Non-Parametric Bayesian Inference for Controlled Branching...

Controlled Branching ProcessesBayesian Inference for Controlled Branching Processes

A Simulation-Based Method using Gibbs SamplerConcluding Remarks and References

Non-Parametric Bayesian Inference forControlled Branching Processes Through

MCMC Methods

M. González, R. Martínez, I. del Puerto, A. Ramos

Department of Mathematics. University of ExtremaduraSpanish Branching Processes Group

18th International Conference in Computational StatisticsPorto, August 2008

M. González Bayesian Inference for Controlled Branching Processes

Contents

1 Controlled Branching Processes

2 Bayesian Inference for Controlled Branching Processes

3 A Simulation-Based Method using Gibbs SamplerIntroducing the MethodDeveloping the MethodSimulated Example

4 Concluding Remarks and ReferencesConcluding RemarksReferences

Contents

Branching Processes

Inside the general context concerning Stochastic Models, BranchingProcesses Theory provides appropriate mathematical models fordescription of the probabilistic evolution of systems whosecomponents (cell, particles, individuals in general), after certain lifeperiod, reproduce and die. Therefore, it can be applied in severalfields (biology, demography, ecology, epidemiology, genetics,medicine,...).

Branching Processes

ExampleZ0 = 1

Z1 = 2Z2 = 7Z3 = 10

Zn+1 =Zn∑

Branching Processes

ExampleZ0 = 1

Z1 = 2Z2 = 7Z3 = 10

Zn+1 =Zn∑

Branching Processes

ExampleZ0 = 1Z1 = 2

Z2 = 7Z3 = 10

Zn+1 =Zn∑

Branching Processes

ExampleZ0 = 1Z1 = 2

Z2 = 7Z3 = 10

Zn+1 =Zn∑

Branching Processes

ExampleZ0 = 1Z1 = 2Z2 = 7

Z3 = 10

Zn+1 =Zn∑

Branching Processes

ExampleZ0 = 1Z1 = 2Z2 = 7

Z3 = 10

Zn+1 =Zn∑

Branching Processes

ExampleZ0 = 1Z1 = 2Z2 = 7Z3 = 10

Zn+1 =Zn∑

Branching Processes

ExampleZ0 = 1Z1 = 2Z2 = 7Z3 = 10

Zn+1 =Zn∑

Branching Processes

ExampleZ0 = 1Z1 = 2Z2 = 7Z3 = 10

Zn+1 =Zn∑

Branching Processes

Main Results for Galton–Watson Branching Processes

Let m = E[X01] and σ2 = Var[X01]

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

Branching Processes

Extinction Problem

Branching Processes

Extinction Problem

Branching Processes

Extinction Problem

Branching Processes

Many monographs about the theory and applications about the branching processeshave been published:

Harris, T. (1963). The Theory of branching processes. Springer-Verlag.

Jagers, P. (1975) Branching processes with Biological Applications, JohnWiley and Sons, Inc.

Asmussen, S. and Hering, H. (1983). Branching processes. Birkhäuser.Boston.

Athreya, K.B. and Jagers, P. (1997). Classical and modern branchingprocesses. 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.

Branching Processes

A Controlled Branching Process is a discrete-time stochastic growthpopulation model in which the individuals with reproductive capacityin each generation are controlled by some function φ. This branchingmodel is well-suited for describing the probabilistic evolution ofpopulations in which, for various reasons of an environmental, socialor other nature, there is a mechanism that establishes the number ofprogenitors who take part in each generation.

Branching Processes

Mathematically: Controlled Branching Process

{Zn}n≥0

Z0 = N, Zn+1 =φ(Zn)∑i=1

Xni, n = 0, 1, . . .

