MCMC and ABC Methodologies in the context of Controlled ...villegas/PDF/GonzalezPuerto.pdf · Contents 1 Controlled Branching Processes 2 MCMC for CBP with Deterministic Control Function

MCMC and ABC Methodologies in the context ofControlled Branching Processes

M. González, I. del Puerto

Department of Mathematics. University of ExtremaduraSpanish Branching Processes Group

Workshop Métodos Bayesianos 11Madrid, November 2011

M. González (Universidad de Extremadura) Bayesian Inference for CBP Workshop Métodos Bayesianos 11 1 / 55

Contents

1 Controlled Branching Processes

2 MCMC for CBP with Deterministic Control FunctionBayesian Inference for Controlled Branching ProcessesA Simulation-Based Method using Gibbs Sampler

3 MCMC for CBP with Random Control FunctionBayesian Inference for Controlled Branching ProcessesA Simulation-Based Method using Gibbs Sampler

4 ABC for CBP with Deterministic and Random Control FunctionsApproximate Bayesian ComputationSimulated Example

5 Concluding Remarks and ReferencesConcluding RemarksReferences


Contents







Contents







Contents







Contents







Branching Processes

Inside the general context concerning Stochastic Models, BranchingProcesses Theory provides appropriate mathematical models for descriptionof 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, Algorithms,...).


Branching Processes

ExampleZ0 = 1

Z1 = 2Z2 = 7Z3 = 10

...

Zn+1 =Zn∑

j=1

Xnj


Branching Processes

ExampleZ0 = 1

Z1 = 2Z2 = 7Z3 = 10

...

Zn+1 =Zn∑

j=1

Xnj


Branching Processes

ExampleZ0 = 1Z1 = 2

Z2 = 7Z3 = 10

...

Zn+1 =Zn∑

j=1

Xnj


Branching Processes

ExampleZ0 = 1Z1 = 2

Z2 = 7Z3 = 10

...

Zn+1 =Zn∑

j=1

Xnj


Branching Processes

ExampleZ0 = 1Z1 = 2Z2 = 7

Z3 = 10

...

Zn+1 =Zn∑

j=1

Xnj


Branching Processes

ExampleZ0 = 1Z1 = 2Z2 = 7

Z3 = 10

...

Zn+1 =Zn∑

j=1

Xnj


Branching Processes

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

...

Zn+1 =Zn∑

j=1

Xnj


Branching Processes

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

...

Zn+1 =Zn∑

j=1

Xnj


Branching Processes

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

...

Zn+1 =Zn∑

j=1

Xnj


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 processes have beenpublished:

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

Jagers, P. (1975). Branching processes with Biological Applications, John Wiley andSons, 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-VerlagNew York, Inc.

Haccou, P., Jagers, P., and Vatutin, V. (2005). Branching Processes: Variation, Growth,and Extinction of Populations. Cambridge University Press.

González, M., del Puerto, I., Martínez, R., Molina, M., Mota, M., Ramos, A. (Editors)(2010). Workshop on Branching Processes and their Applications. Lecture Notes inStatistics, 197. Springer.


Branching Processes

A Controlled Branching Process is a discrete-time stochastic growthpopulation model in which the individuals with reproductive capacity in eachgeneration are controlled by some function φ. This branching model iswell-suited for describing the probabilistic evolution of populations in which,for various reasons of an environmental, social or other nature, there is amechanism that establishes the number of progenitors who take part in eachgeneration.


Branching Processes

Mathematically: Controlled Branching Process {Zn}n≥0

Z0 = N, Zn+1 =φn(Zn)∑

i=1

Xni, n = 0, 1, . . .

Two independent sequences of random variables (r.v.):

{Xni : i = 1, 2, . . . , n = 0, 1, . . .} are i.i.d. r.v.p = {pk : k = 0, 1, . . .} Offspring Distributionm = E[X01], σ2 = Var[X01]{φn(k) : n = 0, 1, . . . ; k = 0, 1, . . .}, where {φn(k)}k≥0 are independentstochastic processes with identical one-dimensional probabilitydistributions, n = 0, 1, . . . Random Control Functionsε(k) = E[φn(k)], σ2(k) = Var[φn(k)].φn(k) = φ(k), k = 0, 1, . . . Deterministic Control Function


Branching Processes

Mathematically: Controlled Branching Process {Zn}n≥0

Z0 = N, Zn+1 =φn(Zn)∑

i=1

Xni, n = 0, 1, . . .

