Particle rolling MCMC · 2018-08-12 · Introduction Particle rolling MCMC Theoretical justi cation of PRMCMC Numerical examples Motivation State space model Rolling parameter and

IntroductionParticle rolling MCMC

Theoretical justification of PRMCMCNumerical examples

Particle rolling MCMC

Yasuhiro Omori

Faculty of Economics, University of Tokyo

with Naoki Awaya

August 28, 2018

1 / 52



Outline

Introduction

Motivation

State space model

Rolling parameter and state estimation

Contribution of the paper

Particle rolling MCMC

Simple particle rolling MCMC

Particle rolling MCMC with double block sampling

Theoretical justification of PRMCMC

Forward block sampling

Backward block sampling

Numerical examples

Example 1 (Linear Gaussian SSM)

Example 2 (Realized stochastic volatility model)2 / 52



MotivationState space modelRolling parameter and state estimationContribution of the paper

1. Introduction

3 / 52




Motivation

We consider modelling economic time series during the long

sample period where we

▶ estimate model parameters,

▶ forecast on the time series

▶ for the risk management and policy simulation ...

but, in the economic time series, it is well-known that

▶ there have been major structural changes (due to such as the

financial crisis in 2008)

▶ parameters are not necessary constant and subject to changes

in the long sample period

4 / 52




Motivation

To deal with the structural change in economic times series, for

example,

▶ divide the sample periods into several subsample periods→1. How many structural changes?

2. When does the structural changes occur?

▶ fix the length of the sample period and implement the rolling

estimation→1. How to decide the length of the sample period?

2. It takes time to estimate parameters and forecast using

MCMC.

5 / 52




State space model

This paper considers the state space model

p(yt | y1:t−1, α1:t , θ) = p(yt | αt , θ) ≡ gθ(yt | αt),

p(αt | α1:t−1, θ) = p(αt | αt−1, θ) ≡ fθ(αt | αt−1),

p(α1 | θ) ≡ µθ(α1),

where

yt : an observation vector at time t,

αt : an unobserved state vector at time t,

θ: a static parameter vector,

ys:t = (ys , . . . , yt) and αs:t = (αs , . . . , αt). The correlation

between yt and αt+1 is also taken into account (e.g. leverage in

stochastic volatility models).6 / 52




Rolling parameter and state estimation

We estimate the posterior distribution of θ and αs:t given the

observations ys:t with t = s + L for s = 1, 2 . . .:

π(θ, αs:t | ys:t)

∝ p(θ)µθ(αs)gθ(ys | αs)

t∏

j=s+1

fθ(αj | αj−1, yj−1)gθ(yj | αj)

∝ p(θ)µθ(αs)

t∏

j=s+1

fθ(αj | αj−1)gθ(yj−1 | αj , αj−1)

gθ(yt | αt),

where p(θ): the prior probability density function of θ.

7 / 52




Contribution of the paper

This paper considers the state space model for economic times

series and

▶ proposes the efficient rolling estimation method for both

parameter θ and states αs:t .

▶ gives the theoretical justification of the algorithm by proving

the marginal density of parameters and states.

▶ the algorithm is based on Particle Gibbs

▶ proposes, as a special case, the sequential Bayesian estimation

method (which is alternative to SMC2 based on Particle MH).

8 / 52



Simple particle rolling MCMCParticle rolling MCMC with double block sampling

2. Particle rolling MCMC (PRMCMC)

9 / 52





Suppose, at time t − 1, we have

▶ (θn, αns−1:t−1): a collection of particles for n = 1, . . . ,N

▶ W ns−1:t−1: the importance weight which is a discrete

approximation of π(θ, αs−1:t−1 | ys−1:t−1).

We update these particles in two steps:

1. Step 1. Add a new observation yt .

(θn, αns−1:t−1) → (θn, αn

s−1:t), Wns−1:t−1 → W n

s−1:t .

2. Step 2. Discard the old observation ys−1.

(θn, αns−1:t) → (θn, αn

s:t), Wns−1:t → W n

s:t .

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-ss − 1

ss

st − 1

st

-� Sample period

Step 1. Generate αnt given (θn, αn

s−1:t−1) and ys−1:t

Update W ns−1:t−1 → W n

s−1:t

time

where

W ns−1:t ∝

fθn(αnt | αn

t−1, yt−1)gθn(yt | αnt )

qt,θn(αnt | αn

t−1, yt)W n

s−1:t−1,

qt,θn(· | αnt−1, yt): proposal density to generate αn

t .

11 / 52





-ss − 1

ss

st − 1

st

-� Sample period

Step 2. Discard αns−1 and construct (θn, αn

s:t) given ys:t

Update W ns−1:t → W n

s:t for n = 1, . . . ,N

time

Ws:t ∝ gθn(ys−1 | αns−1, α

ns )

−1Ws−1:t ,

gθ(ys−1 | αs−1, αs) =µθ(αs−1)fθ(αs | αs−1)gθ(ys−1 | αs−1, αs)

p(αs−1 | αs , θ)µθ(αs)

12 / 52





Remarks.

