Non-Parametric Bayesian Inference for Controlled Branching...
Transcript of 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
Controlled Branching ProcessesBayesian Inference for Controlled Branching Processes
A Simulation-Based Method using Gibbs SamplerConcluding Remarks and References
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
M. González Bayesian Inference for Controlled Branching Processes
Controlled Branching ProcessesBayesian Inference for Controlled Branching Processes
A Simulation-Based Method using Gibbs SamplerConcluding Remarks and References
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
M. González Bayesian Inference for Controlled Branching Processes
Controlled Branching ProcessesBayesian Inference for Controlled Branching Processes
A Simulation-Based Method using Gibbs SamplerConcluding Remarks and References
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
M. González Bayesian Inference for Controlled Branching Processes
Controlled Branching ProcessesBayesian Inference for Controlled Branching Processes
A Simulation-Based Method using Gibbs SamplerConcluding Remarks and References
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
M. González Bayesian Inference for Controlled Branching Processes
Controlled Branching ProcessesBayesian Inference for Controlled Branching Processes
A Simulation-Based Method using Gibbs SamplerConcluding Remarks and References
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,...).
M. González Bayesian Inference for Controlled Branching Processes
Controlled Branching ProcessesBayesian Inference for Controlled Branching Processes
A Simulation-Based Method using Gibbs SamplerConcluding Remarks and References
Branching Processes
ExampleZ0 = 1
Z1 = 2Z2 = 7Z3 = 10
...
Zn+1 =Zn∑
j=1
Xnj
M. González Bayesian Inference for Controlled Branching Processes
Controlled Branching ProcessesBayesian Inference for Controlled Branching Processes
A Simulation-Based Method using Gibbs SamplerConcluding Remarks and References
Branching Processes
ExampleZ0 = 1
Z1 = 2Z2 = 7Z3 = 10
...
Zn+1 =Zn∑
j=1
Xnj
M. González Bayesian Inference for Controlled Branching Processes
Controlled Branching ProcessesBayesian Inference for Controlled Branching Processes
A Simulation-Based Method using Gibbs SamplerConcluding Remarks and References
Branching Processes
ExampleZ0 = 1Z1 = 2
Z2 = 7Z3 = 10
...
Zn+1 =Zn∑
j=1
Xnj
M. González Bayesian Inference for Controlled Branching Processes
Controlled Branching ProcessesBayesian Inference for Controlled Branching Processes
A Simulation-Based Method using Gibbs SamplerConcluding Remarks and References
Branching Processes
ExampleZ0 = 1Z1 = 2
Z2 = 7Z3 = 10
...
Zn+1 =Zn∑
j=1
Xnj
M. González Bayesian Inference for Controlled Branching Processes
Controlled Branching ProcessesBayesian Inference for Controlled Branching Processes
A Simulation-Based Method using Gibbs SamplerConcluding Remarks and References
Branching Processes
ExampleZ0 = 1Z1 = 2Z2 = 7
Z3 = 10
...
Zn+1 =Zn∑
j=1
Xnj
M. González Bayesian Inference for Controlled Branching Processes
Controlled Branching ProcessesBayesian Inference for Controlled Branching Processes
A Simulation-Based Method using Gibbs SamplerConcluding Remarks and References
Branching Processes
ExampleZ0 = 1Z1 = 2Z2 = 7
Z3 = 10
...
Zn+1 =Zn∑
j=1
Xnj
M. González Bayesian Inference for Controlled Branching Processes
Controlled Branching ProcessesBayesian Inference for Controlled Branching Processes
A Simulation-Based Method using Gibbs SamplerConcluding Remarks and References
Branching Processes
ExampleZ0 = 1Z1 = 2Z2 = 7Z3 = 10
...
