Monte Carlo Integration II & Sampling from PDFs

Transcription

1 Monte Carlo Integration II & Sampling from PDFs CS295, Spring 2017 Shuang Zhao Computer Science Department University of California, Irvine CS295, Spring 2017 Shuang Zhao 1

2 Last Lecture Direct illumination Monte Carlo integration I CS295, Spring 2017 Shuang Zhao 2

3 Today s Lecture Monte Carlo integration II Convergence properties Integrals over higher-dimensional domains Sampling from PDFs Inversion method CS295, Spring 2017 Shuang Zhao 3

4 Recap: Monte Carlo Integration Goal: to estimate Pick a probability density function Generate n independent samples: Evaluate for j = 1, 2,, n Return sample mean: CS295, Spring 2017 Shuang Zhao 4

5 Central Limit Theorem Let variables with be a sequence of i.i.d. random for all i Let, then It follows that for any z > 0: CDF of standard normal distrb. CS295, Spring 2017 Shuang Zhao 5

6 Central Limit Theorem CS295, Spring 2017 Shuang Zhao 6

7 Confidence Interval is called a confidence interval (CI) of μ Interpretation: there is a chance for to reside in this CI z is called the critical value Common critical values z = 1.960: 95%-CIs z = 2.576: 99%-CIs CS295, Spring 2017 Shuang Zhao 7

8 Confidence Interval is called a confidence interval (CI) of μ One more problem: the standard deviation is generally unknown Solution: use (corrected) sample standard deviation resulting in CIs of the form CS295, Spring 2017 Shuang Zhao 8

9 Convergence of MC Methods The relative size of a CI, namely is an important indicator for the reliability of estimated result S n Monte Carlo methods generally have a convergence rate of CS295, Spring 2017 Shuang Zhao 9

10 Today s Lecture Monte Carlo integration II Convergence properties Integrals over higher-dimensional domains Sampling from PDFs CS295, Spring 2017 Shuang Zhao 10

11 Background: Measures and Lebesgue Integration Γ is a set specifying the domain of integration μ is a measure function capturing the sizes of Γ and its (measurable) subsets Properties: CS295, Spring 2017 Shuang Zhao 11

12 Background: Measures and Lebesgue Integration Assuming Γ can be partitioned into n disjoint components Γ 1, Γ 2,, Γ n and f(x i ) a i for all x i in Γ i Then, Γ 1 Γ 2 Γ 3 Special case: Γ CS295, Spring 2017 Shuang Zhao 12

13 Background: Measures and Lebesgue Integration In general, Γ can be partitioned into many components Γ 1, Γ 2, so that f becomes (approximately) constant in each component Γ i In the limiting case, CS295, Spring 2017 Shuang Zhao 13

14 Background: Borel Measure and 1D Integrals The Borel measure captures the length of 1D intervals CS295, Spring 2017 Shuang Zhao 14

15 Background: Lebesgue Measure The Lebesgue measure captures the volume of hyper-cubes CS295, Spring 2017 Shuang Zhao 15

16 Background: Solid Angle Measure For a region A on the unit sphere, the solid angle measure captures A s surface area Unit sphere A Unit hemisphere 1 CS295, Spring 2017 Shuang Zhao 16

17 Background: Projected Solid Angle Measure For a region A on the unit sphere, the projected solid angle measure captures the surface area of A (which is A s projection on the plane perpendicular to the normal direction) Unit hemisphere A A CS295, Spring 2017 Shuang Zhao 17

18 Background: Probability Measures Given a sample space (i.e., all possible outcomes) and, denotes the probability for an outcome in to occur Example 1-1: fair coin Example 1-2: Poisson distribution (with parameter λ) CS295, Spring 2017 Shuang Zhao 18

19 Background: Probability Measures Given a sample space (i.e., all possible outcomes) and, denotes the probability for an outcome in to occur Example 2-1: U(0, 1] Example 2-2: N(0, 1) CS295, Spring 2017 Shuang Zhao 19

20 Background: Probability Measures Given a sample space (i.e., all possible outcomes) and, denotes the probability for an outcome in to occur Example 2-3: 1D continuous distrb. with CDF F There is a one-to-one correspondence between CDFs and probability measures CS295, Spring 2017 Shuang Zhao 20

21 Background: High-Dimensional Distributions Consider a sample space measure μ E.g., associated with a Given a probability density function p satisfying the corresponding probability measure P is with CS295, Spring 2017 Shuang Zhao 21

22 Background: General Definition of Expected Values Applies to both discrete and continuous cases Example 1: fair coin (discrete) Consider f(h) = 2, f(t) = -1 (i.e., you win 2 dollars for an H and lose 1 for a T) CS295, Spring 2017 Shuang Zhao 22

23 Background: General Definition of Expected Values Example 2: U(0, 1] (continuous) Consider f(x) = x for all x in (0, 1] CS295, Spring 2017 Shuang Zhao 23

24 Background: General Definition of Expected Values General continuous distribution with: Sample space Ω and associated measure μ Probability density p Classical definition of expected value CS295, Spring 2017 Shuang Zhao 24

25 Monte Carlo Integration over High-Dimensional Domains Let p be a pdf on Γ (with f(x) > 0 implying p(x) > 0) Then Exactly the same as the 1D case! CS295, Spring 2017 Shuang Zhao 25

26 Today s Lecture Monte Carlo integration II Convergence properties Integrals over higher-dimensional domains Sampling from PDFs CS295, Spring 2017 Shuang Zhao 26

27 Sampling from PDFs Monte Carlo integration requires drawing independent samples from arbitrary probability density p(x) What we generally have in practice: Generating random floating numbers from U(0, 1] CS295, Spring 2017 Shuang Zhao 27

28 Inversion Method Given a 1D distribution with CDF F, then follows this given distribution Proof: for all, CS295, Spring 2017 Shuang Zhao 28

29 Inversion Method Also works for discrete distributions with finite outcomes: consider one with CMF F Sampling algorithm Draw Return the minimal integer X satisfying Complexity: CS295, Spring 2017 Shuang Zhao 29

30 Next Lecture Sampling from PDFs II CS295, Spring 2017 Shuang Zhao 30