Non-Parametric Bayesian Inference for Controlled Branching Processes Through MCMC Methods

Transcription

1 Non-Parametric Bayesian Inference for Controlled Branching Processes Through MCMC Methods M. González, R. Martínez, I. del Puerto, A. Ramos Department of Mathematics. University of Extremadura Spanish Branching Processes Group 18th International Conference in Computational Statistics Porto, August 2008 M. González

2 Contents 1 Controlled Branching Processes 2 3 Introducing the Method Developing the Method Simulated Example 4 Concluding Remarks References M. González

3 Contents 1 Controlled Branching Processes 2 3 Introducing the Method Developing the Method Simulated Example 4 Concluding Remarks References M. González

4 Contents 1 Controlled Branching Processes 2 3 Introducing the Method Developing the Method Simulated Example 4 Concluding Remarks References M. González

5 Contents 1 Controlled Branching Processes 2 3 Introducing the Method Developing the Method Simulated Example 4 Concluding Remarks References M. González

6 Branching Processes Inside the general context concerning Stochastic Models, Branching Processes Theory provides appropriate mathematical models for description of the probabilistic evolution of systems whose components (cell, particles, individuals in general), after certain life period, reproduce and die. Therefore, it can be applied in several fields (biology, demography, ecology, epidemiology, genetics, medicine,...). M. González

7 Branching Processes Example Z 0 = 1. Z n Z n+1 = j=1 X nj M. González

8 Branching Processes Example Z 0 = 1. Z n Z n+1 = j=1 X nj M. González

9 Branching Processes Example Z 0 = 1 Z 1 = 2. Z n Z n+1 = j=1 X nj M. González

10 Branching Processes Example Z 0 = 1 Z 1 = 2. Z n Z n+1 = j=1 X nj M. González

11 Branching Processes Example Z 0 = 1 Z 1 = 2 Z 2 = 7. Z n Z n+1 = j=1 X nj M. González

12 Branching Processes Example Z 0 = 1 Z 1 = 2 Z 2 = 7. Z n Z n+1 = j=1 X nj M. González

13 Branching Processes Example Z 0 = 1 Z 1 = 2 Z 2 = 7 Z 3 = 10. Z n Z n+1 = j=1 X nj M. González

14 Branching Processes Example Z 0 = 1 Z 1 = 2 Z 2 = 7 Z 3 = 10. Z n Z n+1 = j=1 X nj M. González

15 Branching Processes Example Z 0 = 1 Z 1 = 2 Z 2 = 7 Z 3 = 10. Z n Z n+1 = j=1 X nj M. González

16 Branching Processes Main Results for Galton Watson Branching Processes Let m = E[X 01 ] and σ 2 = Var[X 01 ] Extinction Problem If m 1 the process dies out with probability 1 If m > 1 there exists a positive probability of non-extinction Asymptotic behaviour Statistical Inference M. González

17 Branching Processes Main Results for Galton Watson Branching Processes Let m = E[X 01 ] and σ 2 = Var[X 01 ] Extinction Problem If m 1 the process dies out with probability 1 If m > 1 there exists a positive probability of non-extinction Asymptotic behaviour Statistical Inference M. González

18 Branching Processes Main Results for Galton Watson Branching Processes Let m = E[X 01 ] and σ 2 = Var[X 01 ] Extinction Problem If m 1 the process dies out with probability 1 If m > 1 there exists a positive probability of non-extinction Asymptotic behaviour Statistical Inference M. González

19 Branching Processes Main Results for Galton Watson Branching Processes Let m = E[X 01 ] and σ 2 = Var[X 01 ] Extinction Problem If m 1 the process dies out with probability 1 If m > 1 there exists a positive probability of non-extinction Asymptotic behaviour Statistical Inference M. González

20 Branching Processes Many monographs about the theory and applications about the branching processes have been published: Harris, T. (1963). The Theory of branching processes. Springer-Verlag. Jagers, P. (1975) Branching processes with Biological Applications, John Wiley and Sons, Inc. Asmussen, S. and Hering, H. (1983). Branching processes. Birkhäuser. Boston. Athreya, K.B. and Jagers, P. (1997). Classical and modern branching processes. Springer-Verlag. Kimmel, M. and Axelrod, D.E. (2002) Branching processes in Biology, Springer-Verlag New York, Inc. Haccou, P., Jagers, P., and Vatutin, V. (2005) Branching Processes: Variation, Growth, and Extinction of Populations. Cambridge University Press. M. González

