lecture 19: monte carlo methods

Transcription

1 lecture 19: monte carlo methods STAT 598z: Introduction to computing for statistics Vinayak Rao Department of Statistics, Purdue University April 3, 2019

2 Monte Carlo integration We want to calculate integrals/summations o ten expectations w.r.t. some probability distribution p(x) µ := E p [f] = f(x)p(x)dx X 1/16

3 Monte Carlo integration We want to calculate integrals/summations o ten expectations w.r.t. some probability distribution p(x) µ := E p [f] = f(x)p(x)dx X What is the prob. a game of patience (solitaire) is solvable? P(Solvable) = 1 Π 1(Π is solvable) Π 1/16

4 Monte Carlo integration We want to calculate integrals/summations o ten expectations w.r.t. some probability distribution p(x) µ := E p [f] = f(x)p(x)dx X If we drop 3 points on the plane, each Gaussian distributed, what is average the area of the resulting triangle? E[A] = A(x 1, x 2, x 3 )p(x 1 )p(x 2 )p(x 3 )dx 1 dx 2 dx 3 1/16

5 Monte Carlo integration We want to calculate integrals/summations o ten expectations w.r.t. some probability distribution p(x) µ := E p [f] = f(x)p(x)dx X For dataset (X, y), what is is average loss if you randomly choose a weight-vector according to some distribution (e.g. rnorm)? E w [L(X, y)] = (y w T X) 2 p(w)dw 1/16

6 Monte Carlo integration We want to calculate integrals: µ := E p [f] = X f(x)p(x)dx 2/16

7 Monte Carlo integration We want to calculate integrals: µ := E p [f] = X f(x)p(x)dx Sampling approximation: rather than visit all points in X, calculate a summation over a finite set. Monte Carlo approximation: Obtain points by sampling from p(x) x i p ˆµ 1 N N f(x i ) i=1 2/16

8 Monte Carlo integration Is this a good idea? Very simple Unbiased 3/16

9 Monte Carlo integration Is this a good idea? Very simple Unbiased If x i p, [ 1 E p [ˆµ] = E p N ] N f(x) = 1 N i=1 N E p [f] = µ i=1 Unbiased estimate 3/16

10 Monte Carlo integration Is this a good idea? Very simple Unbiased If x i p, [ 1 E p [ˆµ] = E p N ] N f(x) = 1 N i=1 N E p [f] = µ i=1 Unbiased estimate Var p [ˆµ] = 1 N Var p[f], Error = StdDev N 1/2 3/16

11 Monte Carlo integration Is this a good idea? In low-dims, use numerical methods like quadrature In high-dims, numerical methods become infeasible 4/16

12 Monte Carlo integration Is this a good idea? In low-dims, use numerical methods like quadrature In high-dims, numerical methods become infeasible E.g. Simpson s rule in d-dimensions, with N grid points: error N 4/d 4/16

13 Monte Carlo integration Is this a good idea? In low-dims, use numerical methods like quadrature In high-dims, numerical methods become infeasible E.g. Simpson s rule in d-dimensions, with N grid points: Monte Carlo integration: error N 4/d error N 1/2 4/16

14 Monte Carlo integration Is this a good idea? In low-dims, use numerical methods like quadrature In high-dims, numerical methods become infeasible E.g. Simpson s rule in d-dimensions, with N grid points: Monte Carlo integration: error N 4/d error N 1/2 Independent of dimensionality! 4/16

15 Generating random variables The simplest useful probability distribution Unif(0, 1). In theory, can be used to generate any other RV. Easy to generate uniform RVs on a deterministic computer? 5/16

16 Generating random variables The simplest useful probability distribution Unif(0, 1). In theory, can be used to generate any other RV. Easy to generate uniform RVs on a deterministic computer? No! 5/16

17 Generating random variables The simplest useful probability distribution Unif(0, 1). In theory, can be used to generate any other RV. Easy to generate uniform RVs on a deterministic computer? No! Instead: pseudorandom numbers. 5/16

