Main obstacles of Bayesian statistics or Bayesian machine learning is computing posterior distribution. In many contexts, computing posterior distribution is intractable. Today, there are two main stream to detour directly computing posterior distribution. One is using sampling method(ex. MCMC) and another is Variational inference. Compared to Variational inference, MCMC takes more time and vulnerable to high-dimensional parameters. However, MCMC has strength in simplicity and guarantees of convergence. I'll briefly introduce several methods people using in application.

1. Sampling Methods for Statistical Inference
2020. 10. 19
Jinhwan Suk
Department of Mathematical Science, KAIST
GROOT
SEMINAR
GROOT SEMINAR
GROOT AI

2. Obstacles in latent modeling
• Latent variable model
• We need posterior
Sampling Methods for Statistical Inference
GROOT
SEMINAR

3. Latent Variable Model
GROOT
SEMINAR
𝑧 𝒟
e.g. RBM, VAE, GAN, HMM, Particle Filter, …
prior
likelihood
Data

4. Latent Variable Model
GROOT
SEMINAR
𝑧 𝒟
e.g. RBM, VAE, GAN, HMM, Particle Filter, …
prior
likelihood
Data

5. Latent Variable Model
GROOT
SEMINAR
𝑧 𝒟
e.g. RBM, VAE, GAN, HMM, Particle Filter, …
prior
likelihood
Data

6. Latent Variable Model
GROOT
SEMINAR
𝑧 𝒟
e.g. RBM, VAE, GAN, HMM, Particle Filter, …
prior
likelihood
Hidden markov model
Data

7. Latent Variable Model
GROOT
SEMINAR
𝑧 𝒟
e.g. RBM, VAE, GAN, HMM, Particle Filter, …
prior
likelihood
Hidden markov model VAE
Data

8. Latent Variable Model
GROOT
SEMINAR
zprior
likelihood Given
Prior : 𝑝𝜃(𝑧) Posterior : 𝑝𝜃(𝒟| 𝑧)
𝒟
Data

9. Latent Variable Model
GROOT
SEMINAR
zprior
likelihood
^𝜃 = 𝑎𝑟𝑔𝑚𝑎𝑥 𝑝𝜃(𝒟)
Inference
Given
Prior : 𝑝𝜃(𝑧) Posterior : 𝑝𝜃(𝒟| 𝑧)
𝒟
Data

10. Latent Variable Model
GROOT
SEMINAR
zprior
likelihood
^𝜃 = 𝑎𝑟𝑔𝑚𝑎𝑥 𝑝𝜃(𝒟)
Inference
𝑝𝜃( 𝒟) =
∫
𝑝𝜃(𝑧) 𝑝 𝜃( 𝒟 𝑧) 𝑑𝑧
Compute
Given
Prior : 𝑝𝜃(𝑧) Posterior : 𝑝𝜃(𝒟| 𝑧)
𝒟
Data

11. Latent Variable Model
GROOT
SEMINAR
zprior
likelihood
^𝜃 = 𝑎𝑟𝑔𝑚𝑎𝑥 𝑝𝜃(𝒟)
Inference
𝑝𝜃( 𝒟) =
∫
𝑝𝜃(𝑧) 𝑝 𝜃( 𝒟 𝑧) 𝑑𝑧
Compute Intractable!
Given
Prior : 𝑝𝜃(𝑧) Posterior : 𝑝𝜃(𝒟| 𝑧)
𝒟
Data

12. Latent Variable Model
GROOT
SEMINAR
zprior
likelihood
^𝜃 = 𝑎𝑟𝑔𝑚𝑎𝑥 𝑝𝜃(𝒟)
Inference
𝑝𝜃( 𝒟) =
∫
𝑝𝜃(𝑧) 𝑝 𝜃( 𝒟 𝑧) 𝑑𝑧
Compute Intractable!
Given
Prior : 𝑝𝜃(𝑧) Posterior : 𝑝𝜃(𝒟| 𝑧)
𝒟
Data
EM Algorithm