21 Branching Processes A Controlled Branching Process is a discrete-time stochastic growth population model in which the individuals with reproductive capacity in each generation are controlled by some function φ. This branching model is well-suited for describing the probabilistic evolution of populations in which, for various reasons of an environmental, social or other nature, there is a mechanism that establishes the number of progenitors who take part in each generation. M. González

22 Branching Processes Mathematically: Controlled Branching Process Z 0 = N, Z n+1 = {Z n } n 0 φ(z n) i=1 X ni, n = 0, 1,... {X ni : i = 1, 2,..., n = 0, 1,...} are i.i.d. random variables. {p k : k S} Offspring Distribution m = E[X 01 ], σ 2 = Var[X 01 ] φ : R + R + is assumed to be integer-valued for integer-valued arguments Control Function M. González

23 Controlled Branching Processes Properties {Z n } n 0 is a Homogeneous Markov Chain Duality Extinction-Explosion: P(Z n 0) + P(Z n ) = 1 Main Topics Investigated Extinction Problem Sevast yanov and Zubkov (1974) Zubkov (1974) Molina, González and Mota (1998) Asymptotic Behaviour: Growth rates Bagley (1986) Molina, González and Mota (1998) González, Molina, del Puerto (2002, 2003, 2004, 2005a,b) M. González

24 Controlled Branching Processes Properties {Z n } n 0 is a Homogeneous Markov Chain Duality Extinction-Explosion: P(Z n 0) + P(Z n ) = 1 Main Topics Investigated Extinction Problem Sevast yanov and Zubkov (1974) Zubkov (1974) Molina, González and Mota (1998) Asymptotic Behaviour: Growth rates Bagley (1986) Molina, González and Mota (1998) González, Molina, del Puerto (2002, 2003, 2004, 2005a,b) M. González

25 Controlled Branching Processes Main Topics Investigated Statistical Inference Dion, J. P. and Essebbar, B. (1995). On the statistics of controlled branching processes. Lecture Notes in Statistics, 99: M. González, R. Martínez, I. Del Puerto (2004). Nonparametric estimation of the offspring distribution and the mean for a controlled branching process. Test, 13(2), M. González, R. Martínez, I. Del Puerto (2005). Estimation of the variance for a controlled branching process. Test, 14(1), T.N. Sriram, A. Bhattacharya, M. González, R. Martínez, I. Del Puerto (2007). Estimation of the offspring mean in a controlled branching process with a random control function. Stochastic Processes and their Applications, 117, M. González

26 Non-Parametric Framework Offspring Distribution: p = {p k : k S} S finite. Sample: The entire family tree up to the current generation or at least {X ki : i = 1,..., φ(z k ), k = 0, 1,..., n} Z n = {Z j (k): k S, j = 0,..., n} where Z j (k) = φ(z j ) i=1 I {X ji =k} = number of parents in the jth-generation which generate exactly k offspring Objetive: Make inference on p M. González

27 Non-Parametric Framework Offspring Distribution: p = {p k : k S} S finite. Sample: The entire family tree up to the current generation or at least {X ki : i = 1,..., φ(z k ), k = 0, 1,..., n} Z n = {Z j (k): k S, j = 0,..., n} where Z j (k) = φ(z j ) i=1 I {X ji =k} = number of parents in the jth-generation which generate exactly k offspring Objetive: Make inference on p M. González

28 Likelihood Function L(p Z n ) n j=0 p Z j(k) k k S Conjugate Class of Distributions: Dirichlet Family Prior Distribution: p D(α k : k S) Posterior Distribution: n p Z n D(α k + Z j (k): k S) j=0 M. González

29 Likelihood Function L(p Z n ) n j=0 p Z j(k) k k S Conjugate Class of Distributions: Dirichlet Family Prior Distribution: p D(α k : k S) Posterior Distribution: n p Z n D(α k + Z j (k): k S) j=0 M. González

30 Setting out the Problem In real problems it is difficult to observe the entire family tree {X ki : i = 1, 2,..., k = 0, 1,..., n} or even the random variables Z n = {Z j (k): k S, j = 0,..., n} Usual Sample Information Z n = {Z j : j = 0,..., n} A Solution We introduce an algorithm to approximate the distribution p Z n using Markov Chain Monte Carlo Methods M. González