18 Generating random variables The simplest useful probability distribution Unif(0, 1). In theory, can be used to generate any other RV. Easy to generate uniform RVs on a deterministic computer? No! Instead: pseudorandom numbers. Map a seed to a random-looking sequence. 5/16

19 Generating random variables The simplest useful probability distribution Unif(0, 1). In theory, can be used to generate any other RV. Easy to generate uniform RVs on a deterministic computer? No! Instead: pseudorandom numbers. Map a seed to a random-looking sequence. Downside: 5/16

20 Generating random variables The simplest useful probability distribution Unif(0, 1). In theory, can be used to generate any other RV. Easy to generate uniform RVs on a deterministic computer? No! Instead: pseudorandom numbers. Map a seed to a random-looking sequence. Downside: Upside: Can use seeds for reproducibility or debugging set.seed(1) 5/16

21 Generating random variables The simplest useful probability distribution Unif(0, 1). In theory, can be used to generate any other RV. Easy to generate uniform RVs on a deterministic computer? No! Instead: pseudorandom numbers. Map a seed to a random-looking sequence. Downside: Upside: Can use seeds for reproducibility or debugging set.seed(1) Careful with batch/parallel processing. 5/16

22 Generating random variables In R we generate uniform random variables via runif Additional functions include rnorm(), rgamma(), rexp(), sample() etc. 6/16

23 Generating random variables In R we generate uniform random variables via runif Additional functions include rnorm(), rgamma(), rexp(), sample() etc. In theory, we can generate any other random variable by transforming a uniform (or any other) random variable: u Unif(0, 1), x = f(u) 6/16

24 Generating random variables In R we generate uniform random variables via runif Additional functions include rnorm(), rgamma(), rexp(), sample() etc. In theory, we can generate any other random variable by transforming a uniform (or any other) random variable: u Unif(0, 1), x = f(u) In practice, finding this f is too hard. Need other approaches. There is a whole subfield of statistics addressing this. 6/16

25 Examples of Monte Carlo sampling x, y are a pair of dice. What is p(x, y)? 7/16

26 Examples of Monte Carlo sampling x, y are a pair of dice. What is p(x, y)? What is E[min(x, y)] = 6 6 i=1 j=1 min(x, y)p(x, y) 7/16

27 Examples of Monte Carlo sampling x, y are a pair of dice. What is p(x, y)? What is E[min(x, y)] = 6 6 i=1 j=1 min(x, y)p(x, y) Don t really need Monte Carlo here (but what if we had a 100 dice?), but let s try it anyway. How? 7/16

28 Examples of Monte Carlo sampling x, y are a pair of dice. What is p(x, y)? What is E[min(x, y)] = 6 6 i=1 j=1 min(x, y)p(x, y) Don t really need Monte Carlo here (but what if we had a 100 dice?), but let s try it anyway. How? Roll a pair of dice N times. Call the ith outcome (x i, y i ). Then E[min(x, y)] 1 N N min(x i, y i ) i=1 7/16

29 A (bad) way of estimating the area of a circle C Let C be the unit disc, i.e. all points (x, y) with x 2 + y 2 1. Its area is A(C) = = x,y C 0 0 dxdy δ C ((x, y)) dxdy Here δ C ((x, y)) = 1 if x 2 + y 2 1, else it equals 0 8/16

30 A (bad) way of estimating the area of a circle C Let C be the unit disc, i.e. all points (x, y) with x 2 + y 2 1. Its area is A(C) = = x,y C 0 0 dxdy δ C ((x, y)) dxdy Here δ C ((x, y)) = 1 if x 2 + y 2 1, else it equals 0 How might we try to approximate this using Monte Carlo? What is the f and p? 8/16