13. Latent Variable Model
GROOT
SEMINAR
zprior
likelihood
^𝜃 = 𝑎𝑟𝑔𝑚𝑎𝑥 𝑝𝜃(𝒟)
Inference
𝑝𝜃( 𝒟) =
∫
𝑝𝜃(𝑧) 𝑝 𝜃( 𝒟 𝑧) 𝑑𝑧
Compute Intractable!
Given
Prior : 𝑝𝜃(𝑧) Posterior : 𝑝𝜃(𝒟| 𝑧)
𝒟
Data
EM Algorithm
log𝑝( 𝒟) =
∫
log 𝑝( 𝒟) 𝑝( 𝑧 𝒟) 𝑑𝑧
=
∫
𝑙𝑜𝑔
𝑝( 𝒟, 𝑧)
𝑝( 𝑧 𝒟)
𝑝( 𝑧 𝒟) 𝑑𝑧

14. Latent Variable Model
GROOT
SEMINAR
zprior
likelihood
^𝜃 = 𝑎𝑟𝑔𝑚𝑎𝑥 𝑝𝜃(𝒟)
Inference
𝑝𝜃( 𝒟) =
∫
𝑝𝜃(𝑧) 𝑝 𝜃( 𝒟 𝑧) 𝑑𝑧
Compute Intractable!
Given
Prior : 𝑝𝜃(𝑧) Posterior : 𝑝𝜃(𝒟| 𝑧)
𝒟
Data
EM Algorithm
log𝑝( 𝒟) =
∫
log 𝑝( 𝒟) 𝑝( 𝑧 𝒟) 𝑑𝑧
=
∫
𝑙𝑜𝑔
𝑝( 𝒟, 𝑧)
𝑝( 𝑧 𝒟)
𝑝( 𝑧 𝒟) 𝑑𝑧
=
∫
log𝑝( 𝒟, 𝑧) 𝑝(𝑧| 𝒟) 𝑑𝑧 −
∫
log𝑝( 𝑧 𝒟) 𝑝( 𝑧 𝒟) 𝑑𝑧

15. Latent Variable Model
GROOT
SEMINAR
zprior
likelihood
^𝜃 = 𝑎𝑟𝑔𝑚𝑎𝑥 𝑝𝜃(𝒟)
Inference
𝑝𝜃( 𝒟) =
∫
𝑝𝜃(𝑧) 𝑝 𝜃( 𝒟 𝑧) 𝑑𝑧
Compute Intractable!
Given
Prior : 𝑝𝜃(𝑧) Posterior : 𝑝𝜃(𝒟| 𝑧)
𝒟
Data
EM Algorithm
log𝑝( 𝒟) =
∫
log 𝑝( 𝒟) 𝑝( 𝑧 𝒟) 𝑑𝑧
=
∫
𝑙𝑜𝑔
𝑝( 𝒟, 𝑧)
𝑝( 𝑧 𝒟)
𝑝( 𝑧 𝒟) 𝑑𝑧
=
∫
log𝑝( 𝒟, 𝑧) 𝑝(𝑧| 𝒟) 𝑑𝑧 −
∫
log𝑝( 𝑧 𝒟) 𝑝( 𝑧 𝒟) 𝑑𝑧
= 𝔼𝑝( 𝑧 𝒟)[log𝑝( 𝒟, 𝑧)] + 𝐻 ≥ 𝔼𝑝( 𝑧 𝒟)[log𝑝( 𝒟, 𝑧)]

16. Latent Variable Model
GROOT
SEMINAR
zprior
likelihood
^𝜃 = 𝑎𝑟𝑔𝑚𝑎𝑥 𝑝𝜃(𝒟)
Inference
𝑝𝜃( 𝒟) =
∫
𝑝𝜃(𝑧) 𝑝 𝜃( 𝒟 𝑧) 𝑑𝑧
Compute Intractable!
Given
Prior : 𝑝𝜃(𝑧) Posterior : 𝑝𝜃(𝒟| 𝑧)
𝒟
Data
EM Algorithm
log𝑝( 𝒟) =
∫
log 𝑝( 𝒟) 𝑝( 𝑧 𝒟) 𝑑𝑧
=
∫
𝑙𝑜𝑔
𝑝( 𝒟, 𝑧)
𝑝( 𝑧 𝒟)
𝑝( 𝑧 𝒟) 𝑑𝑧
=
∫
log𝑝( 𝒟, 𝑧) 𝑝(𝑧| 𝒟) 𝑑𝑧 −
∫
log𝑝( 𝑧 𝒟) 𝑝( 𝑧 𝒟) 𝑑𝑧
= 𝔼𝑝( 𝑧 𝒟)[log𝑝( 𝒟, 𝑧)] + 𝐻 ≥ 𝔼𝑝( 𝑧 𝒟)[log𝑝( 𝒟, 𝑧)]
E-Step