{Xni : i = 1, 2, . . . , n = 0, 1, . . .} are i.i.d. random variables.{pk : k ∈ S} Offspring Distribution m = E[X01], σ2 = Var[X01]

φ : R+ → R+ is assumed to be integer-valued for integer-valuedarguments Control Function

Controlled Branching Processes

Properties

{Zn}n≥0 is a Homogeneous Markov Chain

Duality Extinction-Explosion: P(Zn → 0) + P(Zn →∞) = 1

Main Topics InvestigatedExtinction Problem

Sevast’yanov and Zubkov (1974)Zubkov (1974)Molina, González and Mota (1998)

Asymptotic Behaviour: Growth ratesBagley (1986)Molina, González and Mota (1998)González, Molina, del Puerto (2002, 2003, 2004, 2005a,b)

Properties

{Zn}n≥0 is a Homogeneous Markov Chain

Duality Extinction-Explosion: P(Zn → 0) + P(Zn →∞) = 1

Main Topics InvestigatedExtinction Problem

Sevast’yanov and Zubkov (1974)Zubkov (1974)Molina, González and Mota (1998)

Asymptotic Behaviour: Growth ratesBagley (1986)Molina, González and Mota (1998)González, Molina, del Puerto (2002, 2003, 2004, 2005a,b)

Main Topics InvestigatedStatistical Inference

Dion, J. P. and Essebbar, B. (1995). On the statistics of controlledbranching processes. Lecture Notes in Statistics, 99:14-21.

M. González, R. Martínez, I. Del Puerto (2004). Nonparametricestimation of the offspring distribution and the mean for acontrolled branching process. Test, 13(2), 465-479.

M. González, R. Martínez, I. Del Puerto (2005). Estimation ofthe variance for a controlled branching process. Test, 14(1),199-213.

T.N. Sriram, A. Bhattacharya, M. González, R. Martínez, I. DelPuerto (2007). Estimation of the offspring mean in a controlledbranching process with a random control function. StochasticProcesses and their Applications, 117, 928-946.

Bayesian Inference for Controlled Branching Processes

Non-Parametric FrameworkOffspring Distribution: p = {pk : k ∈ S} S finite.Sample: The entire family tree up to the current generation

{Xki : i = 1, . . . , φ(Zk), k = 0, 1, . . . , n}

or at leastZn = {Zj(k) : k ∈ S, j = 0, . . . , n}

where Zj(k) =∑φ(Zj)

i=1 I{Xji=k} = number of parents in thejth-generation which generate exactly k offspringObjetive: Make inference on p

Non-Parametric FrameworkOffspring Distribution: p = {pk : k ∈ S} S finite.Sample: The entire family tree up to the current generation

{Xki : i = 1, . . . , φ(Zk), k = 0, 1, . . . , n}

or at leastZn = {Zj(k) : k ∈ S, j = 0, . . . , n}

where Zj(k) =∑φ(Zj)

i=1 I{Xji=k} = number of parents in thejth-generation which generate exactly k offspringObjetive: Make inference on p

Likelihood Function

L(p|Zn) ∝∏k∈S

j=0 Zj(k)k

Conjugate Class of Distributions: Dirichlet Family

Prior Distribution: p ∼ D(αk : k ∈ S)Posterior Distribution:

p|Zn ∼ D(αk +n∑

Zj(k) : k ∈ S)

Likelihood Function

L(p|Zn) ∝∏k∈S

j=0 Zj(k)k

Conjugate Class of Distributions: Dirichlet Family

Prior Distribution: p ∼ D(αk : k ∈ S)Posterior Distribution:

p|Zn ∼ D(αk +n∑

Zj(k) : k ∈ S)

Setting out the ProblemIn real problems it is difficult to observe the entire family tree{Xki : i = 1, 2, . . . , k = 0, 1, . . . , n} or even the random variablesZn = {Zj(k) : k ∈ S, j = 0, . . . , n}

Usual Sample Information

Z∗n = {Zj : j = 0, . . . , n}

A SolutionWe introduce an algorithm to approximate the distribution

p|Z∗n

using Markov Chain Monte Carlo Methods

Z∗n = {Zj : j = 0, . . . , n}

p|Z∗n

Z∗n = {Zj : j = 0, . . . , n}

p|Z∗n

Introducing the MethodDeveloping the MethodSimulated Example

Gibbs Sampler: Introducing the Method

Sample: Z∗n = {Zj : j = 0, . . . , n}

The Problem

p|Z∗n

Latent Variables:

