論文紹介 Adaptive metropolis algorithm using variational bayesian

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

Abstract
1
 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.

Index 2
1. Introduction – What’s Statistical Problem?
2. MCMC
3. Adaptive MCMC Methods
4. VB-AKF
5. VBAM
6. Numerical Experiments
7. Conclusion

Index 3
2. MCMC
4. VB-AKF
5. VBAM
7. Conclusion

What’s Statistical Problem? (1)
-Modeling- 4
Linear Regression State Space Model
𝒚 = 𝒘 𝑇 𝒙 + 𝝐
𝒙 𝑡= 𝐴𝒙 𝑡−1 + 𝜼
𝒚 𝑡 = 𝐻𝒙 𝑡 + 𝝐

What’s Statistical Problem? (1)
-Modeling- 5
Linear Regression State Space Model
𝒚 = 𝒘 𝑇 𝒙 + 𝝐
𝒙 𝑡= 𝐹𝒙 𝑡−1 + 𝜼
𝒚 𝑡 = 𝐻𝒙 𝑡 + 𝝐
Parameter 𝜽

What’s Statistical Problem?(2)
-Parameter Estimation- 6
Point Estimation
• Maximum Likelihood
→ EM algorithm
• Maximum a Posteriori (MAP)
Interval Estimation
• Bayes method
Variational Bayes
Marcov Chain Monte Carlo
𝜃
𝜃
posterior distribution

Index 7
2. MCMC
4. VB-AKF
5. VBAM
7. Conclusion

MCMC
(Malkov Chain Monte Carlo Methods) 8
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

Computing Expectation by MCMC 9
𝑬 𝑓 = 𝑓 𝑧 𝑝 𝜃 𝑑𝜃
𝑬 𝑓 ≈
1
𝑛
𝑠=1
𝑛
𝑓(𝜃(𝑠))
Computing Expectation is difficult
Sampling 𝜃(𝑠) by MCMC

Metropolis Algorithm 10
http://visualize-
mcmc.appspot.com/2_metropolis.html
1. Initialize 𝜃0, Σ
2. For 𝑘 = 1,2,3, …
I. 𝜃∗
~𝑁 𝜃 𝑘−1, Σ
II. 𝛼 = min 1,
𝑝 𝜃∗ 𝑧1, … , 𝑧 𝑀
𝑝 𝜃 𝑘−1 𝑧1, … , 𝑧 𝑀
III. 𝑢~𝑈(0,1)
IV. 𝑖𝑓 𝑢 < 𝛼 ∶ set 𝜃 𝑘 = 𝜃∗
𝑒𝑙𝑠𝑒 ∶ set 𝜃 𝑘 = 𝜃 𝑘−1
Metropolis Algorithm :

http://visualize-
mcmc.appspot.com/2_metropolis.html
1. Initialize 𝜃0, Σ
2. For 𝑘 = 1,2,3, …
I. 𝜃∗
~𝑁 𝜃 𝑘−1, Σ
II. 𝛼 = min 1,
𝑝 𝜃∗ 𝑧1, … , 𝑧 𝑀
𝑝 𝜃 𝑘−1 𝑧1, … , 𝑧 𝑀
III. 𝑢~𝑈(0,1)
Metropolis Algorithm :
proposed Gaussian distribution

Index 12
2. MCMC
4. VB-AKF
5. VBAM
7. Conclusion

Adaptive Metropolis algorithm 13
The AM algorithm :
1. Initialize 𝜃0, Σ0
2. For 𝑘 = 1,2,3, …
I. 𝜃∗~𝑁 𝜃 𝑘−1, 𝜆Σk−1
II. 𝛼 = min 1,
𝑝 𝜃∗ 𝑧1, … , 𝑧 𝑀
𝑝 𝜃 𝑘−1 𝑧1, … , 𝑧 𝑀
III. 𝑢~𝑈(0,1)
V. 𝑈𝑝𝑑𝑎𝑡𝑒 Σ using following equation
Σk = cov 𝜃0, 𝜃1, … , 𝜃 𝑘 + 𝜀𝐼
𝜆 = 2.38/𝑑2
where 𝑑 is the dimension of Σ
Introduced by recursion formula

Other adaptive algorithm 14
• 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…

Index 15
2. MCMC
4. VB-AKF
5. VBAM
7. Conclusion

Variational Bayesian Adaptive
The VB-AM algorithm :
2. For 𝑘 = 1,2,3, …
II. 𝛼 = min 1,
𝑝 𝜃∗ 𝑧1, … , 𝑧 𝑀
𝑝 𝜃 𝑘−1 𝑧1, … , 𝑧 𝑀
III. 𝑢~𝑈(0,1)
V. 𝑈𝑝𝑑𝑎𝑡𝑒 Σ using VB-AKF update step

Kalman Filter (1) 17
State Space Model
𝒙 𝑘 = 𝐴 𝑘−1 𝒙 𝑘−1 + 𝜼
𝒚 𝑘 = 𝐻 𝑘 𝒙 𝑘 + 𝝐
⇔
𝒙 𝑘 ~ 𝑁(𝐴 𝑘−1 𝒙 𝑘−1, 𝑄 𝑘−1)
𝒚 𝑘 ~ 𝑁(𝑦 𝑘 𝒙 𝑘−1, Σ 𝑘)

