Approximate Bayesian Computation

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1 Approximate Bayesian Computation Jessi Cisewski Department of Statistics Yale University October 26, 2016 SAMSI Astrostatistics Course

2 Our goals... Approximate Bayesian Computation (ABC) 1

3 Background: Bayesian methods Goal: the posterior distribution of the unknown parameter(s) θ. Posterior distribution Likelihood Prior {}}{{}}{ Data {}}{ f (y θ) π(θ) π(θ y ) = f (y) f (y θ)π(θ) The prior distribution allows you to easily incorporate your beliefs about the parameter(s) of interest Posterior is a distribution on the parameter space given the observed data 2

4 The posterior for θ given observed data x obs : π(θ x obs ) = f (x obs θ)π(θ) f (xobs θ)π(θ)dθ = f (x obs θ)π(θ) f (x obs ) 3

5 The posterior for θ given observed data x obs : π(θ x obs ) = f (x obs θ)π(θ) f (xobs θ)π(θ)dθ = f (x obs θ)π(θ) f (x obs ) Approximate Bayesian Computation Likelihood-free approach to approximating π(θ x obs ) Proceeds via simulation of the forward process 3

6 The posterior for θ given observed data x obs : π(θ x obs ) = f (x obs θ)π(θ) f (xobs θ)π(θ)dθ = f (x obs θ)π(θ) f (x obs ) Approximate Bayesian Computation Likelihood-free approach to approximating π(θ x obs ) Proceeds via simulation of the forward process Why would we not know f (x obs θ)? 1 Physical model too complex to put in analytical form 2 Strong dependency in data 3 Observational limitations 3

7 4

8 ABC for Astronomy cosmoabc: Likelihood-free inference via Population Monte Carlo Approximate Bayesian Computation (Ishida et al., 2015) Approximate Bayesian Computation for Forward Modeling in Cosmology (Akeret et al., 2015) Likelihood-Free Cosmological Inference with Type Ia Supernovae: Approximate Bayesian Computation for a Complete Treatment of Uncertainty (Weyant et al., 2013) Likelihood - free inference in cosmology: potential for the estimation of luminosity functions (Schafer and Freeman, 2012) Approximate Bayesian Computation for Astronomical Model Analysis: A case study in galaxy demographics and morphological transformation at high redshift (Cameron and Pettitt, 2012) 5

9 From Weyant et al. (2013) 6

10 Basic ABC algorithm 7

11 Basic ABC algorithm For the observed data x obs and prior π(θ): Algorithm 1 Sample θ prop from prior π(θ) 2 Generate x prop from forward process f (x θ prop ) 3 Accept θ prop if x obs = x prop 4 Return to step 1 Generates a sample from an approximation of the posterior Introduced in Pritchard et al. (1999) (population genetics) 8

12 Step 3: Accept θ prop if x obs = x prop Waiting for proposals such that x obs = x prop would be computationally prohibitive Instead, accept proposals with (x obs, x prop ) ɛ for some distance and some tolerance threshold ɛ When x is high-dimensional, will have to make ɛ too large in order to keep acceptance probability reasonable. Instead, reduce the dimension by comparing summaries S(x prop ) and S(x obs ) 9

13 Simple examples 10

14 ABC illustration: binomial distribution Data y 1:n iid Bern(θ) where n = sample size, θ = P(Y = 1) Forward process F (x θ) = ( ) n x θ x (1 θ) n x, where x = n (In this case, we use the likelihood) i=1 x i Distance function ρ(y, x) = y x Hence ρ(y, x) = 0 if the generated dataset x has the same number of 1 s as y Tolerance ɛ = 0 Prior π(θ) = Uniform(0,1) Reference: Turner and Zandt (2012) 11

15 Binomial illustration: R code n <- 25 #number of observations N < #generated sample size true.theta <-.75 data <- rbinom(n,1,true.theta) epsilon <- 0 #Tolerance theta <- numeric(n) rho <- function(y,x) abs(sum(y)-sum(x))/n #Distance function for(i in 1:N){ d <- epsilon+1 while(d>epsilon) { proposed.theta <- rbeta(1,1,1) x <- rbinom(n,1,proposed.theta) d <- rho(data,x) } theta[i] <- proposed.theta } #Prior #Forward process 12