Two independent sequences of random variables (r.v.):

{Xni : i = 1, 2, . . . , n = 0, 1, . . .} are i.i.d. r.v.p = {pk : k = 0, 1, . . .} Offspring Distributionm = E[X01], σ2 = Var[X01]{φn(k) : n = 0, 1, . . . ; k = 0, 1, . . .}, where {φn(k)}k≥0 are independentstochastic processes with identical one-dimensional probabilitydistributions, n = 0, 1, . . . Random Control Functionsε(k) = E[φn(k)], σ2(k) = Var[φn(k)].φn(k) = φ(k), k = 0, 1, . . . Deterministic 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 controlled branching processes. LectureNotes in Statistics, 99:14-21.

M. González, R. Martínez, I. Del Puerto (2004). Nonparametric estimation of the offspringdistribution and the mean for a controlled branching process. Test, 13(2), 465-479.

M. González, R. Martínez, I. Del Puerto (2005). Estimation of the variance for a controlledbranching process. Test, 14(1), 199-213.

T.N. Sriram, A. Bhattacharya, M. González, R. Martínez, I. Del Puerto (2007). Estimation of theoffspring mean in a controlled branching process with a random control function. StochasticProcesses and their Applications, 117, 928-946.

R. Martínez, I. del Puerto, M. Mota (2009). On asymptotic posterior normality for controlledbranching processes. Statistics, 43, 367-378.


Bayesian Inference for Controlled Branching Processes

Non-Parametric FrameworkOffspring Distribution: p = {pk : k ∈ S} S finite.Deterministic Control Function: φ(·)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 the jth-generationwhich generate exactly k offspringObjective: Make inference on p




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







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







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






Likelihood Function

f (Zn|p) ∝∏k∈S

p∑n

j=0 Zj(k)

k

Conjugate Class of Distributions: Dirichlet FamilyPrior Distribution: p ∼ D(αk : k ∈ S)Posterior Distribution:

p|Zn ∼ D(αk +n∑

j=0

Zj(k) : k ∈ S)



Likelihood Function

f (Zn|p) ∝∏k∈S

p∑n

j=0 Zj(k)

k

Conjugate Class of Distributions: Dirichlet FamilyPrior Distribution: p ∼ D(αk : k ∈ S)Posterior Distribution:

p|Zn ∼ D(αk +n∑

j=0

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 InformationZ∗n = {Zj : j = 0, . . . , n}

SolutionWe introduce an algorithm to approximate the distribution

p|Z∗n

using Markov Chain Monte Carlo Methods






p|Z∗n







p|Z∗n



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}




Sample: Z∗n = {Zj : j = 0, . . . , n}

The Problem

p|Z∗n

Latent Variables:

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




Sample: Z∗n = {Zj : j = 0, . . . , n}

The Problem

p|Z∗n

Latent Variables:

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




First Conditional Distribution: p|Zn,Z∗n

p|Zn,Z∗n ≡ p|Zn ∼ D(αk +n∑

j=0

Zj(k) : k ∈ S)

For j = 0, . . . , nφ(Zj) =

∑k∈S

Zj(k)

Zj+1 =∑k∈S

kZj(k)





j=0

Zj(k) : k ∈ S)

For j = 0, . . . , nφ(Zj) =

∑k∈S

Zj(k)

Zj+1 =∑k∈S

kZj(k)





j=0

Zj(k) : k ∈ S)

For j = 0, . . . , nφ(Zj) =

∑k∈S

Zj(k)

Zj+1 =∑k∈S

kZj(k)



Second Conditional Distribution: Zn|Z∗n , p

f (Zn|Z∗n , p) =n∏

j=0

f (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)



Second Conditional Distribution: Zn|Z∗n , p

f (Zn|Z∗n , p) =n∏

j=0

f (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)



Second Conditional Distribution: Zn|Z∗n , pp

φ(Z0)Z0(k), k ∈ S

Z1 φ(Z1)Z1(k), k ∈ S

Z2 φ(Z2)...