Zn = {Zj(k) : k ∈ S, j = 0, . . . , n}

Gibbs Sampler:

p|Zn,Z∗n Zn|Z∗n , p

Sample: Z∗n = {Zj : j = 0, . . . , n}

The Problem

p|Z∗n

Latent Variables:

Zn = {Zj(k) : k ∈ S, j = 0, . . . , n}

Gibbs Sampler:

Sample: Z∗n = {Zj : j = 0, . . . , n}

The Problem

p|Z∗n

Latent Variables:

Zn = {Zj(k) : k ∈ S, j = 0, . . . , n}

Gibbs Sampler:

Sample: Z∗n = {Zj : j = 0, . . . , n}

The Problem

p|Z∗n

Latent Variables:

Zn = {Zj(k) : k ∈ S, j = 0, . . . , n}

Gibbs Sampler:

First Conditional Distribution: p|Zn,Z∗n

p|Zn,Z∗n ≡ p|Zn ∼ D(αk +n∑

Zj(k) : k ∈ S)

For j = 0, . . . , nφ(Zj) =

∑k∈S

Zj+1 =∑k∈S

kZj(k)

Zj(k) : k ∈ S)

For j = 0, . . . , nφ(Zj) =

∑k∈S

Zj+1 =∑k∈S

kZj(k)

Zj(k) : k ∈ S)

For j = 0, . . . , nφ(Zj) =

∑k∈S

Zj+1 =∑k∈S

kZj(k)

Second Conditional Distribution: Zn|Z∗n , p

P(Zn|Z∗n , p) =n∏

P(Zj(k) : k ∈ S|Zj,Zj+1, p)

(Zj(k) : k ∈ S)|Zj,Zj+1, p

is obtained from aMultinomial(φ(Zj), p)

normalized by considering the constraint

Zj+1 =∑k∈S

kZj(k)

P(Zn|Z∗n , p) =n∏

P(Zj(k) : k ∈ S|Zj,Zj+1, p)

(Zj(k) : k ∈ S)|Zj,Zj+1, p

is obtained from aMultinomial(φ(Zj), p)

normalized by considering the constraint

Zj+1 =∑k∈S

kZj(k)

pφ(Z0)

Z0(k), k ∈ SZ1 φ(Z1)

Z1(k), k ∈ SZ2 φ(Z2)...

......

Zn φ(Zn)Zn(k), k ∈ S

φ(Zj) =∑k∈S

Zj(k), Zj+1 =∑k∈S

kZj(k)

pφ(Z0)

Z0(k), k ∈ SZ1 φ(Z1)

Z1(k), k ∈ SZ2 φ(Z2)...

......

φ(Zj) =∑k∈S

kZj(k)

pφ(Z0)

Z0(k), k ∈ SZ1 φ(Z1)

Z1(k), k ∈ SZ2 φ(Z2)...

......

φ(Zj) =∑k∈S

kZj(k)

pφ(Z0)

Z0(k), k ∈ SZ1 φ(Z1)

Z1(k), k ∈ SZ2 φ(Z2)...

......

φ(Zj) =∑k∈S

kZj(k)

Gibbs Sampler: Developing the Method

Algorithm

Fixed p(0)

Do l = 1Generate Z(l)

n ∼ Zn|Z∗n , p(l−1)

Generate p(l) ∼ p|Z(l)n

Do l = l + 1

For a run of the sequence {p(l)}l≥0, we choose Q + 1 vectors in theway {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 independentsamples 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 applythe algorithm T times, obtaining a final sample of length T(Q + 1).

Algorithm

Fixed p(0)

n ∼ Zn|Z∗n , p(l−1)

Do l = l + 1

Algorithm

Fixed p(0)

n ∼ Zn|Z∗n , p(l−1)

Do l = l + 1

Algorithm

Fixed p(0)

n ∼ Zn|Z∗n , p(l−1)

Do l = l + 1

Gibbs Sampler: Simulated ExampleOffspring Distribution:

k 0 1 2 3 4pk 0.28398 0.42014 0.233090 0.05747 0.00531

Parameters: m = 1.08, σ2 = 0.7884

Control function: φ(x) = 7 if x ≤ 7; x if 7 < x ≤ 20; 20 if x > 20Simulated Data

0 20 40 60 80 100

Controlled Branching Process

0 20 40 60 80 100

Galton−Watson Branching Process

Gibbs Sampler: Simulated Example

Observed Data: n = 15n 0 1 2 3 4 5 6 7 8 9 10Zn 10 12 17 13 12 12 11 10 11 14 13

n 11 12 13 14 15Zn 15 21 24 20 19

p ∼ D(1/2, . . . , 1/2)

Selection of N, G, Q and TN = 5000, G = 100, Q = 49 and T = 100

Gelman-Rubin-Brooks diagnostic plots.