16 Binomial illustration: ABC posterior 13

17 Binomial illustration: ABC posterior 13

18 Binomial illustration: ABC posterior 13

19 Gaussian illustration Data x obs consists of 25 iid draws from Normal(µ, 1) Summary statistics S(x) = x Distance function (S(x prop ), S(x obs )) = x prop x obs Tolerance ɛ = 0.50 and 0.10 Prior π(µ) = Normal(0,10) 14

20 Gaussian illustration: posteriors for µ Density Tolerance: 0.5, N:1000, n:25 True Posterior ABC Posterior Input value Density Tolerance: 0.1, N:1000, n:25 True Posterior ABC Posterior Input value µ µ 15

21 Some intuition about ABC 16

22 If one wants to generate a draw from the posterior: 1 Draw θ prop from the prior π(θ). 2 Draw x prop from the density f (x θ prop ). 3 Accept θ prop if x prop = x obs. Why? 17

23 If one wants to generate a draw from the posterior: 1 Draw θ prop from the prior π(θ). 2 Draw x prop from the density f (x θ prop ). 3 Accept θ prop if x prop = x obs. Why? Let θ acc denote an accepted θ prop. Then, for any θ, P(θ acc = θ) = P(θ prop = θ x prop = x obs ) / = P(x prop = x obs θ prop = θ)p(θ prop = θ) P(x prop = x obs ) / = f (x obs θ)π(θ) f (x obs ) = π(θ x obs ) in the case where θ is discrete. Illustration from Chad Schafer (CMU) 17

24 If one wants to generate a draw from the posterior: 1 Draw θ prop from the prior π(θ). 2 Draw x prop from the density f (x θ prop ). 3 Accept θ prop if x prop = x obs. Why? Let θ acc denote an accepted θ prop. Then, for any T IR, P(θ acc T ) = P(θ prop T x prop = x obs ) / = P(x prop = x obs θ prop = θ)π(θ) dθ P(x prop = x obs ) T / = f (x obs θ)π(θ) dθ f (x obs ) = π(θ x obs ) dθ in the case where θ is continuous. T T 18

25 If one wants to generate a draw from the posterior: 1 Draw θ prop from the prior π(θ). 2 Draw x prop from the density f (x θ prop ). 3 Accept θ prop if x prop = x obs. Why? Let θ acc denote an accepted θ prop. Then, for any T IR, P(θ acc T ) = P(θ prop T x prop = x obs ) = K P(x prop = x obs θ prop = θ)π(θ) dθ T = K f (x obs θ)π(θ) dθ = π(θ x obs ) dθ in the case where θ is continuous. T T 19

26 If one wants to generate a draw from the posterior: 1 Draw θ prop from the prior π(θ). 2 Draw x prop from the density f (x θ prop ). 3 Accept θ prop if??? Why? Let θ acc denote an accepted θ prop. Then, for any T IR, P(θ acc T ) = P(θ prop T Accept θ prop ) = K P(Accept θ prop θ prop = θ)π(θ) dθ T? = K f (x obs θ)π(θ) dθ = π(θ x obs ) dθ in the case where θ is continuous. T T 20

27 The Point: θ acc is a draw from the posterior if P(Accept θ prop θ prop = θ) f (x obs θ) (the likelihood) This creates a basis for assessing the quality of the approximation, irrespective of the prior. To achieve this, we could accept θ prop if x prop = x obs. Of course, this is not practical. 21

28 Clear alternative is to accept θ prop if x prop is close to x obs using some chosen distance metric. 22

29 Clear alternative is to accept θ prop if x prop is close to x obs using some chosen distance metric. What is the price of this approximation? 22