......

Zn φ(Zn)Zn(k), k ∈ S

Zn+1

φ(Zj) =∑k∈S

Zj(k), Zj+1 =∑k∈S

kZj(k)




φ(Z0)Z0(k), k ∈ S

Z1 φ(Z1)Z1(k), k ∈ S

Z2 φ(Z2)...

......


Zn+1

φ(Zj) =∑k∈S


kZj(k)




φ(Z0)Z0(k), k ∈ S

Z1 φ(Z1)Z1(k), k ∈ S

Z2 φ(Z2)...

......


Zn+1

φ(Zj) =∑k∈S


kZj(k)




φ(Z0)Z0(k), k ∈ S

Z1 φ(Z1)Z1(k), k ∈ S

Z2 φ(Z2)...

......


Zn+1

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 the way{p(N), p(N+G), . . . , p(N+QG)}, where N is the burn-in period and G is a batchsize.

The vectors {p(N), p(N+G), . . . , p(N+QG)} are considered independent samplesfrom 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 thealgorithm 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 Example

Offspring 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 > 20

Simulated Data

0 20 40 60 80 100

510

1520

25

Controlled Branching Process

0 20 40 60 80 100

020

040

060

0

Galton−Watson Branching Process



Observed Data: n = 40

0 10 20 30 405

1015

2025

Generations

Indi

vidu

als

p ∼ D(1/2, . . . , 1/2)

Selection of N, G, Q and TGelman-Rubin-Brooks diagnostic plots.

Estimated potential scale reduction factor.

Autocorrelation values.



Gelman-Rubin-Brooks diagnostic plots (CODA package for R)

0 2000 4000 6000 8000 10000

1.0

1.5

2.0

2.5

3.0

last iteration in chain

shrin

k fa

ctor

median97.5%

0 2000 4000 6000 8000 10000

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5


shrin

k fa

ctor

median97.5%

0 2000 4000 6000 8000 10000

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5


shrin

k fa

ctor

median97.5%

0 2000 4000 6000 8000 10000

1.0

1.5

2.0

2.5

3.0

3.5


shrin

k fa

ctor

median97.5%

0 2000 4000 6000 8000 10000

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4


shrin

k fa

ctor

median97.5%


Gibbs Sampler: Simulated Examples

p ∼ D(1/2, . . . , 1/2)

Selection of N, G, Q and TN = 1000, G = 350, Q = 25 and T = 200

Gelman-Rubin-Brooks diagnostic plots.





Sample Information: Z∗nN = 1000, G = 350, Q = 25 and T = 200 (Sample Size: 5200)

p0

Den

sity

0.1 0.2 0.3 0.4 0.5 0.6

01

23

4

hpd 95% hpd 95%p0 p̂0

p1

Den

sity

0.0 0.2 0.4 0.6 0.8

0.0

0.5

1.0

1.5

2.0

hpd 95% hpd 95%p1p̂1

p2

Den

sity

0.0 0.1 0.2 0.3 0.4 0.5

01

23

45

6


p3

Den

sity

0.00 0.05 0.10 0.15 0.20 0.25 0.30

05

1015

20


p4

Den

sity

0.00 0.05 0.10 0.15

010

2030

4050

60

hpd 95% hpd 95%p4 p̂4




Offspring Mean

Den

sity

0.90 0.95 1.00 1.05 1.10 1.15 1.20

02

46

810

hpd 95% hpd 95%mm̂

Offspring Variance

Den

sity

0.5 1.0 1.5 2.0

0.0

0.5

1.0

1.5

2.0

hpd 95% hpd 95%σ2σ̂2

Algorithm’s EfficiencyMEAN SD MCSE TSSE

m 1.051518 0.042049 0.000583 0.000551σ2 0.965560 0.219196 0.003040 0.002793




0 10 20 30 40

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Generations

Est

imat

es

0 10 20 30 40

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Generations

Est

imat

es



Non-Parametric/Parametric FrameworkOffspring Distribution: p = {pk : k ∈ S} S finite.Random Control Function: Power series family distributions, i.e.

