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
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.
3Index
1. Introduction – What’s Statistical Problem?
2. MCMC
3. Adaptive MCMC Methods
4. VB-AKF
5. VBAM
6. Numerical Experiments
7. Conclusion
4Index
1. Introduction – What’s Statistical Problem?
2. MCMC
3. Adaptive MCMC Methods
4. VB-AKF
5. VBAM
6. Numerical Experiments
7. Conclusion
5What’s Statistical Problem? (1)-Modeling-
Linear Regression State Space Model
𝒚=𝒘𝑇 𝒙+𝝐 {𝒙𝑡=𝐴𝒙 𝑡−1+𝜼𝒚 𝑡=𝐻 𝒙𝑡+𝝐
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
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
8Index
1. Introduction – What’s Statistical Problem?
2. MCMC
3. Adaptive MCMC Methods
4. VB-AKF
5. VBAM
6. Numerical Experiments
7. Conclusion
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
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
11Metropolis Algorithm
http://visualize-mcmc.appspot.com/2_metropolis.html
1. Initialize 2. For
I. set set
Metropolis Algorithm :
12Metropolis Algorithm
http://visualize-mcmc.appspot.com/2_metropolis.html
1. Initialize 2. For
I. set set
Metropolis Algorithm :
proposed Gaussian distribution
13Index
1. Introduction – What’s Statistical Problem?
2. MCMC
3. Adaptive MCMC Methods
4. VB-AKF
5. VBAM
6. Numerical Experiments
7. Conclusion
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
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…
16Index
1. Introduction – What’s Statistical Problem?
2. MCMC
3. Adaptive MCMC Methods
4. VB-AKF
5. VBAM
6. Numerical Experiments
7. Conclusion
17Variational Bayesian Adaptive Metropolis Algorithm
The VB-AM algorithm :
1. Initialize 2. For
I. set set
II. using VB-AKF update step
18Kalman Filter (1)
State Space Model
⇔
𝒙𝑘 𝑁 (𝐴𝑘−1𝒙𝑘−1 ,𝑄𝑘−1)𝒚 𝑘 𝑁 (𝑦𝑘 𝒙𝑘−1 , Σ𝑘)
19Kalman Filter (2)
true data observation
Gaussian Noise
Estimate by Kalman Filter
20Kalman Filter(3)
known )objective : }
Prediction Step
(8)
Update Step
(9)
Initialize
algorithm
IterateFor
21Recursive Least Square(RLS)
If the Kalman filter reduces to Recursive Least Square
𝑥
𝑦
22VB-AKF(Variational Bayes Adaptive Kalman Filter)
objective :
known unknown
Initialize
Prediction Step
Update Step Iterateuntil is convergence
IterateFor
algorithm
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¿
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)
25VB-AKF(Variational Bayes Adaptive Kalman Filter)
objective :
known unknown
Initialize
Prediction Step
Update Step Iterateuntil is convergence
IterateFor
algorithm
26Index
1. Introduction – What’s Statistical Problem?
2. MCMC
3. Adaptive MCMC Methods
4. VB-AKF
5. VBAM
6. Numerical Experiments
7. Conclusion
27Variational Bayesian Adaptive Metropolis Algorithm
The VB-AM algorithm :
1. Initialize 2. For
I. set set
II. using VB-AKF update step
28Numerical Experiment 5.1
29Numerical Experiment 5.2
30Numerical Experiment 5.3
31Numerical Experiment 5.4
32Numerical Experiment 5.5
33Index
1. Introduction – What’s Statistical Problem?
2. MCMC
3. Adaptive MCMC Methods
4. VB-AKF
5. VBAM
6. Numerical Experiments
7. Conclusion
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.
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.
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.
37State Space Model(1)
State Space Model
𝑥2 𝑥𝑇𝑥1
𝑦 1 𝑦 2 𝑦 𝑇
latent variable
observed variable
sys-eq
obs-eq