論文紹介 Adaptive metropolis algorithm using variational bayesian
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Transcript of 論文紹介 Adaptive metropolis algorithm using variational bayesian
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Mbalawata, Isambi S., et al.
Computational Statistics & Data Analysis 83 (2015): 101-115.
Adaptive Metropolis Algorithm Using Variational Bayesian Adaptive Kalman Filter
Presenter : Shuuji Mihara
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Abstract2
This Paper propose a new adaptive MCMC algorithm called Variational Bayesian adaptive Metropolis(VBAM).
The VBAM algorithm updates the proposal covariance matrix using the Variational Bayesian adaptive Kalman filter(VB-AKF).
In the simulated experiments, VBAM perform better than the AM algorithm of Harrio et al.
In the real data examples, VBAM produced results consisted with results reported literature.
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3Index
1. Introduction – What’s Statistical Problem?
2. MCMC
3. Adaptive MCMC Methods
4. VB-AKF
5. VBAM
6. Numerical Experiments
7. Conclusion
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4Index
1. Introduction – What’s Statistical Problem?
2. MCMC
3. Adaptive MCMC Methods
4. VB-AKF
5. VBAM
6. Numerical Experiments
7. Conclusion
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5What’s Statistical Problem? (1)-Modeling-
Linear Regression State Space Model
𝒚=𝒘𝑇 𝒙+𝝐 {𝒙𝑡=𝐴𝒙 𝑡−1+𝜼𝒚 𝑡=𝐻 𝒙𝑡+𝝐
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6What’s Statistical Problem? (1)-Modeling-
Linear Regression State Space Model
𝒚=𝒘𝑇 𝒙+𝝐 {𝒙𝑡=𝐹 𝒙𝑡 −1+𝜼𝒚 𝑡=𝐻 𝒙 𝑡+𝝐
Parameter
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7What’s Statistical Problem?(2)-Parameter Estimation-
Point Estimation
• Maximum Likelihood
→ EM algorithm
• Maximum a Posteriori (MAP)
Interval Estimation
• Bayes method
Variational Bayes
Marcov Chain Monte Carlo
𝜃
𝜃posterior distribution
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8Index
1. Introduction – What’s Statistical Problem?
2. MCMC
3. Adaptive MCMC Methods
4. VB-AKF
5. VBAM
6. Numerical Experiments
7. Conclusion
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9MCMC(Malkov Chain Monte Carlo Methods)
For many models of practical interest, it will be infeasible to evaluate the posterior distribution or indeed to compute expectations .
We need approximate scheme = MCMC
Metropolis Algorithm
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10Computing Expectation by MCMC
𝑬 [ 𝑓 ]=∫ 𝑓 (𝑧 )𝑝 (𝜃 )𝑑𝜃
𝑬 [ 𝑓 ]≈ 1𝑛∑𝑠=1
𝑛
𝑓 (𝜃(𝑠))
Computing Expectation is difficult
Sampling by MCMC
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11Metropolis Algorithm
http://visualize-mcmc.appspot.com/2_metropolis.html
1. Initialize 2. For
I. set set
Metropolis Algorithm :
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12Metropolis Algorithm
http://visualize-mcmc.appspot.com/2_metropolis.html
1. Initialize 2. For
I. set set
Metropolis Algorithm :
proposed Gaussian distribution
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13Index
1. Introduction – What’s Statistical Problem?
2. MCMC
3. Adaptive MCMC Methods
4. VB-AKF
5. VBAM
6. Numerical Experiments
7. Conclusion
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14Adaptive Metropolis algorithm
The AM algorithm :
1. Initialize 2. For
I. set set
II. using following equation
/ where is the dimension of
Introduced by recursion formula
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15Other adaptive algorithm
• regeneration-based adaptive algorithm [Gilks et al 1998]
• adaptive independent Metropolis-Hastings algorithm[Holden et al 2009]
• Robust adaptive Metropolis(RAM) [Vihola 2012]
• MCMC integrated with differential evolution [Vrugt et al 2009]