P(φn(k) = j) = ak(j)θj/Ak(θ), j = 0, 1, . . . , θ ∈ Θ, k = 1, 2, . . .

ak(j) ≥ 0 known values, Ak(θ) =∑∞

j=0 ak(j)θj, Θ = {θ > 0 : Ak(θ) <∞}open subset of R.• Regularity assumption:∏

k∈B

Ak(θ) = A∑k∈B k(θ), for every B ⊆ N, θ ∈ Θ.

Sample: The entire family tree up to the current generation, Zn.Objective: Make inference on (p, θ)



Non-Parametric/Parametric FrameworkOffspring Distribution: p = {pk : k ∈ S} S finite.Random Control Function: Power series family distributions, i.e.

P(φn(k) = j) = ak(j)θj/Ak(θ), j = 0, 1, . . . , θ ∈ Θ, k = 1, 2, . . .

ak(j) ≥ 0 known values, Ak(θ) =∑∞

j=0 ak(j)θj, Θ = {θ > 0 : Ak(θ) <∞}open subset of R.• Regularity assumption:∏

k∈B

Ak(θ) = A∑k∈B k(θ), for every B ⊆ N, θ ∈ Θ.

Sample: The entire family tree up to the current generation, Zn.Objective: Make inference on (p, θ)



Likelihood Function

f (Zn|p, θ) ∝∏k∈S

pZ∗n,kk θY∗n /AYn(θ)

with Z∗n,k =∑n−1

l=0 Zl(k), k ∈ S; Yn =∑n−1

j=0 Zj and Y∗n =∑n−1

j=0 φj(Zj).

Conjugate Class of DistributionsPrior Distribution: (p, θ) ∼ p⊗ θ with p ∼ D(αk : k ∈ S) and

π(θ) = ϕ(a, b)−1θa/Ab(θ), where ϕ(a, b) =∫

Θθa/Ab(θ)dθ.

Posterior Distribution: (p, θ)|Zn ∼ p|Zn ⊗ θ|Zn withp|Zn ∼ D(αk + Z∗n,k : k ∈ S) and

π(θ|Zn) = ϕ(a + Y∗n , b + Yn)−1θa+Y∗n /Ab+Yn(θ)



Likelihood Function

f (Zn|p, θ) ∝∏k∈S

pZ∗n,kk θY∗n /AYn(θ)

with Z∗n,k =∑n−1

l=0 Zl(k), k ∈ S; Yn =∑n−1

j=0 Zj and Y∗n =∑n−1

j=0 φj(Zj).

Conjugate Class of DistributionsPrior Distribution: (p, θ) ∼ p⊗ θ with p ∼ D(αk : k ∈ S) and

π(θ) = ϕ(a, b)−1θa/Ab(θ), where ϕ(a, b) =∫

Θθa/Ab(θ)dθ.

Posterior Distribution: (p, θ)|Zn ∼ p|Zn ⊗ θ|Zn withp|Zn ∼ D(αk + Z∗n,k : k ∈ S) and




Usual Sample Information: Z∗n = {Zj : j = 0, . . . , n}

The Problem

(p, θ)|Z∗n

Latent Variables:

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

Gibbs Sampler:

(p, θ)|Zn,Z∗n Zn|Z∗n , p, θ



First Conditional Distribution: (p, θ)|Zn,Z∗n(p, θ)|Zn,Z∗n ≡ (p, θ)|Zn ≡ p|Zn ⊗ θ|Zn

p|Zn ∼ D(αk + Z∗n,k : k ∈ S)


For j = 0, . . . , n

φj(Zj) =∑k∈S

Zj(k) Zj+1 =∑k∈S

kZj(k)



First Conditional Distribution: (p, θ)|Zn,Z∗n(p, θ)|Zn,Z∗n ≡ (p, θ)|Zn ≡ p|Zn ⊗ θ|Zn

p|Zn ∼ D(αk + Z∗n,k : k ∈ S)


For j = 0, . . . , n

φj(Zj) =∑k∈S

Zj(k) Zj+1 =∑k∈S

kZj(k)



Second Conditional Distribution: Zn|Z∗n , p, θ

f (Zn|Z∗n , p, θ) =n∏

j=0

f (Zj(k) : k ∈ S|Zj,Zj+1, p, θ)

P(Zl(k) = zl(k), k ∈ S | Zl = zl,Zl+1 = zl+1, p, θ)

=1

pzl,zl+1

φ∗l !∏k∈S zl(k)!

