]; Density estimation: ˆπ(g(θ)); Bayes factors: f (x M 0 )/f (x M 1 ); Decision: max d U(d, θ)π(θ)dθ.

8 The objective here is to compute moments E π [h(θ)] = h(θ)π(θ)dθ If θ 1,..., θ n is a random sample from π( ) then h mc = 1 n h(θ i ) E π [h(θ)] as n. n i=1 If, additionally, E π [h 2 (θ)] <, then V π [ h mc ] = 1 {h(θ) E π [h(θ)]} 2 π(θ)dθ n and v mc = 1 n 2 n (h(θ i ) h mc ) 2 V π [ h mc ] as n. i=1

9 The objective here is to compute 1 p = 1 0 Example i. [cos(50θ) + sin(20θ)] 2 dθ by noticing that the above integral can be rewritten as E π [h(θ)] = h(θ)π(θ)dθ where h(θ) = [cos(50θ) + sin(20θ)] 2 and π(θ) = 1 is the density of a U(0, 1). Therefore ˆp = 1 n n h(θ i ) i=1 where θ 1,..., θ n are i.i.d. from U(0, 1). 1 True value is

10 h(θ) θ p Sample size (log10)

11 The objective is still the same, ie to compute E π [h(θ)] = h(θ)π(θ)dθ by noticing that h(θ)π(θ) E π [h(θ)] = q(θ)dθ q(θ) where q( ) is an importance function.

12 If θ 1,..., θ n is a random sample from q( ) then as n. h is = 1 n Ideally, q( ) should be n i=1 h(θ i )π(θ i ) q(θ i ) As close as possible to h( )π( ), and Easy to sample from. E π [h(θ)]

13 The objective here is to estimate p = Pr(θ > 2) = 2 Example ii. 1 π(1 + θ 2 dθ = ) where θ is a standard Cauchy random variable. A natural estimator of p is ˆp 1 = 1 n n I {θ i (2, )} i=1 where θ 1,..., θ n Cauchy(0,1).

14 A more elaborated estimator based on a change of variables from θ to u = 1/θ is ˆp 2 = 1 n n i=1 where u 1,..., u n U(0, 1/2). u 2 i 2π[1 + u 2 i ]

15 The true value is p = n ˆp 1 ˆp 2 v 1/2 1 v 1/ With only n = 1000 draws, ˆp 2 has roughly the same precision that ˆp 1, which is based on 1000n draws, ie. three orders of magnitude.

16 The objective is to draw from a target density π(θ) = c π π(θ) when only draws from an auxiliary density q(θ) = c q q(θ) is available, for normalizing constants c π and c q. If there exist a constant A < such that 0 π(θ) 1 for all θ A q(θ) then q(θ) becomes a blanketing density or an envelope and A the envelope constant.

17 Blanket distribution pi (target) q (proposal) Aq

18 Bad draw pi (target) q (proposal) Aq pi/aq=

19 Good draw pi (target) q (proposal) Aq pi/aq=

20 Acceptance probability Acceptance probability θ

21 Drawing from π(θ). 1 Draw θ from q( ); 2 Draw u from U(0, 1); 3 Accept θ if u π(θ ) A q(θ ) ; Algorithm 4 Repeat 1, 2 and 3 until n draws are accepted. Normalizing constants c π and c q are not needed. The theoretical acceptance rate is cq Ac π. The smaller the A, the larger the acceptance rate.

22 Enveloping the standard normal density π(θ) = 1 2π exp{ 0.5θ 2 } Example iii. by a Cauchy density q C (θ) = 1/(π(1 + θ 2 )), or a uniform density q U (θ) = 0.05 for θ ( 10, 10). Bad proposal:the maximum of π(θ)/q U (θ) is roughly A U = 7.98 for θ ( 10, 10). The theoretical acceptance rate is 12.53%. Good proposal: The max of π(θ)/q C (θ) is equal to A C = 2π/e The theoretical acceptance rate is 65.35%.

23 UNIFORM PROPOSAL CAUCHY PROPOSAL Empirical rates: (Uniform) and (Cauchy) Theoretical rates: (Uniform) and (Cauchy)

24 No need to rely on the existance of A! Algorithm 1 Draw θ1,..., θ n from q( ) 2 Compute (unnormalized) weights ω i = π(θi )/q(θi ) i = 1,..., n 3 Sample θ from {θ1,..., θ n} such that Pr(θ = θi ) ω i i = 1,..., n. 4 Repeat m times step 3. Rule of thumb: n/m = 20. Ideally, ω i = 1/n and Var(ω) = 0.

25 UNIFORM PROPOSAL Example iii. revisited CAUCHY PROPOSAL Fraction of redraws: (Uniform) and (Cauchy) Variance of weights: (Uniform) and (Cauchy)

26 Example iv. Assume that we are interested in sampling from π(θ) = α 1 p N (θ; µ 1, Σ 1 ) + α 2 p N (θ; µ 2, Σ 2 ) + α 3 p N (θ; µ 3, Σ 3 ) where p N ( ; µ, Σ) is the density of a bivariate normal distribution with mean vector µ and covariance matrix Σ. The mean vectors are µ 1 = (1, 4) µ 2 = (4, 2) µ 3 = (6.5, 2), the covariance matrices are ( ) Σ 1 = and Σ = Σ 3 = ( ), and weights α 1 = α 2 = α 3 = 1/3.

27 Target π(θ) θ θ 1

28 Target π(θ) f(theta) theta2 5 theta

29 θ 2 Proposal q(θ) θ 1 q(θ) N(µ, Σ) where µ 2 = (4, 2) and Σ = 9 ( )

30 θ 1 θ θ 1 θ 2 Acceptance rate: 9.91% of n = 10, 000 draws.

31 θ 1 θ θ 1 θ 2 Fraction of redraws: 29.45% of (n = 10, 000, m = 2, 000).

32 & SIR θ 1 θ θ 1 θ 2

33 Example v. Let us now assume that π(θ) = α 1 p N (θ; µ 1, Σ 1 ) + α 3 p N (θ; µ 3, Σ 3 ) where mean vectors are the covariance matrices are ( ) Σ 1 = µ 1 = (1, 4) µ 3 = (6.5, 2), and Σ 3 = and weights α 1 = 1/3 and α 3 = 2/3. ( ),

34 Target π(θ) θ θ 1

35 Target π(θ) f(theta) theta2 5 theta

36 Proposal q(θ) θ θ 1

37 θ 1 θ θ 1 θ 2 Acceptance rate: 10.1% of n = 10, 000 draws.

38 θ 1 θ θ 1 θ 2 Fraction of redraws: 37.15% of (n = 10, 000, m = 2, 000).

39 & SIR θ 1 θ θ 1 θ 2

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