etc…
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16Index
1. Introduction – What’s Statistical Problem?
2. MCMC
3. Adaptive MCMC Methods
4. VB-AKF
5. VBAM
6. Numerical Experiments
7. Conclusion
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17Variational Bayesian Adaptive Metropolis Algorithm
The VB-AM algorithm :
1. Initialize 2. For
I. set set
II. using VB-AKF update step
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18Kalman Filter (1)
State Space Model
⇔
𝒙𝑘 𝑁 (𝐴𝑘−1𝒙𝑘−1 ,𝑄𝑘−1)𝒚 𝑘 𝑁 (𝑦𝑘 𝒙𝑘−1 , Σ𝑘)
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19Kalman Filter (2)
true data observation
Gaussian Noise
Estimate by Kalman Filter
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20Kalman Filter(3)
known )objective : }
Prediction Step
(8)
Update Step
(9)
Initialize
algorithm
IterateFor
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21Recursive Least Square(RLS)
If the Kalman filter reduces to Recursive Least Square
𝑥
𝑦
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22VB-AKF(Variational Bayes Adaptive Kalman Filter)
objective :
known unknown
Initialize
Prediction Step
Update Step Iterateuntil is convergence
IterateFor
algorithm
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23Variational Bayes
Computing is intractable.
free-form Variational Bayesian approximation
𝑝 (𝑥𝑘 , Σ𝑘|𝑦1 :𝑘¿≈𝑄𝑥 (𝑥𝑘)𝑄Σ(Σ𝑘)
¿𝑵 (𝒙𝑘|𝒎𝑘 ,𝑷𝑘 ¿ 𝑰𝑾 (Σ𝑘|𝑣𝑘 ,𝑽 𝑘¿
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24heuristic dynamic for the covariances
Such kind of dynamical model is hard to construct explicitly
et al. proposed heuristic dynamic for the covariances
𝑣𝑘−=𝜌 (𝑣𝑘− 1−𝑑−1 )+𝑑+1
Σ𝑘−=𝑩Σ𝑘− 1
− 𝑩𝑇(21)
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25VB-AKF(Variational Bayes Adaptive Kalman Filter)
objective :
known unknown
Initialize
Prediction Step
Update Step Iterateuntil is convergence
IterateFor
algorithm
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26Index
1. Introduction – What’s Statistical Problem?
2. MCMC
3. Adaptive MCMC Methods
4. VB-AKF
5. VBAM
6. Numerical Experiments
7. Conclusion
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27Variational Bayesian Adaptive Metropolis Algorithm
The VB-AM algorithm :
1. Initialize 2. For
I. set set
II. using VB-AKF update step
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28Numerical Experiment 5.1
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29Numerical Experiment 5.2
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30Numerical Experiment 5.3
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31Numerical Experiment 5.4
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32Numerical Experiment 5.5
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33Index
1. Introduction – What’s Statistical Problem?
2. MCMC
3. Adaptive MCMC Methods
4. VB-AKF
5. VBAM
6. Numerical Experiments
7. Conclusion
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Conclusion34
This Paper propose a new adaptive MCMC algorithm called Variational Bayesian adaptive Metropolis(VBAM).
The VBAM algorithm updates the proposal covariance matrix using the Variational Bayesian adaptive Kalman filter(VB-AKF).
In the simulated experiments, VBAM perform better than the AM algorithm of Harrio et al.
In the real data examples, VBAM produced results consisted with results reported literature.
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35Discussion
The advantage of the proposed method is that it has more parameters to tune , which gives more freedom.
The computational requirements of VBAM method while the usual AM is .
But these operations are still quite cheap compared with MCMC sampling.
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36Future works
replacing Kalman filter with non-linear Kalman filters (ex) extended Kalman filter , Unscented Kalman filter
particle filter (Rao-Blackwellized) could be used for estimate the noise covariance.
Compare proposal adaptation method using different kinds of filters.
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37State Space Model(1)
State Space Model
𝑥2 𝑥𝑇𝑥1
𝑦 1 𝑦 2 𝑦 𝑇
latent variable
observed variable
sys-eq
obs-eq