∏k∈S

pzl(k)k al (φ∗l ) θφ

∗l /Al(θ)

zl =∑

k∈S zl(k), zl+1 =∑

k∈S kzl(k), φ∗l =∑

k∈S zl(k) andpzl,zl+1 = P(Zl+1 = zl+1 | Zl = zl, p, θ)



Second Conditional Distribution: Zn|Z∗n , p, θ

f (Zn|Z∗n , p, θ) =n∏

j=0

f (Zj(k) : k ∈ S|Zj,Zj+1, p, θ)

P(Zl(k) = zl(k), k ∈ S | Zl = zl,Zl+1 = zl+1, p, θ)

=1

pzl,zl+1

φ∗l !∏k∈S zl(k)!

∏k∈S

pzl(k)k al (φ∗l ) θφ

∗l /Al(θ)

zl =∑

k∈S zl(k), zl+1 =∑

k∈S kzl(k), φ∗l =∑

k∈S zl(k) andpzl,zl+1 = P(Zl+1 = zl+1 | Zl = zl, p, θ)



Second Conditional Distribution: Zn|Z∗n , p, θ(p, θ)

Z0 φ0(Z0)Z0(k), k ∈ S

Z1 φ1(Z1)Z1(k), k ∈ S

Z2 φ2(Z2)...

......

Zn φn(Zn)Zn(k), k ∈ S

Zn+1

φj(Zj) =∑k∈S


kZj(k)




Z0 φ0(Z0)Z0(k), k ∈ S

Z1 φ1(Z1)Z1(k), k ∈ S

Z2 φ2(Z2)...

......


Zn+1

φj(Zj) =∑k∈S


kZj(k)




Z0 φ0(Z0)Z0(k), k ∈ S

Z1 φ1(Z1)Z1(k), k ∈ S

Z2 φ2(Z2)...

......


Zn+1

φj(Zj) =∑k∈S


kZj(k)




Z0 φ0(Z0)Z0(k), k ∈ S

Z1 φ1(Z1)Z1(k), k ∈ S

Z2 φ2(Z2)...

......


Zn+1

φj(Zj) =∑k∈S


kZj(k)




Z0 φ0(Z0)Z0(k), k ∈ S

Z1 φ1(Z1)Z1(k), k ∈ S

Z2 φ2(Z2)...

......


Zn+1

φj(Zj) =∑k∈S


kZj(k)




Z0 φ0(Z0)Z0(k), k ∈ S

Z1 φ1(Z1)Z1(k), k ∈ S

Z2 φ2(Z2)...

......


Zn+1

φj(Zj) =∑k∈S


kZj(k)




Z0 φ0(Z0)Z0(k), k ∈ S

Z1 φ1(Z1)Z1(k), k ∈ S

Z2 φ2(Z2)...

......


Zn+1

φj(Zj) =∑k∈S


kZj(k)




Z0 φ0(Z0)Z0(k), k ∈ S

Z1 φ1(Z1)Z1(k), k ∈ S

Z2 φ2(Z2)...

......


Zn+1

φj(Zj) =∑k∈S


kZj(k)



AlgorithmInitialize l = 0Generate p(0) ∼ Dirichlet(α)Generate θ(0) from π(θ) = ϕ(a, b)−1θa/Ab(θ)Iterate

l = l + 1Generate Z(l)

n ∼ f (Zn | Z∗n , p(l−1), θ(l−1))Generate (p(l), θ(l)) ∼ π(p, θ | Z(l)

n )

For a run of the sequence {(θ, p)(l)}l≥0, we choose Q + 1 vectors in the way{(θ, p)(N), (θ, p)(N+G)), . . . , (θ, p)(N+QG)}, where N is a burning period and Gis a batch size.