31 Setting out the Problem In real problems it is difficult to observe the entire family tree {X ki : i = 1, 2,..., k = 0, 1,..., n} or even the random variables Z n = {Z j (k): k S, j = 0,..., n} Usual Sample Information Z n = {Z j : j = 0,..., n} A Solution We introduce an algorithm to approximate the distribution p Z n using Markov Chain Monte Carlo Methods M. González

32 Setting out the Problem In real problems it is difficult to observe the entire family tree {X ki : i = 1, 2,..., k = 0, 1,..., n} or even the random variables Z n = {Z j (k): k S, j = 0,..., n} Usual Sample Information Z n = {Z j : j = 0,..., n} A Solution We introduce an algorithm to approximate the distribution p Z n using Markov Chain Monte Carlo Methods M. González

33 Introducing the Method Developing the Method Simulated Example Gibbs Sampler: Introducing the Method Sample: Z n = {Z j : j = 0,..., n} The Problem Latent Variables: p Z n Z n = {Z j (k): k S, j = 0,..., n} Gibbs Sampler: p Z n, Z n Z n Z n, p M. González

34 Introducing the Method Developing the Method Simulated Example Gibbs Sampler: Introducing the Method Sample: Z n = {Z j : j = 0,..., n} The Problem Latent Variables: p Z n Z n = {Z j (k): k S, j = 0,..., n} Gibbs Sampler: p Z n, Z n Z n Z n, p M. González

35 Introducing the Method Developing the Method Simulated Example Gibbs Sampler: Introducing the Method Sample: Z n = {Z j : j = 0,..., n} The Problem Latent Variables: p Z n Z n = {Z j (k): k S, j = 0,..., n} Gibbs Sampler: p Z n, Z n Z n Z n, p M. González

36 Introducing the Method Developing the Method Simulated Example Gibbs Sampler: Introducing the Method Sample: Z n = {Z j : j = 0,..., n} The Problem Latent Variables: p Z n Z n = {Z j (k): k S, j = 0,..., n} Gibbs Sampler: p Z n, Z n Z n Z n, p M. González

37 Introducing the Method Developing the Method Simulated Example Gibbs Sampler: Introducing the Method First Conditional Distribution: p Z n, Zn n p Z n, Zn p Z n D(α k + Z j (k): k S) j=0 For j = 0,..., n φ(z j ) = k S Z j (k) Z j+1 = k S kz j (k) M. González

38 Introducing the Method Developing the Method Simulated Example Gibbs Sampler: Introducing the Method First Conditional Distribution: p Z n, Zn n p Z n, Zn p Z n D(α k + Z j (k): k S) j=0 For j = 0,..., n φ(z j ) = k S Z j (k) Z j+1 = k S kz j (k) M. González

39 Introducing the Method Developing the Method Simulated Example Gibbs Sampler: Introducing the Method First Conditional Distribution: p Z n, Zn n p Z n, Zn p Z n D(α k + Z j (k): k S) j=0 For j = 0,..., n φ(z j ) = k S Z j (k) Z j+1 = k S kz j (k) M. González

40 Introducing the Method Developing the Method Simulated Example Gibbs Sampler: Introducing the Method Second Conditional Distribution: Z n Zn, p n P(Z n Zn, p) = P(Z j (k): k S Z j, Z j+1, p) j=0 (Z j (k): k S) Z j, Z j+1, p is obtained from a Multinomial(φ(Z j ), p) normalized by considering the constraint Z j+1 = kz j (k) k S M. González

41 Introducing the Method Developing the Method Simulated Example Gibbs Sampler: Introducing the Method Second Conditional Distribution: Z n Zn, p n P(Z n Zn, p) = P(Z j (k): k S Z j, Z j+1, p) j=0 (Z j (k): k S) Z j, Z j+1, p is obtained from a Multinomial(φ(Z j ), p) normalized by considering the constraint Z j+1 = kz j (k) k S M. González

42 Introducing the Method Developing the Method Simulated Example Gibbs Sampler: Introducing the Method Second Conditional Distribution: Z n Z n, p p φ(z 0 ) Z 0 (k), k S Z 1 φ(z 1 ) Z 1 (k), k S Z 2 φ(z 2 )... Z n φ(z n ) Z n (k), k S Z n+1 φ(z j ) = k S Z j (k), Z j+1 = k S kz j (k) M. González

