An Introduction to Monte Carlo

Transcription

1 1/28 An Introduction to Monte Carlo Joshua Lande Stanford February 23, 2011

2 Monte Carlo - Only SASS Topic to Win an Academy Award? 2/28 As part of the 73rd Scientific and Technical Academy Awards ceremony presentation on March 3, 2001, The Academy of Motion Picture Arts and Sciences Board of Governors honored Ed Catmull, Loren Carpenter, and Rob Cook, with an Academy Award of Merit (Oscar) for significant advancements to the field of motion picture rendering as exemplified in Pixars RenderMan. This was the first Oscar awarded to a software package for its outstanding contributions to the field.

3 3/28 Motivation for Monte Carlo Example, What is the expected distance between two points distributed randomly on a rectangle There is a simple integral to determine this d = 1 (dx1 (ab) 2 dx 2 ) 2 + (dy 1 dy 2 ) 2 )dx 1 dy 1 dx 2 dy 2 There is acutally an analytic solution: d = 1 15 [ ) a 3 (3 b 2 + b3 a 2 + d a2 b 2 b2 a ( b 2 2 a log a + b d Where d = a 2 + b 2 + a2 b log b + d )] a

4 4/28 Average Distance Between Points If a = b = 1, then d = With Monte Carlo, can calculate with 3 lines of code: >>> from pylab import >>> x1, x2, y1, y2=rand (4,1 e6 ) >>> average ( s q r t ( ( x2 x1 ) 2+(y2 y1 ) 2 ) ) Impressively, we can calculate a 1σ error: >>> std ( s q r t ( ( x2 x1 ) 2+(y2 y1 ) 2 ) ) / s q r t (1 e6 ) My value is off by 0.5σ.

5 5/28 Monte Carlo Benefits The analytic solution is fragile to symmetries that don t exist in the real world What if there is a hole somewhere in the box... In real life, there is always measurement error and model error which is probably larger than any introduced statistical error. The Monte Carlo solution is dead simple to calculate and generalize.

6 6/28 Another Cute Example Want to estimate π a fraction π/4 will fall within a circle enscribed in the box. Formally, we want to solve the integral π = ( ) δ x2 + y 2 < 1 dxdy Monte-Carlo_method_pi.svg Analytic solution to the integral won t give you a number easy to solve with Monte Carlo >>> x, y=rand (2,1 e6 ) >>> 4 sum( s q r t ( x 2+y 2 <1))/1 e

7 7/28 Lots of Monte Carlo Applications Learn about a system by random sampling from it The Laws of physics are probabilistic, physics models inherently requires Monte Carlo sampling. Real life simulations are always high dimensional A particle detector s performance might be a function of energy, incident angle, incident momentum, conversion location,... Trying to calculate something like an expectation value using a straightforward integral over all all the dimensions of a model would be hopeless Testing out analysis code for bugs and errors Blind analysis Test applicability of statistical theorems

8 8/28 Primary Uses of Monte Carlo Numerically integrating functions b a f(x)dx 1 n n f(x i ) As with all experimental data, convergence goes as O(1/ n) There are better one dimensional integration algorithms, but they all grow exponentially slower as the dimesnions of the integral grow On the other hand, the O(1/ n) behavior is independent of dimensions! As d, Monte Carlo will always be the best integration method A central limit theorem can be used to estimate errors on integral i=1

9 9/28 Random vs. PseudoRandom One could imagine all sorts of methods to generate truly random numbers Flip a coin Counts from radioactive decay Measure spin True random numbers are almost never used Measuring devices can introduce correlations into random measurements Random numbers are incompressible slow to generate hard to store