The vectors {(θ, p)(N), (θ, p)(N+G), . . . , (θ, p)(N+QG)} are consideredindependent samples from (θ, p)|Z∗n if G and N are large enough.

Since these vectors could be affected by the initial state (θ, p)(0), we apply thealgorithm T times, obtaining a final sample of length T(Q + 1).



AlgorithmInitialize l = 0Generate p(0) ∼ Dirichlet(α)Generate θ(0) from π(θ) = ϕ(a, b)−1θa/Ab(θ)Iterate

l = l + 1Generate Z(l)

n ∼ f (Zn | Z∗n , p(l−1), θ(l−1))Generate (p(l), θ(l)) ∼ π(p, θ | Z(l)

n )

For a run of the sequence {(θ, p)(l)}l≥0, we choose Q + 1 vectors in the way{(θ, p)(N), (θ, p)(N+G)), . . . , (θ, p)(N+QG)}, where N is a burning period and Gis a batch size.

The vectors {(θ, p)(N), (θ, p)(N+G), . . . , (θ, p)(N+QG)} are consideredindependent samples from (θ, p)|Z∗n if G and N are large enough.

Since these vectors could be affected by the initial state (θ, p)(0), we apply thealgorithm T times, obtaining a final sample of length T(Q + 1).




Parameters: m = 2.8, σ2 = 0.84

Random Control function: φn(k) ∼ Binom(k, θ), k = 0, 1, . . .; θ = 0.35

Simulated Data

0 20 40 60 80 100

2040

6080

100

120

CBP with Binomial Control

generations




0 10 20 30 4010

2030

4050

60

CBP with Binomial Control

generations

p ∼ D(1/2, . . . , 1/2)

Selection of N, G, Q and TGelman-Rubin-Brooks diagnostic plots.





Gelman-Rubin-Brooks diagnostic plots (CODA package for R)

0 10000 20000 30000 40000

12

34

5


shrin

k fa

ctor

median97.5%

0 10000 20000 30000 40000

12

34

5

p0


shrin

k fa

ctor

median97.5%

0 10000 20000 30000 40000

24

68

p1


shrin

k fa

ctor

median97.5%

0 10000 20000 30000 40000

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

p2


shrin

k fa

ctor

median97.5%

0 10000 20000 30000 40000

1.0

1.5

2.0

2.5

3.0

3.5

4.0

p3


shrin

k fa

ctor

median97.5%

0 10000 20000 30000 40000

12

34

p4


shrin

k fa

ctor

median97.5%


Gibbs Sampler: Simulated Examples

p ∼ D(1/2, . . . , 1/2)

Selection of N, G, Q and TN = 10000, G = 1000, Q = 30 and T = 59

Gelman-Rubin-Brooks diagnostic plots.






Den

sity

1.0 1.5 2.0 2.5

0.0

0.5

1.0

1.5

mm̂

Den

sity

0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0

0.5

1.0

1.5

2.0

θ̂θ

Den

sity

0.5 1.0 1.5 2.0 2.5

0.0

0.5

1.0

1.5

mθ mθ̂

Algorithm’s EfficiencyMEAN SD MCSE TSSE

m 1.4021 0.3100 0.0073 0.0077θ 0.7518 0.1492 0.0035 0.0036

mθ 1.0551 0.3203 0.0075 0.0074


Approximate Bayesian Computation

Marin, J.M., Pudlo,P., Robert,C.P., Ryder,R.J. (2011). Approximate Bayesiancomputational methods. Statistics and Computing.DOI 10.1007/s11222-011-9288-2

Likelihood-free rejection sampler: πε(p, θ|Z∗n )

for i = 1 to N dorepeat

Generate p′ ∼ Dirichlet(α)Generate θ′ from π(θ) = ϕ(a, b)−1θa/Ab(θ)Generate Z ′n from the likelihood f (Zn|p′, θ′)

until ρ(S(Z ′n),S(Zn)) ≤ εset (pi, θi) = (p′, θ′)

end for

S(·) a function on Zn defining a summary statistic: S(Zn) = Z∗n .ρ is a metric on S(Zn).ε a tolerance level.