43 Introducing the Method Developing the Method Simulated Example Gibbs Sampler: Introducing the Method Second Conditional Distribution: Z n Z n, p p φ(z 0 ) Z 0 (k), k S Z 1 φ(z 1 ) Z 1 (k), k S Z 2 φ(z 2 )... Z n φ(z n ) Z n (k), k S Z n+1 φ(z j ) = k S Z j (k), Z j+1 = k S kz j (k) M. González

44 Introducing the Method Developing the Method Simulated Example Gibbs Sampler: Introducing the Method Second Conditional Distribution: Z n Z n, p p φ(z 0 ) Z 0 (k), k S Z 1 φ(z 1 ) Z 1 (k), k S Z 2 φ(z 2 )... Z n φ(z n ) Z n (k), k S Z n+1 φ(z j ) = k S Z j (k), Z j+1 = k S kz j (k) M. González

45 Introducing the Method Developing the Method Simulated Example Gibbs Sampler: Introducing the Method Second Conditional Distribution: Z n Z n, p p φ(z 0 ) Z 0 (k), k S Z 1 φ(z 1 ) Z 1 (k), k S Z 2 φ(z 2 )... Z n φ(z n ) Z n (k), k S Z n+1 φ(z j ) = k S Z j (k), Z j+1 = k S kz j (k) M. González

46 Introducing the Method Developing the Method Simulated Example Gibbs Sampler: Developing the Method Algorithm Fixed p (0) Do l = 1 Generate Z n (l) Z n Zn, p (l 1) Generate p (l) p Z n (l) Do l = l + 1 For a run of the sequence {p (l) } l 0, we choose Q + 1 vectors in the way {p (N), p (N+G),..., p (N+QG) }, where G is a batch size. The vectors {p (N), p (N+G),..., p (N+QG) } are considered independent samples from p Z n if G and N are large enough (Tierney (1994)). Since these vectors could be affected by the initial state p (0), we apply the algorithm T times, obtaining a final sample of length T(Q + 1). M. González

47 Introducing the Method Developing the Method Simulated Example Gibbs Sampler: Developing the Method Algorithm Fixed p (0) Do l = 1 Generate Z n (l) Z n Zn, p (l 1) Generate p (l) p Z n (l) Do l = l + 1 For a run of the sequence {p (l) } l 0, we choose Q + 1 vectors in the way {p (N), p (N+G),..., p (N+QG) }, where G is a batch size. The vectors {p (N), p (N+G),..., p (N+QG) } are considered independent samples from p Z n if G and N are large enough (Tierney (1994)). Since these vectors could be affected by the initial state p (0), we apply the algorithm T times, obtaining a final sample of length T(Q + 1). M. González

48 Introducing the Method Developing the Method Simulated Example Gibbs Sampler: Developing the Method Algorithm Fixed p (0) Do l = 1 Generate Z n (l) Z n Zn, p (l 1) Generate p (l) p Z n (l) Do l = l + 1 For a run of the sequence {p (l) } l 0, we choose Q + 1 vectors in the way {p (N), p (N+G),..., p (N+QG) }, where G is a batch size. The vectors {p (N), p (N+G),..., p (N+QG) } are considered independent samples from p Z n if G and N are large enough (Tierney (1994)). Since these vectors could be affected by the initial state p (0), we apply the algorithm T times, obtaining a final sample of length T(Q + 1). M. González

49 Introducing the Method Developing the Method Simulated Example Gibbs Sampler: Developing the Method Algorithm Fixed p (0) Do l = 1 Generate Z n (l) Z n Zn, p (l 1) Generate p (l) p Z n (l) Do l = l + 1 For a run of the sequence {p (l) } l 0, we choose Q + 1 vectors in the way {p (N), p (N+G),..., p (N+QG) }, where G is a batch size. The vectors {p (N), p (N+G),..., p (N+QG) } are considered independent samples from p Z n if G and N are large enough (Tierney (1994)). Since these vectors could be affected by the initial state p (0), we apply the algorithm T times, obtaining a final sample of length T(Q + 1). M. González