▶ If some degeneracy criterion is fulfilled (such as ESS < 0.5N),

resample all the particles by implementing MCMC algorithm.

▶ We often use a prior density fθ(αt |αt−1, yt−1) as a proposal

density qt,θ to generate αt in Step 1. It is known to cause the

weight degeneracy problem since the prior density does not

take account of the new observation yt .

▶ Even if we are able to use a fully adapted proposal density,

qt,θ(αt | αt−1, ys−1:t) = p(αt | αt−1, yt , θ), we still have the

weight degeneracy phenomenon.

13 / 52





Example 1.

yt = αt + ϵt , ϵt ∼ N (0, σ2), t = 1, . . . , 2000

αt+1 = µ+ 0.25(αt − µ) + ηt , ηt ∼ N (0, 2σ2), t = 1, . . . , 2000,

α1 = µ+η0√

1− 0.252, η0 ∼ N (0, 2σ2),

where θ = (µ, σ2)′, µ | σ2 ∼ N (0, 10σ2), σ2 ∼ IG(5/2, 0.05/2).The window size = 1000. In order to evaluate the weight

degeneracy in Steps 1a and 2a, we define two ratios:

R1t =ESSs−1:t

ESSs−1:t−1, R2t =

ESSs:tESSs−1:t

, ESSs:t =1∑N

n=1(Wns:t)

2.

14 / 52





1000 1200 1400 1600 1800 2000

0.2

0.4

0.6

0.8

1.0R1t

R2t

1000 1200 1400 1600 1800 2000

0.2

0.4

0.6

0.8

1.0 R2t

0.00 0.25 0.50 0.75 1.00

100

200

300

400R1t

0.00 0.25 0.50 0.75 1.00

100

200

300

400

Figure: Rolling window estimation of p(θ, αt−999:t | yt−999:t),

t = 1001, . . . , 2000 in the linear Gaussian state space model. The fully

adapted proposal density is used for qt,θ.15 / 52





We update these particles in two steps:

Step 1. Add a new obs yt (forward block sampling)

1. Generate αnt−K :t given (θn, αn

s−1:t−K−1) and ys−1:t

2. Construct a collection of particles (θn, αns−1:t) with the

importance weight W ns−1:t

3. Implement the particle simulation smoother to improve the

mixing property.

Step 2. Remove the old obs ys−1 (backward block sampling)

1. Generate αns−1:s+K−1 given (θn, αn

s+K :t) and ys:t .

2. Construct a collection of particles (θn, αns:t) with the

importance weight W ns:t

3. Discard αns−1 and implement the particle simulation smoother.

16 / 52





-ss − 1

ss

st − K

st − 1

st

-s� s


Particle simulation smoother

Step 1. Generate αnt−K :t given (θn, αn

s−1:t−K−1) and ys−1:t

Update W ns−1:t−1 → W n

s−1:t

time

where

W ns−1:t ∝ p(yt | ys−1:t−1, α

nt−K−1, θ

n)×W ns−1:t−1.

17 / 52





-ss − 1

ss

ss + K − 1

st − 1

st

-s� sBackward block sampling


Step 2. Generate αns−1:s+K−1 given (θn, αn

s+K :t) and ys:t

Discard αns−1 and update W n

s−1:t → W ns:t

time

where

W ns:t ∝

1

p(ys−1 | ys:t , αns+K , θ

n)W n

s−1:t

18 / 52






1. ‘Move’ the particles αnt−K :t−1 using the conditional SMC

update for each n = 1, . . . ,N based on particle Gibbs(Andrieu et al. (2010)).

(1) Generate a number of candidates αn,mt−K :t−1 (m = 1, . . . ,M)

where the current ‘lineage’ αnt−K :t−1 is fixed as one of

candidates.

(2) The αn,mt is also generated conditional on αn,m

t−K :t−1.

2. Determine which lineage is appropriate as αnt−k:t and compute

its importance weight for each n = 1, . . . ,N.

19 / 52





Figure: (θn, αns−1:t−1) is fixed and shown in the rectangle. Other state

variables in the circle are generated.

20 / 52






1. Generate a cloud of particles of αn,ms+K−1, α

n,ms+K−2, . . . , α

n,ms−1

(m = 1, . . . ,M) given (αns+K :t , θ

n) and ys:t targeting

π(θ, αs−1:t | ys:t)2. Stochastically determine the new values of αn

s:t (discarding

αns−1) with the updated importance weight.

21 / 52





Figure: (θn, αns−1:t−1) is fixed and shown in the rectangle. Other state

variables in the circle are generated.

22 / 52




Initialization of the rolling MCMC

1. In order to sample from the initial posterior distribution, it is

straightforward to use MCMC methods as in the warm-up

period for the practical filtering in Polson et al. (2008).

2. SMC-based methods are preferred when we need to compute

the marginal likelihood p(y1:s−1).