10 10/28 Aside

11 Amazon reviews are funny The bulk of each page seems random enough. However at the lower left and lower right of alternate pages, the number is found to increment directly. I took a class in statistics in college. I used this book to help me select random phone numbers for a poll I was conducting for my class project... One of those phone calls was answered by the woman who is now my wife. We ve been happily married for ten years! Thank you, RAND. Sure, you think you can just open up your spreadsheet and generate random numbers at will. But did you know that those are only pseudo-random numbers?... While that may be good enough for everyday use, on those special occasions when only the real thing will do, this is the book you need. Truly random digits, and in copious quantities. This is the mother load of true randomness on earth. I wrote a report on this book last semester. My roommate failed to tell me he had the Cliffs! Can you believe that? Thanks, Brad! 11/28

12 12/28 Generating PseudoRandom Numbers Typically use computer algorithms which approximate generating random numbers Easy to store Results completely reproducible Require only an input seed One of the most famous PRNG is X n+1 = (ax n + b) mod m As far as I can tell, simpler PRNGs can have issues with repeated patters, correlations, and measures of randomness...

13 Mersenne Twister Most modern languages (python, matlab, R) use the Mersenn Twister From Wikipedia 1 It has a very long period of While a long period is not a guarantee of quality in a random number generator, short periods (such as the 232 common in many software packages) can be problematic It is k-distributed to 32-bit accuracy for every 1 k 623 It passes numerous tests for statistical randomness, including the Diehard tests. It passes most, but not all, of the even more stringent TestU01 Crush randomness tests When doing a Monte Carlo, floating point rounding errors should probably be a bigger concern than RNG errors /28

14 14/28 Inverse Transform Sampling Can sample from U(0 x 1) Want to sample from any arbitrary probability distribution f(x) Try Inverse Transform Sampling Let F be the CDF F (x) = x F (y)dy Then y = F 1 (u) F Proof: P r(y x) = P r(f 1 (u) x) = P r(u F (x)) = F (x) Since P r(y x) = F (x), y is distributed as f

15 Inverse Transform Sampling Qualitativly, when f(x) is large, CDF increases rapidly. A bigger range of u which will be mapped to a given x More likely to pick that x sampling_method/ 15/28

16 Examples: Uniform and Exponential Sample Uniformly between a and b { f(x) = 1 1 a x b b a 0 x < a or x > b 0 x a x a F (x) = b a a x b 1 x b F 1 (p) = a + (b a)p for 0 p 1 So if u U(0, 1), then x = a + (b a)u U(a, b) To sample from an exponential distribution: f(x) = λe λx F (x) = 1 e λx F 1 (x) = ln(1 x)/λ So x = ln(1 u)/λ or x = ln u/λ will have an exponential distribution 16/28

17 17/28 Important Reference in the Field There is a great book with lots of standard distributions The book can be viewed free online rnbookindex.html Preface to the Web Edition... I have asked Springer to print more copies, but they flatly refused, unless I was willing to publish a second edition with them in the near future. Burnt once, why would I trust them with a second edition? Also, I figured that since Springer had gross income about 500,000 US dollars from my books with them, that they would be more generous with their royalties and more responsive to demands for second printings. The contrary is true in fact: royalties are decreasing (they stand now at 7.5% per book)

18 18/28 One More: Gaussian Distribution How to sample from a Gaussian distribution N(0, 1) f(x) = 1 2π e t2 /2 dt F (x) = 1 x e t2 /2 dt = 1 2π 2 F 1 (x) probit(p) = 2erf 1 (2p 1) [ 1 + erf(x/ ] (2)) Although erf 1 can t be written in terms of simpler functions, it has a well defined taylor expansion erf 1 (z) = 1 π (z + π ) 2 12 z3 + 7π2 480 z π z π z Which is good enough for a computer

19 19/28 Box-Muller Transform For special distributions, there are faster algorithms A pair of indepnedent Gaussian varaibles z 1 and z 2 can be generated using the Box-Muller transform z 1 = r cos θ = 2 ln u 1 cos(2πu 2 ) (1) z 2 = r sin θ = 2 ln u 1 sin(2πu 2 ) (2) Motivation: In a Two-dimensional Cartesian system with x and y described by two independent and normally distributed random variables, r 2 = x 2 + y 2 and θ are also independent r 2 has a chi squared distribution with two degrees of freedom which is an exponential distribution θ is just uniform between 0 and 2π So we can generate r 2 = 2 ln u 1 and θ = 2πu 2