end for









end for








end for

ρ is a metric on S(Zn). Wilkinson (2008)

ρ(Z∗n ,Z ′∗n) =

∣∣∣∣∑ni=1 Z′i∑ni=1 Zi

− 1∣∣∣∣+

12

n∑j=1

∣∣∣∣ Zj∑ni=1 Zi

−Z′j∑ni=1 Z′i

∣∣∣∣M. González (Universidad de Extremadura) Bayesian Inference for CBP Workshop Métodos Bayesianos 11 39 / 55

ABC: Simulated Example


Parameters: m = 1.08, σ2 = 0.7884

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


0 10 20 30 40

510

1520

25

Generations

Indi

vidu

als



Sample Information Z∗n , p ∼ D(1/2, . . . , 1/2), N = 20 millionsGeneration 10

●

●

●

●

●

●

●

●

●

●

●

0.10 0.12 0.14 0.16 0.18 0.20

5000

1000

015

000

2000

025

000

3000

0

Tolerance

effe

ctiv

e sa

mpl

e si

ze o

ver

5 m

illio

ns o

f sim

ulat

ions




0.8 1.0 1.2 1.4

01

23

45

Tolerance: 0.11

Den

sity

0.6 0.8 1.0 1.2 1.4

01

23

45

Tolerance: 0.12

Den

sity

0.6 0.8 1.0 1.2 1.40

12

34

5

Tolerance: 0.13

Den

sity

0.6 0.8 1.0 1.2 1.4

01

23

45

Tolerance: 0.14

Den

sity

0.6 0.8 1.0 1.2 1.4

01

23

45

Tolerance: 0.15

Den

sity

0.6 0.8 1.0 1.2 1.4 1.6

01

23

45

Tolerance: 0.16

Den

sity

0.6 0.8 1.0 1.2 1.4 1.6

01

23

45

Tolerance: 0.17

Den

sity

0.6 0.8 1.0 1.2 1.4 1.6

01

23

45

Tolerance: 0.18

Den

sity

0.6 0.8 1.0 1.2 1.4 1.6

01

23

45

Tolerance: 0.19

Den

sity

0.6 0.8 1.0 1.2 1.4 1.6

01

23

45

Tolerance: 0.2

Den

sity



Sample Information: Z∗n , p ∼ D(1/2, . . . , 1/2), N = 20 millionsGeneration 10

● ●

●

●

●

●

●

●

●

●

0.12 0.14 0.16 0.18 0.20

0.05

0.06

0.07

0.08

0.09

0.10

0.11

Tolerance

ISE




●

●

●

●

●

●

●

●

●

●

●

0.10 0.12 0.14 0.16 0.18 0.20

2000

040

000

6000

080

000

Tolerance

effe

ctiv

e sa

mpl

e si

ze o

ver

10 m

illio

ns o

f sim

ulat

ions




0.9 1.0 1.1 1.2 1.3 1.4

02

46

8

Tolerance: 0.11

Den

sity

0.9 1.0 1.1 1.2 1.3 1.4

02

46

8

Tolerance: 0.12

Den

sity

0.9 1.0 1.1 1.2 1.3 1.40

24

68

Tolerance: 0.13

Den

sity

0.9 1.0 1.1 1.2 1.3 1.4 1.5

02

46

8

Tolerance: 0.14

Den

sity

0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5

02

46

8

Tolerance: 0.15

Den

sity

0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5

02

46

8

Tolerance: 0.16

Den

sity

0.8 1.0 1.2 1.4

02

46

8

Tolerance: 0.17

Den

sity

0.8 1.0 1.2 1.4

02

46

8

Tolerance: 0.18

Den

sity

0.8 1.0 1.2 1.4

02

46

8

Tolerance: 0.19

Den

sity

0.8 1.0 1.2 1.4

02

46

8

Tolerance: 0.2

Den

sity




●

●●

●

●

●

●

●

●

●

0.12 0.14 0.16 0.18 0.20

0.25

0.30

0.35

0.40

0.45

0.50

Tolerance

ISE




● ● ●●

●

●

●

●

●

●

●

0.10 0.12 0.14 0.16 0.18 0.20

010

000

2000

030

000

4000

050

000

6000

0

Tolerance

effe

ctiv

e sa

mpl

e si

ze o

ver

20 m

illio

ns o

f sim

ulat

ions




0.9 1.0 1.1 1.2 1.3

02

46

8

Tolerance: 0.11

Den

sity

0.9 1.0 1.1 1.2 1.3

02

46

8

Tolerance: 0.12

Den

sity

0.9 1.0 1.1 1.20

24

68

Tolerance: 0.13

Den

sity

0.9 1.0 1.1 1.2 1.3

02

46

8

Tolerance: 0.14

Den

sity

0.9 1.0 1.1 1.2 1.3

02

46

8

Tolerance: 0.15

Den

sity

0.9 1.0 1.1 1.2 1.3

02

46

8

Tolerance: 0.16

Den

sity

0.9 1.0 1.1 1.2 1.3