17. Latent Variable Model
GROOT
SEMINAR
zprior
likelihood
^𝜃 = 𝑎𝑟𝑔𝑚𝑎𝑥 𝑝𝜃(𝒟)
Inference
𝑝𝜃( 𝒟) =
∫
𝑝𝜃(𝑧) 𝑝 𝜃( 𝒟 𝑧) 𝑑𝑧
Compute Intractable!
Given
Prior : 𝑝𝜃(𝑧) Posterior : 𝑝𝜃(𝒟| 𝑧)
𝒟
Data
EM Algorithm
log𝑝( 𝒟) =
∫
log 𝑝( 𝒟) 𝑝( 𝑧 𝒟) 𝑑𝑧
=
∫
𝑙𝑜𝑔
𝑝( 𝒟, 𝑧)
𝑝( 𝑧 𝒟)
𝑝( 𝑧 𝒟) 𝑑𝑧
=
∫
log𝑝( 𝒟, 𝑧) 𝑝(𝑧| 𝒟) 𝑑𝑧 −
∫
log𝑝( 𝑧 𝒟) 𝑝( 𝑧 𝒟) 𝑑𝑧
= 𝔼𝑝( 𝑧 𝒟)[log𝑝( 𝒟, 𝑧)] + 𝐻 ≥ 𝔼𝑝( 𝑧 𝒟)[log𝑝( 𝒟, 𝑧)]
E-Step
𝑝( 𝑧 𝒟) =
𝑝( 𝒟, 𝑧)
𝑝(𝒟)

18. Latent Variable Model
GROOT
SEMINAR
zprior
likelihood
^𝜃 = 𝑎𝑟𝑔𝑚𝑎𝑥 𝑝𝜃(𝒟)
Inference
𝑝𝜃( 𝒟) =
∫
𝑝𝜃(𝑧) 𝑝 𝜃( 𝒟 𝑧) 𝑑𝑧
Compute Intractable!
Given
Prior : 𝑝𝜃(𝑧) Posterior : 𝑝𝜃(𝒟| 𝑧)
𝒟
Data
EM Algorithm
log𝑝( 𝒟) =
∫
log 𝑝( 𝒟) 𝑝( 𝑧 𝒟) 𝑑𝑧
=
∫
𝑙𝑜𝑔
𝑝( 𝒟, 𝑧)
𝑝( 𝑧 𝒟)
𝑝( 𝑧 𝒟) 𝑑𝑧
=
∫
log𝑝( 𝒟, 𝑧) 𝑝(𝑧| 𝒟) 𝑑𝑧 −
∫
log𝑝( 𝑧 𝒟) 𝑝( 𝑧 𝒟) 𝑑𝑧
= 𝔼𝑝( 𝑧 𝒟)[log𝑝( 𝒟, 𝑧)] + 𝐻 ≥ 𝔼𝑝( 𝑧 𝒟)[log𝑝( 𝒟, 𝑧)]
E-Step
𝑝( 𝑧 𝒟) =
𝑝( 𝒟, 𝑧)
𝑝(𝒟)
Intractable!

19. Latent Variable Model
GROOT
SEMINAR
zprior
likelihood
^𝜃 = 𝑎𝑟𝑔𝑚𝑎𝑥 𝑝𝜃(𝒟)
Inference
𝑝𝜃( 𝒟) =
∫
𝑝𝜃(𝑧) 𝑝 𝜃( 𝒟 𝑧) 𝑑𝑧
Compute Intractable!
Given
Prior : 𝑝𝜃(𝑧) Posterior : 𝑝𝜃(𝒟| 𝑧)
Posterior inference
𝔼[ 𝑧 𝒟] 𝔼[ 𝑓(𝑧) 𝒟]
𝒟
Data