50 Introducing the Method Developing the Method Simulated Example Gibbs Sampler: Simulated Example Offspring Distribution: k p k Parameters: m = 1.08, σ 2 = Control function: φ(x) = 7 if x 7; x if 7 < x 20; 20 if x > 20 Simulated Data Controlled Branching Process Galton Watson Branching Process M. González

51 Introducing the Method Developing the Method Simulated Example Gibbs Sampler: Simulated Example Observed Data: n = 15 n Z n n Z n p D(1/2,..., 1/2) Selection of N, G, Q and T N = 5000, G = 100, Q = 49 and T = 100 Gelman-Rubin-Brooks diagnostic plots. Estimated potential scale reduction factor. Autocorrelation values. M. González

52 Introducing the Method Developing the Method Simulated Example Gibbs Sampler: Simulated Example Gelman-Rubin-Brooks diagnostic plots (CODA package for R) p 0 p 1 p 2 shrink factor median 97.5% shrink factor median 97.5% shrink factor median 97.5% last iteration in chain last iteration in chain last iteration in chain p 3 p 4 shrink factor median 97.5% shrink factor median 97.5% last iteration in chain last iteration in chain M. González

53 Introducing the Method Developing the Method Simulated Example Gibbs Sampler: Simulated Examples p D(1/2,..., 1/2) Selection of N, G, Q and T N = 5000, G = 100, Q = 49 and T = 100 Gelman-Rubin-Brooks diagnostic plots. Estimated potential scale reduction factor. Autocorrelation values. Computation Time: 60.10s for each chain on an Intel(R) Core(TM)2 Duo CPU T7500 running at 2.20GHz with 2038 MB RAM. M. González

54 Introducing the Method Developing the Method Simulated Example Gibbs Sampler: Simulated Example Sample Information: Z n N = 5000, G = 100, Q = 49 and T = 100 (Sample Size: 5000) p 0 hpd 95% p^0 p 0 hpd 95% p 3 hpd 95% p^3 p 0 hpd 95% Offspring Mean hpd 95% mm^ hpd 95% Density Density Density Algorithm s Efficiency MEAN SD MCSE TSSE M. González

55 Concluding Remarks Concluding Remarks References In a non-parametric Bayesian framework we can make inference on the offspring distribution of Controlled Branching Processes, and consequently on the rest of offspring parameters, without observing the entire family tree, but only considering the total number of individuals in each generation. We use a MCMC method (Gibbs sampler) and the statistical software and programming environment R, in order to give a "likely" approach to family trees. M. González

56 Concluding Remarks Concluding Remarks References In a non-parametric Bayesian framework we can make inference on the offspring distribution of Controlled Branching Processes, and consequently on the rest of offspring parameters, without observing the entire family tree, but only considering the total number of individuals in each generation. We use a MCMC method (Gibbs sampler) and the statistical software and programming environment R, in order to give a "likely" approach to family trees. M. González

57 References Concluding Remarks References Bagley, J. H. (1986). On the almost sure convergence of controlled branching processes. Journal of Applied Probability, 23: M. González, M. Molina, I. del Puerto (2002). On the class of controlled branching process with random control functions. Journal of Applied Probability, 39 (4), M. González, M. Molina, I. del Puerto (2003). On the geometric growth in controlled branching processes with random control function. Journal of Applied Probability, 40(4), M. González, M. Molina, I. del Puerto (2004). Limiting distribution for subcritical controlled branching processes with random control function. Statistics and Probability Letters, 67(3), M. González, M. Molina, I. del Puerto (2005a). Asymptotic behaviour of critical controlled branching process with random control function. Journal of Applied Probability, 42(2), M. González, M. Molina, I. del Puerto (2005b). On the L2-convergence of controlled branching processes with random control function. Bernoulli, 11(1), M. Molina, M. González, M. Mota (1998). Some theoretical results about superadditive controlled Galton-Watson branching processes. Proceedings of the International Conference Prague Stochastics98, 2, Sevast yanov, B. A. and Zubkov, A. (1974). Controlled branching processes. Theory of Probability and its Applications, 19: Tierney, L. (1994). Markov chains for exploring posterior distributions. Annals of Statistics, 22, Zubkov, A. M. (1974). Analogies between Galton-Watson processes and φ-branching processes. Theory of Probability and its Applications,19: M. González

58 Concluding Remarks References Thank you very much! M. González