30 Clear alternative is to accept θ prop if x prop is close to x obs using some chosen distance metric. What is the price of this approximation? Define: φ ɛ (x prop, x obs ) = { 1, if (xprop, x obs ) < ɛ 0, if (x prop, x obs ) ɛ In other words, φ ɛ (x prop, x obs ) is an indicator as to whether or not x prop is close to x obs. 22

31 Hence, P(Accept θ prop θ prop = θ) = P( (x prop, x obs ) < ɛ θ prop = θ) = φ ɛ (x, x obs )f (x θ) dx Kf (x obs θ) as ɛ 0 Hence, for ɛ small, P(Accept θ prop θ prop = θ) Kf (x obs θ) 23

32 Toy Example: Assume that x obs is Gaussian with mean θ and variance one. 24

33 xxx x Depicts the convolution φ ɛ (x, x obs )f (x θ) dx = P(Accept θ prop θ prop = θ) for case where x obs = 1, θ = 0, ɛ =

34 xxx x Depicts the convolution φ ɛ (x, x obs )f (x θ) dx = P(Accept θ prop θ prop = θ) for case where x obs = 1, θ = 1, ɛ =

35 xxx Likelihood Acceptance Prob Compare these quantities for all θ. Case where x obs = 1, ɛ = 0.1. θ 27

36 xxx x Depicts the convolution φ ɛ (x, x obs )f (x θ) dx = P(Accept θ prop θ prop = θ) for case where x obs = 1, θ = 0, ɛ =

37 xxx x Depicts the convolution φ ɛ (x, x obs )f (x θ) dx = P(Accept θ prop θ prop = θ) for case where x obs = 1, θ = 1, ɛ =

38 xxx Likelihood Acceptance Prob Compare these quantities for all θ. Case where x obs = 1, ɛ = 0.4. θ 30

39 xxx Likelihood Acceptance Prob. Case where x obs = 1, ɛ = θ 31

40 Data summaries Ideally, S x is sufficient, i.e. f (x S x, θ) = f (x S x ) as this implies that f (x θ) = f (x θ, S x )f (S x θ) f (S x θ) Examples: 1 Ȳ = n 1 n i=1 Y i is a sufficient statistic for µ where Y i are iid N(µ, 1), i = 1,..., n 2 Y (n) = max{y 1,..., Y n } is a sufficient statistic for θ where Y i are iid U(0, θ), i = 1,..., n 32

41 In a nutshell The basic idea behind ABC is that using a representative (enough) summary statistic η coupled with a small (enough) tolerance ɛ should produce a good (enough) approximation to the posterior... Marin et al. (2012) 33

42 In a nutshell The basic idea behind ABC is that using a representative (enough) summary statistic η coupled with a small (enough) tolerance ɛ should produce a good (enough) approximation to the posterior... Marin et al. (2012) How to pick a tolerance, ɛ? 33

43 Sequential ABC 34

44 Sequential ABC Main idea Instead of starting the ABC algorithm over with a smaller tolerance (ɛ), use the already sampled particle system as a proposal distribution rather than drawing from the prior distribution. Particle system: (1) retained sampled values, (2) importance weights Some references: Beaumont et al. (2009); Moral et al. (2011); Bonassi and West (2004) 35

45 Monte Carlo integration Importance sampling 36

46 MC Integration General idea Monte Carlo methods are a form of stochastic integration used to approximate expectations by invoking the law of large numbers. I = b a h(y)dy = b a w(y)f (y)dy = E f (w(y )) where f (y) = 1 b a and w(y) = h(y) (b a) f (y) = 1 b a is the pdf of a U(a,b) random variable By the LLN, if we take an iid sample of size N from U(a, b), we can estimate I as Î = N 1 N i=1 w(y i ) E(w(Y )) = I 37

47 MC Integration: Gaussian CDF example Goal: estimate F Y (y) = P(Y y) = E [ I (,y) (Y ) ] where Y N(0, 1): y 1 F (Y y) = e t2 /2 1 dt = h(t) e t2 /2 dt 2π 2π where h(t) = 1 if t < y and h(t) = 0 if t y 38