20. Latent Variable Model
GROOT
SEMINAR
zprior
likelihood
^𝜃 = 𝑎𝑟𝑔𝑚𝑎𝑥 𝑝𝜃(𝒟)
Inference
𝑝𝜃( 𝒟) =
∫
𝑝𝜃(𝑧) 𝑝 𝜃( 𝒟 𝑧) 𝑑𝑧
Compute Intractable!
Given
Prior : 𝑝𝜃(𝑧) Posterior : 𝑝𝜃(𝒟| 𝑧)
Posterior inference
𝔼[ 𝑧 𝒟] 𝔼[ 𝑓(𝑧) 𝒟]
𝒟
Data

21. Latent Variable Model
GROOT
SEMINAR
zprior
likelihood
^𝜃 = 𝑎𝑟𝑔𝑚𝑎𝑥 𝑝𝜃(𝒟)
Inference
𝑝𝜃( 𝒟) =
∫
𝑝𝜃(𝑧) 𝑝 𝜃( 𝒟 𝑧) 𝑑𝑧
Compute Intractable!
Given
Prior : 𝑝𝜃(𝑧) Posterior : 𝑝𝜃(𝒟| 𝑧)
Posterior inference
𝔼[ 𝑧 𝒟] 𝔼[ 𝑓(𝑧) 𝒟]
𝒟
Data
“Bayesian inference is all about posterior inference”

22. Latent Variable Model
GROOT
SEMINAR
zprior
likelihood
^𝜃 = 𝑎𝑟𝑔𝑚𝑎𝑥 𝑝𝜃(𝒟)
Inference
𝑝𝜃( 𝒟) =
∫
𝑝𝜃(𝑧) 𝑝 𝜃( 𝒟 𝑧) 𝑑𝑧
Compute Intractable!
Given
Prior : 𝑝𝜃(𝑧) Posterior : 𝑝𝜃(𝒟| 𝑧)
Posterior inference
𝔼[ 𝑧 𝒟] 𝔼[ 𝑓(𝑧) 𝒟]
𝒟
Data
Direct computing is impossible
“Bayesian inference is all about posterior inference”

23. Latent Variable Model
GROOT
SEMINAR
zprior
likelihood
^𝜃 = 𝑎𝑟𝑔𝑚𝑎𝑥 𝑝𝜃(𝒟)
Inference
𝑝𝜃( 𝒟) =
∫
𝑝𝜃(𝑧) 𝑝 𝜃( 𝒟 𝑧) 𝑑𝑧
Compute Intractable!
Given
Prior : 𝑝𝜃(𝑧) Posterior : 𝑝𝜃(𝒟| 𝑧)
Posterior inference
𝔼[ 𝑧 𝒟] 𝔼[ 𝑓(𝑧) 𝒟]
𝒟
Data
Direct computing is impossible
Approximation! But…how?
“Bayesian inference is all about posterior inference”

24. Latent Variable Model
GROOT
SEMINAR
zprior
likelihood
^𝜃 = 𝑎𝑟𝑔𝑚𝑎𝑥 𝑝𝜃(𝒟)
Inference
𝑝𝜃( 𝒟) =
∫
𝑝𝜃(𝑧) 𝑝 𝜃( 𝒟 𝑧) 𝑑𝑧
Compute Intractable!
Given
Prior : 𝑝𝜃(𝑧) Posterior : 𝑝𝜃(𝒟| 𝑧)
Posterior inference
𝔼[ 𝑧 𝒟] 𝔼[ 𝑓(𝑧) 𝒟]
𝒟
Data
Direct computing is impossible
Approximation! But…how?
“Bayesian inference is all about posterior inference”
Let target distribution denoted by 𝑝(𝑥)

25. Two ways of approximation
GROOT
SEMINAR

26. Two ways of approximation
GROOT
SEMINAR
1.

27. Two ways of approximation
GROOT
SEMINAR
1.
2.

28. Two ways of approximation
GROOT
SEMINAR
두 분포 사이의 거리를 줄이자
1.
2.
Variational Inference

29. Two ways of approximation
GROOT
SEMINAR
두 분포 사이의 거리를 줄이자
1.
2.
Variational Inference
Optimization problem

30. Two ways of approximation
GROOT
SEMINAR
두 분포 사이의 거리를 줄이자
1.
2. 표본을 추출하자
Variational Inference
Optimization problem

31. Two ways of approximation
GROOT
SEMINAR
두 분포 사이의 거리를 줄이자
1.
2. 표본을 추출하자
Variational Inference
Monte Carlo Method
Optimization problem