Zn+1 =Zn∑
j=1
Xnj
M. González Bayesian Inference for Controlled Branching Processes
Controlled Branching ProcessesBayesian Inference for Controlled Branching Processes
A Simulation-Based Method using Gibbs SamplerConcluding Remarks and References
Branching Processes
ExampleZ0 = 1Z1 = 2Z2 = 7Z3 = 10
...
Zn+1 =Zn∑
j=1
Xnj
M. González Bayesian Inference for Controlled Branching Processes
Controlled Branching ProcessesBayesian Inference for Controlled Branching Processes
A Simulation-Based Method using Gibbs SamplerConcluding Remarks and References
Branching Processes
ExampleZ0 = 1Z1 = 2Z2 = 7Z3 = 10
...
Zn+1 =Zn∑
j=1
Xnj
M. González Bayesian Inference for Controlled Branching Processes
Controlled Branching ProcessesBayesian Inference for Controlled Branching Processes
A Simulation-Based Method using Gibbs SamplerConcluding Remarks and References
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
M. González Bayesian Inference for Controlled Branching Processes
Controlled Branching ProcessesBayesian Inference for Controlled Branching Processes
A Simulation-Based Method using Gibbs SamplerConcluding Remarks and References
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
M. González Bayesian Inference for Controlled Branching Processes
Controlled Branching ProcessesBayesian Inference for Controlled Branching Processes
A Simulation-Based Method using Gibbs SamplerConcluding Remarks and References
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
M. González Bayesian Inference for Controlled Branching Processes
Controlled Branching ProcessesBayesian Inference for Controlled Branching Processes
A Simulation-Based Method using Gibbs SamplerConcluding Remarks and References
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
M. González Bayesian Inference for Controlled Branching Processes
Controlled Branching ProcessesBayesian Inference for Controlled Branching Processes
A Simulation-Based Method using Gibbs SamplerConcluding Remarks and References
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.
M. González Bayesian Inference for Controlled Branching Processes
Controlled Branching ProcessesBayesian Inference for Controlled Branching Processes
A Simulation-Based Method using Gibbs SamplerConcluding Remarks and References
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.
M. González Bayesian Inference for Controlled Branching Processes
Controlled Branching ProcessesBayesian Inference for Controlled Branching Processes
A Simulation-Based Method using Gibbs SamplerConcluding Remarks and References
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
M. González Bayesian Inference for Controlled Branching Processes
Controlled Branching ProcessesBayesian Inference for Controlled Branching Processes
A Simulation-Based Method using Gibbs SamplerConcluding Remarks and References
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)
M. González Bayesian Inference for Controlled Branching Processes
Controlled Branching ProcessesBayesian Inference for Controlled Branching Processes
A Simulation-Based Method using Gibbs SamplerConcluding Remarks and References
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)
M. González Bayesian Inference for Controlled Branching Processes
Controlled Branching ProcessesBayesian Inference for Controlled Branching Processes
A Simulation-Based Method using Gibbs SamplerConcluding Remarks and References
Controlled Branching Processes
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.