48 MC Integration: Gaussian CDF example Goal: estimate F Y (y) = P(Y y) = E [ I (,y) (Y ) ] where Y N(0, 1): y 1 F (Y y) = e t2 /2 1 dt = h(t) e t2 /2 dt 2π 2π where h(t) = 1 if t < y and h(t) = 0 if t y Draw an iid sample Y 1,..., Y N from a N(0, 1), then the estimator is N Î = N 1 h(y i ) = # draws < x N i=1 Example 24.2 of Wasserman (2004) 38

49 Importance Sampling: motivation Standard Monte Carlo integration is great if you can sample from the target distribution (i.e. the desired distribution) But what if you can t sample from the target? 39

50 Importance Sampling: motivation Standard Monte Carlo integration is great if you can sample from the target distribution (i.e. the desired distribution) But what if you can t sample from the target? Idea of importance sampling: draw the sample from a proposal distribution and re-weight the integral using importance weights so that the correct distribution is targeted Target Proposal Target and Proposal Density X X X 39

51 MC Integration Importance Sampling I = h(y)f (y)dy h is some function and f is the probability density function of Y When the density f is difficult to sample from, importance sampling can be used 40

52 MC Integration Importance Sampling I = h(y)f (y)dy h is some function and f is the probability density function of Y When the density f is difficult to sample from, importance sampling can be used Rather than sampling from f, you specify a different probability density function, g, as the proposal distribution. I = h(y)f (y)dy = h(y) f (y) g(y) g(y)dy = h(y)f (y) g(y)dy g(y) 40

53 Importance Sampling [ ] h(y)f (y) h(y )f (Y ) I = E f [h(y )] = g(y)dy = E g g(y) g(y ) Hence, given an iid sample Y 1,..., Y N from g, our estimator of I becomes Î = N 1 N i=1 h(y i )f (Y i ) g(y i ) [ ] h(y )f (Y ) E g = I g(y ) 41

54 Importance sampling: Illustration Goal: estimate P(Y < 0.3) where Y f Try two proposal distributions: U(0,1) and U(0,4) 42

55 If we take 1000 samples of size 100, and find the IS estimates, we get the following estimated expected values and variances. Expected Value Variance Truth g 1 : U(0,1) g 2 : U(0,4)

56 Extensions of Importance Sampling Sequential Importance Sampling Sequential Monte Carlo (Particle Filtering) See Doucet et al. (2001) Approximate Bayesian Computation See Turner and Zandt (2012) for a tutorial, and Cameron and Pettitt (2012); Weyant et al. (2013) for applications to astronomy 44

57 Importance sampling can be used in ABC to improve the computational efficiency. 45

58 Algorithm 1 ABC - Population Monte Carlo algorithm 1: At iteration t = 1 2: Algorithm 1: Basic ABC sampler to obtain {θ (i) 1 }N i=1 3: Set importance weights W (i) 4: for t = 2 to T do 5: Set τ 2 t = 2 var 1 = 1/N for i = 1,..., N ) ( {θ (i) t 1, W (i) t 1 }N i=1 6: for i = 1 to N do 7: while ρ (S(y 1:n), S(x 1:n)) > ɛ t do 8: 9: Draw θ 0 from {θ (i) t 1 }N i=1 with probabilities {W (i) t 1 }N i=1 Propose θ N(θ 0, τt 2 ) 10: Generate x 1:n from F (x θ ) 11: Calculate summary statistics {S y, S x} 12: end while 13: θ (i) t θ 14: W (i) t ( π Nj=1 W (j) t 1 φ [ θ (i) t ) τt 1 (θ (i) t θ (j) ] t 1 ) 15: end for 16: {W (i) t } N i=1 { W (i) t } N i=1/ N 17: end for i=1 W (i) t Decreasing tolerances ɛ 1 ɛ T, φ( ) is the density function of a N(0, 1) From Beaumont et al. (2009) 46

59 Gaussian illustration: sequential posteriors N:1000, n:25 Density True Posterior ABC Posterior Input value Tolerance sequence, ɛ 1:10 : µ 47