Kalman Filter (2) 18
true data 𝑥1:𝑇 observation 𝑦1:𝑇
Gaussian Noise
𝜀~𝑁(0, Σ)
Estimate by Kalman Filter

Kalman Filter(3) 19
known 𝜃 = (𝐴, 𝐻, 𝑄, Σ)objective : assume 𝑥 ⇔ {𝑚, 𝑃}
Prediction Step
𝑚 𝑘
−
= 𝐴 𝑘−1 𝑚 𝑘−1
𝑃𝑘
−
= 𝐴 𝑘−1 𝑃𝑘−1 𝐴 𝑘−1
𝑇
+ 𝑄 𝑘−1 (8)
Update Step
𝑆 𝑘 = 𝐻 𝑘 𝑃𝑘
−
𝐻 𝑘
𝑇
+ Σ 𝑘
𝐾𝑘 = 𝑃𝑘
−
𝐻 𝑘 𝑆 𝑘
−1
𝑚 𝑘 = 𝑚 𝑘
−
+ 𝐾𝑘 𝑦 𝑘 − 𝐻 𝑘 𝑚 𝑘
−
𝑃𝑘 = 𝑃𝑘
−
− 𝐾𝑘 𝑆 𝑘 𝐾𝑘
𝑇
(9)
Initialize 𝑚0, 𝑃0
algorithm
Iterate
For 𝑘 = 1,2, …

Recursive Least Square(RLS) 20
If 𝐴 𝑘−1 = 𝐼 𝑎𝑛𝑑 𝑄 𝑘−1 = 0 the Kalman filter reduces to
Recursive Least Square
𝑥
𝑦

VB-AKF
(Variational Bayes Adaptive Kalman Filter) 21
objective : assume 𝑥 ⇔ {𝑚, 𝑃} and tuning Σ
known 𝜃 = (𝐴, 𝐻, 𝑄, Σ) Σ:unknown
Initialize 𝑚0, 𝑃0, Σ0, 𝑣0
Prediction Step
Update Step Iterate
until Σ 𝑘 is convergence
Iterate
For 𝑘 = 1,2, …
algorithm

Variational Bayes 22
Computing 𝑝 𝑥 𝑘, Σ 𝑘 𝑦1:𝑘) is intractable.
free-form Variational Bayesian approximation
𝑝 𝑥 𝑘, Σ 𝑘 𝑦1:𝑘) ≈ 𝑄 𝑥 𝑥 𝑘 𝑄Σ(Σ 𝑘)
= 𝑵 𝒙 𝑘 𝒎 𝑘, 𝑷 𝑘)𝑰𝑾 Σ 𝑘 𝑣 𝑘, 𝑽 𝑘)

heuristic dynamic for the covariances
23
Such kind of dynamical model is hard to construct explicitly
Sarkka et al. proposed heuristic dynamic for the covariances
𝑣 𝑘
−
= 𝜌 𝑣 𝑘−1 − 𝑑 − 1 + 𝑑 + 1
Σ 𝑘
−
= 𝑩Σ 𝑘−1
−
𝑩 𝑇
(21)

VB-AKF
(Variational Bayes Adaptive Kalman Filter) 24
objective : assume 𝑥 ⇔ {𝑚, 𝑃} and tuning Σ
known 𝜃 = (𝐴, 𝐻, 𝑄, Σ) Σ:unknown
Initialize 𝑚0, 𝑃0, Σ0, 𝑣0
Prediction Step
Update Step Iterate
until Σ 𝑘 is convergence
Iterate
For 𝑘 = 1,2, …
algorithm

Index 25
2. MCMC
4. VB-AKF
5. VBAM
7. Conclusion

Variational Bayesian Adaptive
The VB-AM algorithm :
2. For 𝑘 = 1,2,3, …
II. 𝛼 = min 1,
𝑝 𝜃∗ 𝑧1, … , 𝑧 𝑀
𝑝 𝜃 𝑘−1 𝑧1, … , 𝑧 𝑀
III. 𝑢~𝑈(0,1)
V. 𝑈𝑝𝑑𝑎𝑡𝑒 Σ using VB-AKF update step

Index 32
2. MCMC
4. VB-AKF
5. VBAM
7. Conclusion

Conclusion
33
 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.

Discussion 34
 The advantage of the proposed method is that it has
more parameters to tune , which gives more freedom.
 The computational requirements of VBAM method
𝑂 𝑑3 , while the usual AM is 𝑂(𝑑2).
 But these operations are still quite cheap compared
with MCMC sampling.

Future works 35
 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.

State Space Model(1) 36
State Space Model
𝑥2 𝑥 𝑇𝑥1
𝑦1 𝑦2 𝑦 𝑇
latent variable
observed variable
sys-eq
obs-eq

論文紹介 Adaptive metropolis algorithm using variational bayesian

Recommended

More Related Content

What's hot (20)

Similar to 論文紹介 Adaptive metropolis algorithm using variational bayesian (20)

Recently uploaded (20)

論文紹介 Adaptive metropolis algorithm using variational bayesian

Editor's Notes