M. González Bayesian Inference for Controlled Branching Processes
Controlled Branching ProcessesBayesian Inference for Controlled Branching Processes
A Simulation-Based Method using Gibbs SamplerConcluding Remarks and References
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
M. González Bayesian Inference for Controlled Branching Processes
Controlled Branching ProcessesBayesian Inference for Controlled Branching Processes
A Simulation-Based Method using Gibbs SamplerConcluding Remarks and References
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
M. González Bayesian Inference for Controlled Branching Processes
Controlled Branching ProcessesBayesian Inference for Controlled Branching Processes
A Simulation-Based Method using Gibbs SamplerConcluding Remarks and References
Bayesian Inference for Controlled Branching Processes
Likelihood Function
L(p|Zn) ∝∏k∈S
p∑n
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∑
j=0
Zj(k) : k ∈ S)
M. González Bayesian Inference for Controlled Branching Processes
Controlled Branching ProcessesBayesian Inference for Controlled Branching Processes
A Simulation-Based Method using Gibbs SamplerConcluding Remarks and References
Bayesian Inference for Controlled Branching Processes
Likelihood Function
L(p|Zn) ∝∏k∈S
p∑n
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∑
j=0
Zj(k) : k ∈ S)
M. González Bayesian Inference for Controlled Branching Processes
Controlled Branching ProcessesBayesian Inference for Controlled Branching Processes
A Simulation-Based Method using Gibbs SamplerConcluding Remarks and References
Bayesian Inference for Controlled Branching Processes
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
M. González Bayesian Inference for Controlled Branching Processes
Controlled Branching ProcessesBayesian Inference for Controlled Branching Processes
A Simulation-Based Method using Gibbs SamplerConcluding Remarks and References
Bayesian Inference for Controlled Branching Processes
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
M. González Bayesian Inference for Controlled Branching Processes
Controlled Branching ProcessesBayesian Inference for Controlled Branching Processes
A Simulation-Based Method using Gibbs SamplerConcluding Remarks and References
Bayesian Inference for Controlled Branching Processes
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
M. González Bayesian Inference for Controlled Branching Processes
Controlled Branching ProcessesBayesian Inference for Controlled Branching Processes
A Simulation-Based Method using Gibbs SamplerConcluding Remarks and References
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
M. González Bayesian Inference for Controlled Branching Processes
Controlled Branching ProcessesBayesian Inference for Controlled Branching Processes
A Simulation-Based Method using Gibbs SamplerConcluding Remarks and References
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
M. González Bayesian Inference for Controlled Branching Processes
Controlled Branching ProcessesBayesian Inference for Controlled Branching Processes
A Simulation-Based Method using Gibbs SamplerConcluding Remarks and References
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
M. González Bayesian Inference for Controlled Branching Processes
Controlled Branching ProcessesBayesian Inference for Controlled Branching Processes
A Simulation-Based Method using Gibbs SamplerConcluding Remarks and References
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
M. González Bayesian Inference for Controlled Branching Processes
Controlled Branching ProcessesBayesian Inference for Controlled Branching Processes
A Simulation-Based Method using Gibbs SamplerConcluding Remarks and References
Introducing the MethodDeveloping the MethodSimulated Example
Gibbs Sampler: Introducing the Method
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)
M. González Bayesian Inference for Controlled Branching Processes
Controlled Branching ProcessesBayesian Inference for Controlled Branching Processes
A Simulation-Based Method using Gibbs SamplerConcluding Remarks and References
Introducing the MethodDeveloping the MethodSimulated Example
Gibbs Sampler: Introducing the Method
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)
M. González Bayesian Inference for Controlled Branching Processes
Controlled Branching ProcessesBayesian Inference for Controlled Branching Processes
A Simulation-Based Method using Gibbs SamplerConcluding Remarks and References
Introducing the MethodDeveloping the MethodSimulated Example
Gibbs Sampler: Introducing the Method
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)
M. González Bayesian Inference for Controlled Branching Processes
Controlled Branching ProcessesBayesian Inference for Controlled Branching Processes
A Simulation-Based Method using Gibbs SamplerConcluding Remarks and References
Introducing the MethodDeveloping the MethodSimulated Example
Gibbs Sampler: Introducing the Method
Second Conditional Distribution: Zn|Z∗n , p
P(Zn|Z∗n , p) =n∏
j=0
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)
M. González Bayesian Inference for Controlled Branching Processes
Controlled Branching ProcessesBayesian Inference for Controlled Branching Processes
A Simulation-Based Method using Gibbs SamplerConcluding Remarks and References
Introducing the MethodDeveloping the MethodSimulated Example
Gibbs Sampler: Introducing the Method
Second Conditional Distribution: Zn|Z∗n , p
P(Zn|Z∗n , p) =n∏
j=0
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)
M. González Bayesian Inference for Controlled Branching Processes
Controlled Branching ProcessesBayesian Inference for Controlled Branching Processes
A Simulation-Based Method using Gibbs SamplerConcluding Remarks and References
Introducing the MethodDeveloping the MethodSimulated Example
Gibbs Sampler: Introducing the Method
Second Conditional Distribution: Zn|Z∗n , p
pφ(Z0)
Z0(k), k ∈ SZ1 φ(Z1)
Z1(k), k ∈ SZ2 φ(Z2)...