60 Stellar Initial Mass Function 48

61 Stellar Initial Mass Function: the distribution of star masses after a star formation event within a specified volume of space Molecular cloud Protostars Stars Image: adapted from 49

62 Broken power-law (Kroupa, 2001) Φ(M) M α i, M 1i M M 2i α 1 = 0.3 α 2 = 1.3 for 0.01 M/MSun 0.08 [Sub-stellar] for 0.08 M/M Sun 0.50 α 3 = 2.3 for 0.50 M/M Sun M max Many other models, e.g. Salpeter (1955); Chabrier (2003) 1 M Sun = 1 Solar Mass (the mass of our Sun) 50

63 ABC for the Stellar Initial Mass Function Frequency Mass Image (left): Adapted from 51

64 IMF Likelihood Start with a power-law distribution: each star s mass is independently drawn from a power law distribution with density ( ) 1 α f (m) = Mmax 1 α M 1 α m α, m (M min, M max ) min Then the likelihood is ( ) ntot 1 α n tot L(α m 1:ntot ) = Mmax 1 α M 1 α min n tot = total number of stars in cluster i=1 m α i 52

Observational limitations: aging Lifecycle of star depends on mass more massive stars die faster Cluster age of τ Myr only observe stars with masses < T age τ 2/5 10 8/5 If age = 30 Myr so the aging

65 Observational limitations: aging Lifecycle of star depends on mass more massive stars die faster Cluster age of τ Myr only observe stars with masses < T age τ 2/5 10 8/5 If age = 30 Myr so the aging cutoff is T age 10 M Sun Then the likelihood is ( L(α m 1:nobs, n tot) = T 1 α age 1 α M 1 α min ) ( nobs nobs i=1 m α i ) P(M > T age) ntot n obs n tot = # of stars in cluster n obs = # stars observed in cluster Image: 53

66 Observational limitations: completeness Completeness function: 0, m < C min m C P(observing star m) = min C max C min, m [C min, C max ] 1, m > C max Probability of observing a particular star given its mass Depends on the flux limit, stellar crowding, etc. Image: NASA, J. Trauger (JPL), J. Westphal (Caltech) 54

67 Observational limitations: measurement error Incorporating log-normal measurement error gives our final likelihood: L(α m 1:nobs, n tot ) = ( ( 1 α P(M > T age ) + Mmax 1 α M 1 α min n { obs Tage (2πσ 2 ) 1 2 m 1 i e 1 i=1 2 ( ( ) M Cmin I {M > C max} + C max C min ) Cmax C min M α 2σ 2 (log(m i ) log(m))2 ( I {C min M C max} ( 1 M C ) ) n tot n obs min dm C max C min ) 1 α Mmax 1 α M 1 α M α min ) } dm 55

68 IMF With aging, completeness, and error Density Density Mass Mass Sample size = 1000 stars, [C min, C max ] = [2, 4], σ =

69 Simulation Study: forward model Draw from ( f (m) = Aged 30 Myrs 1 α 60 1 α 2 1 α ) m α, m (2, 60) Observational completeness: 0, m < 4 P(obs m) = m 2 2, m [2, 4] 1, m > 4. Uncertainty: log M = log m η (with η N(0, 1)) Prior: α U[0, 6] 57

70 Simulation Study: summary statistics We want to account for the following with our summary statistics and distance functions: 1. Shape of the observed Mass Function [ } ] 2 1/2 ρ 1 (m sim, m obs ) = {ˆf log (x) ˆf msim log (x) mobs dx 58

71 2. Number of stars observed ρ 2 (m sim, m obs ) = 1 n sim /n obs m sim = masses of the stars simulated from the forward model m obs = masses of observed stars n sim = number of stars simulated from the forward model n obs = number of observed stars 59

72 Simulation Study 1 Draw n = 10 3 stars 2 IMF slope α = 2.35 with M min = 2 and M max = 60 3 N = 10 3 particles 4 T = 30 sequential time steps Observed Mass Function Density Mass 60