Estimated potential scale reduction factor.

Autocorrelation values.

Gibbs Sampler: Simulated ExampleGelman-Rubin-Brooks diagnostic plots (CODA package for R)

0 2000 4000 6000 8000 10000

last iteration in chain

median97.5%

0 2000 4000 6000 8000 10000

median97.5%

0 2000 4000 6000 8000 10000

median97.5%

0 2000 4000 6000 8000 10000

median97.5%

0 2000 4000 6000 8000 10000

median97.5%

Gibbs Sampler: Simulated Examples

p ∼ D(1/2, . . . , 1/2)

Selection of N, G, Q and TN = 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)2Duo CPU T7500 running at 2.20GHz with 2038 MB RAM.

Gibbs Sampler: Simulated ExampleSample Information: Z∗nN = 5000, G = 100, Q = 49 and T = 100 (Sample Size: 5000)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

hpd 95% hpd 95%p0p̂0

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

hpd 95% hpd 95%p0p̂3

Offspring Mean

0.9 1.0 1.1 1.2 1.3

hpd 95% hpd 95%mm̂

Algorithm’s Efficiency

MEAN SD MCSE TSSE1.0766931 0.0579821 0.0008200 0.0008179

Concluding RemarksReferences

Concluding Remarks

In a non-parametric Bayesian framework we can make inferenceon the offspring distribution of Controlled Branching Processes,and consequently on the rest of offspring parameters, withoutobserving the entire family tree, but only considering the totalnumber of individuals in each generation.

We use a MCMC method (Gibbs sampler) and the statisticalsoftware and programming environment R, in order to give a"likely" approach to family trees.

Concluding Remarks

In a non-parametric Bayesian framework we can make inferenceon the offspring distribution of Controlled Branching Processes,and consequently on the rest of offspring parameters, withoutobserving the entire family tree, but only considering the totalnumber of individuals in each generation.

We use a MCMC method (Gibbs sampler) and the statisticalsoftware and programming environment R, in order to give a"likely" approach to family trees.

References

Bagley, J. H. (1986). On the almost sure convergence of controlled branching processes. Journal of AppliedProbability, 23:827-831.

M. González, M. Molina, I. del Puerto (2002). On the class of controlled branching process with random controlfunctions. Journal of Applied Probability, 39 (4), 804-815.

M. González, M. Molina, I. del Puerto (2003). On the geometric growth in controlled branching processes withrandom control function. Journal of Applied Probability, 40(4), 995-1006.

M. González, M. Molina, I. del Puerto (2004). Limiting distribution for subcritical controlled branching processes withrandom control function. Statistics and Probability Letters, 67(3), 277-284.

M. González, M. Molina, I. del Puerto (2005a). Asymptotic behaviour of critical controlled branching process withrandom control function. Journal of Applied Probability, 42(2), 463-477.

M. González, M. Molina, I. del Puerto (2005b). On the L2-convergence of controlled branching processes withrandom control function. Bernoulli, 11(1), 37-46.

M. Molina, M. González, M. Mota (1998). Some theoretical results about superadditive controlled Galton-Watsonbranching processes. Proceedings of the International Conference Prague Stochastics98, 2, 413-418.

Sevast’yanov, B. A. and Zubkov, A. (1974). Controlled branching processes. Theory of Probability and itsApplications, 19:14-24.

Tierney, L. (1994). Markov chains for exploring posterior distributions. Annals of Statistics, 22, 1701-1762.

Zubkov, A. M. (1974). Analogies between Galton-Watson processes and φ-branching processes. Theory of Probabilityand its Applications,19:309-331.

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