31 A (bad) way of estimating the area of a circle One way: choose some probability distribution p(x, y) (e.g. both x and y are Gaussian distributed) Then: A(C) = = N N i=1 0 δ C ((x, y)) dxdy δ C ((x, y)) p(x, y)dxdy p(x, y) δ C ((x i, y i )) p(x i, y i ) (Monte Carlo, with (x i, y i ) p) 9/16

32 A (bad) way of estimating the area of a circle One way: choose some probability distribution p(x, y) (e.g. both x and y are Gaussian distributed) Then: A(C) = = N N i=1 0 δ C ((x, y)) dxdy δ C ((x, y)) p(x, y)dxdy p(x, y) δ C ((x i, y i )) p(x i, y i ) (Monte Carlo, with (x i, y i ) p) In words, sample N points (x i, y i ) from some distribution p, and plug them into the last equation above 9/16

33 A (bad) way of estimating the area of a circle N < # Number of Monte Carlo simulations x <- rnorm(n); y <- rnorm(n) # Sample N Gaussian pairs (x,y) px <- dnorm(x); py <- dnorm(y) pp <- px * py # Calculate their probabiliy dd <- sqrt(x^2+y^2) mc_est <- 1/N * sum(1/pp[dd<1]) # Monte Carlo estimate 10/16

34 Rare event simulation: Let X = (x 1,..., x 100 ) be a hundred dice. What is p( 100 d=1 x d 450)? 11/16

35 Rare event simulation: Let X = (x 1,..., x 100 ) be a hundred dice. What is p(sum(x) 450)? (where Sum(X) = 100 d=1 x d) 12/16

36 Rare event simulation: Let X = (x 1,..., x 100 ) be a hundred dice. What is p(sum(x) 450)? (where Sum(X) = 100 d=1 x d) p(sum(x) 450) = δ(sum(x) 450)p(X) = E p [δ(sum(x) 450)] δ( ) is the indicator function δ(condition) = 1 if condition is true, else 0. 12/16

37 Naive Monte Carlo sampling Propose from p(x) Calculate 1 N N i=1 δ(sum(x i)) 13/16

38 Naive Monte Carlo sampling Propose from p(x) Calculate 1 N N i=1 δ(sum(x i)) Most δ(sum(x i )) terms will be 0 High variance 13/16

39 Importance sampling Importance sampling: assigns importance weights to samples. 14/16

40 Importance sampling Importance sampling: assigns importance weights to samples. Scheme: Draw a proposal x i from q( ) Assign it a weight w i = p(x i )/q(x i ) 14/16

41 Importance sampling Importance sampling: assigns importance weights to samples. Scheme: Draw a proposal x i from q( ) Assign it a weight w i = p(x i )/q(x i ) µ = X f(x)p(x)dx = X f(x) p(x) q(x) q(x)dx 1 N N w i f(x i ) := µ imp i=1 14/16

42 Importance Sampling (contd) When does this make sense? Sometimes it s easier to simulate from q(x) than p(x) 15/16

43 Importance Sampling (contd) When does this make sense? Sometimes it s easier to simulate from q(x) than p(x) Sometimes it s better to simulate from q(x) than p(x)! To reduce variance. E.g. rare event simulation. 15/16

44 Importance sampling For 100 dice, what is p(sum > 450)? A better choice might be to bias the dice. E.g. q(x i = v) v (for v {1,... 6}) 16/16

45 Importance sampling For 100 dice, what is p(sum > 450)? A better choice might be to bias the dice. E.g. q(x i = v) v (for v {1,... 6}) Propose from q(x) Calculate weights w(x i ) = p(x i )/q(x i ) Calculate 1 N N i=1 w(x i)δ(sum(x i )) 16/16

46 Importance sampling For 100 dice, what is p(sum > 450)? A better choice might be to bias the dice. E.g. q(x i = v) v (for v {1,... 6}) Propose from q(x) Calculate weights w(x i ) = p(x i )/q(x i ) Calculate 1 N N i=1 w(x i)δ(sum(x i )) Gives a better estimate of p(sum(x) 500) = δ(sum(x) 500)p(X) 16/16