73 IMF Observed MF Density Density N = 1000 Bandwidth = N = 510 Bandwidth = Sample size = 1000 stars, [C min, C max ] = [2, 4], σ =

74 Simulation Study results 1 Draw n = 10 3 stars 2 IMF slope α = 2.35 with M min = 2 and M max = 60 3 N = 10 3 particles 4 T = 30 sequential time steps Log10(f M) (A) Observed Mass Function Observed MF 95% Credible band Posterior Median Log10(M) Density (B) Posteriors Initial ABC posterior ABC Posterior... True Posterior Intermediate ABC posterior Input value... Final ABC Posterior α 62

75 Summary ABC can be a useful tool when data are too complex to define a reasonable likelihood Selection of good summary statistics is crucial for ABC posterior to be meaningful 63

76 THANK YOU!!! 64

77 Bibliography I Akeret, J., Refregier, A., Amara, A., Seehars, S., and Hasner, C. (2015), Approximate Bayesian computation for forward modeling in cosmology, Journal of Cosmology and Astroparticle Physics, 2015, 043. Beaumont, M. A., Cornuet, J.-M., Marin, J.-M., and Robert, C. P. (2009), Adaptive approximate Bayesian computation, Biometrika, 96, Bonassi, F. V. and West, M. (2004), Sequential Monte Carlo with Adaptive Weights for Approximate Bayesian Computation, Bayesian Analysis, 1, Cameron, E. and Pettitt, A. N. (2012), Approximate Bayesian Computation for Astronomical Model Analysis: A Case Study in Galaxy Demographics and Morphological Transformation at High Redshift, Monthly Notices of the Royal Astronomical Society, 425, Chabrier, G. (2003), The galactic disk mass function: Reconciliation of the hubble space telescope and nearby determinations, The Astrophysical Journal Letters, 586, L133. Doucet, A., De Freitas, N., and Gordon, N. (2001), Sequential Monte Carlo Methods in Practice, Statistics for Engineering and Information Science, New York: Springer-Verlag. Ishida, E. E. O., Vitenti, S. D. P., Penna-Lima, M., Cisewski, J., de Souza, R. S., Trindade, A. M. M., Cameron, E., and Busti, V. C. (2015), cosmoabc: Likelihood-free inference via Population Monte Carlo Approximate Bayesian Computation, Astronomy and Computing, 13, Kroupa, P. (2001), On the variation of the initial mass function, Monthly Notices of the Royal Astronomical Society, 322, Marin, J.-M., Pudlo, P., Robert, C. P., and Ryder, R. J. (2012), Approximate Bayesian computational methods, Statistics and Computing, 22, Moral, P. D., Doucet, A., and Jasra, A. (2011), An adaptive sequential Monte Carlo method for approximate Bayesian computation, Statistics and Computing, 22, Pritchard, J. K., Seielstad, M. T., and Perez-Lezaun, A. (1999), Population Growth of Human Y Chromosomes: A study of Y Chromosome Microsatellites, Molecular Biology and Evolution, 16, Salpeter, E. E. (1955), The luminosity function and stellar evolution. The Astrophysical Journal, 121,

78 Bibliography II Schafer, C. M. and Freeman, P. E. (2012), Statistical Challenges in Modern Astronomy V, Springer, chap. 1, Lecture Notes in Statistics, pp Turner, B. M. and Zandt, T. V. (2012), A tutorial on approximate Bayesian computation, Journal of Mathematical Psychology, 56, Wasserman, L. (2004), All of statistics: a concise course in statistical inference, Springer. Weyant, A., Schafer, C., and Wood-Vasey, W. M. (2013), Likeihood-free cosmological inference with type Ia supernovae: approximate Bayesian computation for a complete treatment of uncertainty, The Astrophysical Journal, 764,

Approximate Bayesian Computation for Astrostatistics

Approximate Bayesian Computation for Astrostatistics Jessi Cisewski Department of Statistics Yale University October 24, 2016 SAMSI Undergraduate Workshop Our goals Introduction to Bayesian methods Likelihoods,