......
Zn φ(Zn)Zn(k), k ∈ S
Zn+1
φ(Zj) =∑k∈S
Zj(k), Zj+1 =∑k∈S
kZj(k)
M. González Bayesian Inference for Controlled Branching Processes
Controlled Branching ProcessesBayesian Inference for Controlled Branching Processes
A Simulation-Based Method using Gibbs SamplerConcluding Remarks and References
Introducing the MethodDeveloping the MethodSimulated Example
Gibbs Sampler: Introducing the Method
Second Conditional Distribution: Zn|Z∗n , p
pφ(Z0)
Z0(k), k ∈ SZ1 φ(Z1)
Z1(k), k ∈ SZ2 φ(Z2)...
......
Zn φ(Zn)Zn(k), k ∈ S
Zn+1
φ(Zj) =∑k∈S
Zj(k), Zj+1 =∑k∈S
kZj(k)
M. González Bayesian Inference for Controlled Branching Processes
Controlled Branching ProcessesBayesian Inference for Controlled Branching Processes
A Simulation-Based Method using Gibbs SamplerConcluding Remarks and References
Introducing the MethodDeveloping the MethodSimulated Example
Gibbs Sampler: Introducing the Method
Second Conditional Distribution: Zn|Z∗n , p
pφ(Z0)
Z0(k), k ∈ SZ1 φ(Z1)
Z1(k), k ∈ SZ2 φ(Z2)...
......
Zn φ(Zn)Zn(k), k ∈ S
Zn+1
φ(Zj) =∑k∈S
Zj(k), Zj+1 =∑k∈S
kZj(k)
M. González Bayesian Inference for Controlled Branching Processes
Controlled Branching ProcessesBayesian Inference for Controlled Branching Processes
A Simulation-Based Method using Gibbs SamplerConcluding Remarks and References
Introducing the MethodDeveloping the MethodSimulated Example
Gibbs Sampler: Introducing the Method
Second Conditional Distribution: Zn|Z∗n , p
pφ(Z0)
Z0(k), k ∈ SZ1 φ(Z1)
Z1(k), k ∈ SZ2 φ(Z2)...
......
Zn φ(Zn)Zn(k), k ∈ S
Zn+1
φ(Zj) =∑k∈S
Zj(k), Zj+1 =∑k∈S
kZj(k)
M. González Bayesian Inference for Controlled Branching Processes
Controlled Branching ProcessesBayesian Inference for Controlled Branching Processes
A Simulation-Based Method using Gibbs SamplerConcluding Remarks and References
Introducing the MethodDeveloping the MethodSimulated Example
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).
M. González Bayesian Inference for Controlled Branching Processes
Controlled Branching ProcessesBayesian Inference for Controlled Branching Processes
A Simulation-Based Method using Gibbs SamplerConcluding Remarks and References
Introducing the MethodDeveloping the MethodSimulated Example
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).
M. González Bayesian Inference for Controlled Branching Processes
Controlled Branching ProcessesBayesian Inference for Controlled Branching Processes
A Simulation-Based Method using Gibbs SamplerConcluding Remarks and References
Introducing the MethodDeveloping the MethodSimulated Example
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).
M. González Bayesian Inference for Controlled Branching Processes
Controlled Branching ProcessesBayesian Inference for Controlled Branching Processes
A Simulation-Based Method using Gibbs SamplerConcluding Remarks and References
Introducing the MethodDeveloping the MethodSimulated Example
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).