02

46

8

Tolerance: 0.17

Den

sity

0.9 1.0 1.1 1.2 1.3

02

46

8

Tolerance: 0.18

Den

sity

0.8 0.9 1.0 1.1 1.2 1.3 1.4

02

46

8

Tolerance: 0.19

Den

sity

0.8 0.9 1.0 1.1 1.2 1.3 1.4

02

46

8

Tolerance: 0.2

Den

sity




●

●

●

●

●

●

●

●

●

●

0.12 0.14 0.16 0.18 0.20

0.05

0.10

0.15

0.20

0.25

0.30

Tolerance

ISE




● ● ● ●●

●

●

●

●

●

●

0.10 0.12 0.14 0.16 0.18 0.20

050

0010

000

1500

0

Tolerance

effe

ctiv

e sa

mpl

e si

ze o

ver

20 m

illio

ns o

f sim

ulat

ions




0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20

02

46

8

Tolerance: 0.13

Den

sity

0.9 1.0 1.1 1.2

02

46

8

Tolerance: 0.14

Den

sity

0.9 1.0 1.1 1.2

02

46

8

Tolerance: 0.15

Den

sity

0.9 1.0 1.1 1.2 1.3

02

46

8

Tolerance: 0.16

Den

sity

0.9 1.0 1.1 1.2 1.3

02

46

8

Tolerance: 0.17

Den

sity

0.9 1.0 1.1 1.2 1.3

02

46

8

Tolerance: 0.18

Den

sity

0.9 1.0 1.1 1.2 1.3

02

46

8

Tolerance: 0.19

Den

sity

0.8 0.9 1.0 1.1 1.2 1.3

02

46

8

Tolerance: 0.2

Den

sity




●

●

●

●

●

●

●

●

0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20

0.26

0.28

0.30

0.32

0.34

0.36

0.38

Tolerance

ISE


Concluding Remarks

In a non-parametric Bayesian framework we can make inference on theoffspring distribution of CBP, and consequently on the rest of offspringparameters, without observing the entire family tree, but onlyconsidering the total number of individuals in each generation.

We use a MCMC method (Gibbs sampler) in order to give a "likely"approach to family trees, for both CBP with deterministic and withrandom control function.

We take advantage of the ABC methodology to make inference on themain parameters of the model by simulating.

The ABC approach shows a quite good behaviour, being a goodalternative to the MCMC approach.

We have developed the above methodologies using statistical softwareand programming environment R.


Concluding Remarks







Concluding Remarks







Concluding Remarks







Concluding Remarks







References

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

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

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

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

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

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

M. Molina, M. González, M. Mota (1998). Some theoretical results about superadditive controlled Galton-Watson branchingprocesses. 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 its Applications, 19:14-24.

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

Wilkinson, R.D. (2008). Approximate Bayesian computation (ABC) gives exact results under the assumption of model error.Technical Report. arXiv:0811.3355.

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


Thank you very much!


MCMC and ABC Methodologies in the context of Controlled ...villegas/PDF/GonzalezPuerto.pdf · Contents 1 Controlled Branching Processes 2 MCMC for CBP with Deterministic Control Function

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