20 20/28 Box-Muller (cont) >>> u1, u2=rand (2,1 e6 ) >>> data=append ( s q r t ( 2 l o g ( u1 ) ) cos (2 pi u2 ), s q r t ( 2 l o g ( u1 ) ) s i n (2 pi u2 ) )... >>> h i s t ( data, normed=true, bins =100, h i s t t y p e = step ) >>> s a v e f i g ( p l o t. pdf )

21 21/28 Acceptance Rejection Sampling Suppose we want to sample from a distribution f(x). Suppose that we have a distribution g(x) such that f(x) ag(x). Acceptance rejection samples from f(x) by sampling some point x f(x) (3) A second uniform data point u U(0, 1) is generated and the data point x is kept only if u < f(x)/ag(x) (4) the probability of a point being accepted is P r(accept) = 1/a. Nice if g(x) is easy to sample from. Only have to evaluate f(x)

22 Graphically Acceptance Rejection sampling is sampling points from the top curve and keeping them only if they are below the lower curve. Try to pick a function g(x) which squeezes f(x) tightly. If f(x) is very costly to evaluate, can modify algorithm with a lower bounding function h(x) so that f(x) is evaluated only if u > h(x)/ag(x) 22/28

23 23/28 Sampling from a Multidimensional Distribution Sampling a scaler from a probability distribution is basically a solved problem Sampling from a multidimensional distribution is still generally unsolved and an active field of research Of course, if the variables are independent, then it is easy. But hard to get the correlations Curse of Dimensionality

24 Multivariate Inverse Transform Sampling We can generally write out the CDF conditionally F X1,X 2,...,X d (x 1, x 2,..., x d ) = F X1 (x 1 )F X2 X 1 (x 2 x 1 )... F Xd X 1X 2...X d 1 (x d x 1, x 2,..., x d 1) And then given d random uniform variables u 1, u 2,..., u d x 1 = F 1 X 1 (u 1 ) x 2 = F 1 X 2 X 1 (u 2 ) x d = F 1 X d X 1X 2...X d 1 (u d) Doable in simple situations, calculating the conditional inverse CDF can be horrible Curse of Dimensionality! 24/28

25 25/28 Acceptance Rejection in Multiple Dimensions Acceptance Rejection generalizes readily to many dimensions Exceptionally hard to find a function which cover s your function tightly in higher dimensional Most data points end up being rejected Curse of Dimensionality!

26 26/28 Special Case: Multivariate Gaussian Distribution Want to sample from a Multivariate Gaussian Distribution with mean µ and covariance Σ: f(x) = 1 (2π) k/2 Σ exp ( 1 1/2 2 (x µ)t Σ 1 (x µ) ) one can simulate a vector using the eigenvalue decomposition of the covariance matrix Σ = DΛD T If y N p (0, 1) (p independent normal variables), then X = Dλ 1/2 y + µ N(µ, Σ) has the desired Multivariate Gaussian Distribution

27 27/28 Markov Chain Monte Carlo (MCMC) Want to sample from a high dimensional probability distribution P (x) Sometimes Multivariate Inverse Transform Sampling doesn t work Acceptance Rejection doesn t really work MCMC is an attempt to solve this problem. Fundamental insight is to generate data points which are not independent. Use a Markov Process to generate one point from the previous Markov chain that has the desired distribution as its equilibrium distribution State of the chain after many steps is used to sample the distribution

28 MetropolisHastings Algorithm Want to sample from P (x ) Need rule to move from x t to x t+1 Given a proposal density Q(x ; x t ) Pick a place to go x with probability Q(x ; x t ). Pick α U(0, 1). If then x t+1 = x Otherwise, x t+1 = x t α < P (x )Q(x t ; x ) P (x t )Q(x ; x t ) Overal normalization of P (x) not needed Generally hard to pick Q Hard to know when you have well sampeled probability space I think MCMC is a huge field of research 28/28