M. González Bayesian Inference for Controlled Branching Processes
Controlled Branching ProcessesBayesian Inference for Controlled Branching Processes
A Simulation-Based Method using Gibbs SamplerConcluding Remarks and References
Introducing the MethodDeveloping the MethodSimulated Example
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
510
1520
25
Controlled Branching Process
0 20 40 60 80 100
020
040
060
0
Galton−Watson Branching Process
M. González Bayesian Inference for Controlled Branching Processes
Controlled Branching ProcessesBayesian Inference for Controlled Branching Processes
A Simulation-Based Method using Gibbs SamplerConcluding Remarks and References
Introducing the MethodDeveloping the MethodSimulated Example
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.
M. González Bayesian Inference for Controlled Branching Processes
Controlled Branching ProcessesBayesian Inference for Controlled Branching Processes
A Simulation-Based Method using Gibbs SamplerConcluding Remarks and References
Introducing the MethodDeveloping the MethodSimulated Example
Gibbs Sampler: Simulated ExampleGelman-Rubin-Brooks diagnostic plots (CODA package for R)
0 2000 4000 6000 8000 10000
1.0
1.5
2.0
2.5
p0
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
p1
last iteration in chain
shrin
k fa
ctor
median97.5%
0 2000 4000 6000 8000 10000
1.0
1.5
2.0
2.5
p2
last iteration in chain
shrin
k fa
ctor
median97.5%
0 2000 4000 6000 8000 10000
1.0
1.5
2.0
2.5
p3
last iteration in chain
shrin
k fa
ctor
median97.5%
0 2000 4000 6000 8000 10000
1.0
1.2
1.4
1.6
1.8
p4
last iteration in chain
shrin
k fa
ctor
median97.5%
M. González Bayesian Inference for Controlled Branching Processes
Controlled Branching ProcessesBayesian Inference for Controlled Branching Processes
A Simulation-Based Method using Gibbs SamplerConcluding Remarks and References
Introducing the MethodDeveloping the MethodSimulated Example
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.
M. González Bayesian Inference for Controlled Branching Processes
Controlled Branching ProcessesBayesian Inference for Controlled Branching Processes
A Simulation-Based Method using Gibbs SamplerConcluding Remarks and References
Introducing the MethodDeveloping the MethodSimulated Example
Gibbs Sampler: Simulated ExampleSample Information: Z∗nN = 5000, G = 100, Q = 49 and T = 100 (Sample Size: 5000)
p0
Den
sity
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
01
23
45
hpd 95% hpd 95%p0p̂0
p3
Den
sity
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
05
1015
2025
30
hpd 95% hpd 95%p0p̂3
Offspring Mean
Den
sity
0.9 1.0 1.1 1.2 1.3
02
46
hpd 95% hpd 95%mm̂
Algorithm’s Efficiency
MEAN SD MCSE TSSE1.0766931 0.0579821 0.0008200 0.0008179
M. González Bayesian Inference for Controlled Branching Processes
Controlled Branching ProcessesBayesian Inference for Controlled Branching Processes
A Simulation-Based Method using Gibbs SamplerConcluding Remarks and References
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.
M. González Bayesian Inference for Controlled Branching Processes
Controlled Branching ProcessesBayesian Inference for Controlled Branching Processes
A Simulation-Based Method using Gibbs SamplerConcluding Remarks and References
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.
M. González Bayesian Inference for Controlled Branching Processes
Controlled Branching ProcessesBayesian Inference for Controlled Branching Processes
A Simulation-Based Method using Gibbs SamplerConcluding Remarks and References
Concluding RemarksReferences
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.
M. González Bayesian Inference for Controlled Branching Processes
Controlled Branching ProcessesBayesian Inference for Controlled Branching Processes
A Simulation-Based Method using Gibbs SamplerConcluding Remarks and References
Concluding RemarksReferences
Thank you very much!
M. González Bayesian Inference for Controlled Branching Processes