32. Two ways of approximation
GROOT
SEMINAR
두 분포 사이의 거리를 줄이자
1.
2. 표본을 추출하자
Variational Inference
Monte Carlo Method
Optimization problem
Sampling method

33. Monte Carlo Method
GROOT
SEMINAR

34. Monte Carlo Method
GROOT
SEMINAR
The Law of Large Numbers
𝑋1 + 𝑋2 + ⋯ + 𝑋𝑛
𝑛
→ 𝔼[𝑋𝑖]𝑋1, 𝑋2, …, 𝑋𝑛 : 𝑖 . 𝑖 . 𝑑

35. Monte Carlo Method
GROOT
SEMINAR
The Law of Large Numbers
𝑋1 + 𝑋2 + ⋯ + 𝑋𝑛
𝑛
→ 𝔼[𝑋𝑖]𝑋1, 𝑋2, …, 𝑋𝑛 : 𝑖 . 𝑖 . 𝑑
But…

36. Monte Carlo Method
GROOT
SEMINAR
The Law of Large Numbers
𝑋1 + 𝑋2 + ⋯ + 𝑋𝑛
𝑛
→ 𝔼[𝑋𝑖]𝑋1, 𝑋2, …, 𝑋𝑛 : 𝑖 . 𝑖 . 𝑑
But… Yes, How do we sample from ?𝑝(𝑥)

37. Monte Carlo Method
GROOT
SEMINAR
The Law of Large Numbers
𝑋1 + 𝑋2 + ⋯ + 𝑋𝑛
𝑛
→ 𝔼[𝑋𝑖]𝑋1, 𝑋2, …, 𝑋𝑛 : 𝑖 . 𝑖 . 𝑑
But… Yes, How do we sample from ?𝑝(𝑥)
Rejection Sampling

38. Monte Carlo Method
GROOT
SEMINAR
The Law of Large Numbers
𝑋1 + 𝑋2 + ⋯ + 𝑋𝑛
𝑛
→ 𝔼[𝑋𝑖]𝑋1, 𝑋2, …, 𝑋𝑛 : 𝑖 . 𝑖 . 𝑑
But… Yes, How do we sample from ?𝑝(𝑥)
: proposal distribution𝑞(𝑥)
Rejection Sampling
Easy to compute, easy to sample

39. Monte Carlo Method
GROOT
SEMINAR
The Law of Large Numbers
𝑋1 + 𝑋2 + ⋯ + 𝑋𝑛
𝑛
→ 𝔼[𝑋𝑖]𝑋1, 𝑋2, …, 𝑋𝑛 : 𝑖 . 𝑖 . 𝑑
But… Yes, How do we sample from ?𝑝(𝑥)
: proposal distribution𝑞(𝑥)
Rejection Sampling
Easy to compute, easy to sample
𝑀𝑞(𝑥) ≥ ~𝑝(𝑥)
If , we reject the sample, otherwise we accept it𝑢 >
~𝑝(𝑥)
𝑀𝑞(𝑥)

40. Monte Carlo Method
GROOT
SEMINAR
The Law of Large Numbers
𝑋1 + 𝑋2 + ⋯ + 𝑋𝑛
𝑛
→ 𝔼[𝑋𝑖]𝑋1, 𝑋2, …, 𝑋𝑛 : 𝑖 . 𝑖 . 𝑑
But… Yes, How do we sample from ?𝑝(𝑥)
: proposal distribution𝑞(𝑥)
Rejection Sampling
Easy to compute, easy to sample
𝑀𝑞(𝑥) ≥ ~𝑝(𝑥)
If , we reject the sample, otherwise we accept it𝑢 >
~𝑝(𝑥)
𝑀𝑞(𝑥)

41. Monte Carlo Method
GROOT
SEMINAR

42. Monte Carlo Method
GROOT
SEMINAR
𝜇 = 𝔼𝑝(𝑥)[ 𝑓(𝑥)] =
∫
𝑓(𝑥)
~𝑝(𝑥)
𝑍
𝑑𝑥 =
1
𝑍 ∫
𝑓(𝑥)
~𝑝(𝑥)
𝑞(𝑥)
𝑞(𝑥)𝑑𝑥
Importance Samping

43. Monte Carlo Method
GROOT
SEMINAR
𝜇 = 𝔼𝑝(𝑥)[ 𝑓(𝑥)] =
∫
𝑓(𝑥)
~𝑝(𝑥)
𝑍
𝑑𝑥 =
1
𝑍 ∫
𝑓(𝑥)
~𝑝(𝑥)
𝑞(𝑥)
𝑞(𝑥)𝑑𝑥
Importance Samping
𝔼𝑝(𝑥)[ 𝑓(𝑥)] ≈
1
𝑛𝑍 ∑
𝑓( 𝑥𝑖)
~𝑝( 𝑥𝑖)
𝑞( 𝑥𝑖)
=
1
𝑛𝑍 ∑
𝑤( 𝑥𝑖) 𝑓(𝑥𝑖)
𝑍 =
∫
~𝑝(𝑥)𝑑𝑥 =
∫
~𝑝(𝑥)
𝑞(𝑥)
𝑞(𝑥)𝑑𝑥 ≈
1
𝑛 ∑
𝑤(𝑥𝑖)
𝔼𝑝(𝑥)[ 𝑓(𝑥)] ≈
∑
~𝑤( 𝑥𝑖) 𝑓(𝑥𝑖)

44. Monte Carlo Method
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𝜇 = 𝔼𝑝(𝑥)[ 𝑓(𝑥)] =
∫
𝑓(𝑥)
~𝑝(𝑥)
𝑍
𝑑𝑥 =
1
𝑍 ∫
𝑓(𝑥)
~𝑝(𝑥)
𝑞(𝑥)
𝑞(𝑥)𝑑𝑥
Importance Samping
𝔼𝑝(𝑥)[ 𝑓(𝑥)] ≈
1
𝑛𝑍 ∑
𝑓( 𝑥𝑖)
~𝑝( 𝑥𝑖)
𝑞( 𝑥𝑖)
=
1
𝑛𝑍 ∑
𝑤( 𝑥𝑖) 𝑓(𝑥𝑖)
𝑍 =
∫
~𝑝(𝑥)𝑑𝑥 =
∫
~𝑝(𝑥)
𝑞(𝑥)
𝑞(𝑥)𝑑𝑥 ≈
1
𝑛 ∑
𝑤(𝑥𝑖)
𝔼𝑝(𝑥)[ 𝑓(𝑥)] ≈
∑
~𝑤( 𝑥𝑖) 𝑓(𝑥𝑖)
i.i.d sampling is very vulnerable in high-dimensional spaces

45. Markov Chain Monte Carlo
• Gibbs Sampling
• Metropolis-Hestings Algorithm
Sampling Methods for Statistical Inference
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46. Basic idea of MCMC
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Markov Chain :
𝑝( 𝑥(𝑡+1)
𝑥(1), 𝑥(2)
, …, 𝑥(𝑡)
) = 𝑝( 𝑥(𝑡+1)
𝑥(𝑡)
)
is a sequence of random variables. It forms a Markov chain if𝑥(1)
, 𝑥(2)
, …
A Markov chain can be specified by
1. Initial distribution
2. Transition probability
𝑝1(𝑥) = 𝑝( 𝑥(1)
)
𝑇 𝑘(𝑥′, 𝑥) = 𝑝( 𝑥(𝑡+1)
= 𝑥′ 𝑥(𝑡)
= 𝑥)
Ergodicity
, regardless of the initial distributionlim
𝑛→∞
𝑇 𝑛
𝑝1 = 𝜋 𝑝1
1. Build a Markov chain having
as an invariant distribution
2. Sample from the chain
3. Compute
𝑝(𝑥)
( 𝑥(𝑡)
)𝑡≥1
𝔼𝑝(𝑥)[ 𝑓(𝑥)] ≈ 𝔼 𝑇 𝑛 𝑝1(𝑥)[ 𝑓(𝑥)]
≈
1
𝑛 ∑
𝑓( 𝑥(𝑡)
)

47. MCMC Algorithms
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𝑥 𝑠+1
𝑖 ∼ 𝑝(𝑥𝑖 | 𝒙−𝒊)Gibbs Sampling
Metropolis-Hastings algorithm Hamiltonian MCMC
Sequential MCMC
Stochastic gradient MCMC
Particle MCMC

48. Thank you
Sampling Methods for Statistical Inference
2020. 10. 19
Jinhwan Suk
Department of Mathematical Science, KAIST
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