3. Thus we implement only forward block sampling (with particle

simulation smoother) to sample α1:L+1 and θ where L = t − s.

# If some degeneracy criterion is fulfilled (such as ESS < 0.5N),

resample all the particles by implementing MCMC algorithm.

23 / 52




Estimation of the marginal likelihood

We compute

p(ys:t) =

∫p(ys:t | αs:t , θ)p(αs:t | θ)p(θ)dαs:tdθ

=p(yt | ys−1:t−1)

p(ys−1 | ys:t)p(ys−1:t−1),

using the estimates

p(yt | ys−1:t−1) =N∑

n=1

W ns−1:t−1p(yt | ys−1:t−1, α

nt−K−1, θ

n),

p(ys−1 | ys:t) =N∑

n=1

W ns−1:t p(ys−1 | ys:t , αn

s+K , θn).

24 / 52




Estimation of the marginal likelihood

The initial estimate is given by

p(y1:L+1) = p(y1)L+1∏j=2

p(yj | y1:j−1),

where

p(y1) =N∑

n=1

p(y1 | θn),

p(yj | y1:j−1) =N∑

n=1

W nj−1p(yj | y1:j−1, α

nj−K−1, θ

n),

25 / 52



Forward block samplingBackward block sampling

3. Theoretical justification of PRMCMC

26 / 52





� �Proposition 4.1

The marginal density of (θ, αs−1:t−K−1, αk∗t−K

t−K , . . . , αk∗t

t ) for the

artificial target density

π(θ, αs−1:t−K−1, α1:Mt−K :t , a

1:Mt−K :t−1, k

∗t | ys−1:t)

is π(θ, αs−1:t−K−1, αk∗t−K

t−K , . . . , αk∗t

t | ys−1:t) for the forward

block sampling where aj : ‘parent’ index variable and k∗j : (ran-

dom) index (j = t − K , . . . , t).� �27 / 52






E [p(yt | ys−1:t−1, αt−K−1, θ)|ys−1:t ]

= E [p(yt | ys−1:t−1, αt−K−1, θ)|ys−1:t , αt−K−1, θ]

= p(yt | ys−1:t−1).� �

28 / 52





� �Proposition 4.3 The joint conditional density of (k∗t−K , . . . , k

∗t )

is given by

π(k∗t−K , . . . , k∗t |θ, αs−1:t−K−1, α

1:Mt−K :t , a

1:Mt−K :t−1, ys−1:t)

= π(k∗t |θ, αs−1:t−K−1, α1:Mt−K :t , a

1:Mt−K :t−1, ys−1:t)

×t−K∏

t0=t−1

π(k∗t0 |θ, αs−1:t−K−1, α1:Mt−K :t0 , a

1:Mt−K :t0−1, α

k∗t0+1

t0+1 , . . . ,

αk∗t

t , k∗t0+1:t , ys−1:t),� �as in Whitely et al. (2010) and the discussion of Whitely following

Andrieu et al. (2010).

29 / 52






The marginal density of (θ, αk∗s

s , . . . , αk∗s+K−1

s+K−1, αs+K :t) for the

artificial target density

π(θ, α1:Ms−1:s+K−1, αs+K :t , a

1:Ms:s+K−1, ks−1, k

∗s | ys:t)

is π(θ, αk∗s

s , . . . , αk∗s+K−1

s+K−1, αs+K :t | ys−1:t) for the backward block

sampling where aj : ‘parent’ index variable and k∗j : (random)

index (j = t − K , . . . , t).� �30 / 52






E [p(ys−1 | ys:t , αs+K , θ)−1]

= E [p(ys−1 | ys:t , αs+K , θ)−1 | ys:t , αs+K , θ]

= p(ys−1 | ys:t , θ)−1.� �

31 / 52



Example 1 (Linear Gaussian SSM)Example 2 (Realized stochastic volatility model)

4. Numerical examples

32 / 52




Example 1 (Continued)

yt = αt + ϵt , ϵt ∼ N (0, σ2), t = 1, . . . , 2000

αt+1 = µ+ 0.25(αt − µ) + ηt , ηt ∼ N (0, 2σ2), t = 1, . . . , 2000,

α1 = µ+η0√

1− 0.252, η0 ∼ N (0, 2σ2),

where θ = (µ, σ2)′, µ | σ2 ∼ N (0, 10σ2), σ2 ∼ IG(5/2, 0.05/2).▶ The window size = 1000

▶ The number of particles: N = 1000.

▶ The size of block K = 1, 2, 3, 5

▶ For each n = 1, . . . ,N, we generate M particle paths in the

forward and backward block sampling.

33 / 52





Table: # resampling steps in PRMCMC with block sampling ((2))

(1) Fully (2) M (3) SimpleEstimation period K adapted 100 300 500 sampling

1 30 54 37 32 184

Initial estimation 2 10 39 21 15

(t = 1, . . . , 1000) 3 8 38 19 15

5 6 37 18 15

10 8 38 18 14

1 48 104 71 61 1027

Rolling estimation 2 8 74 33 23

(t = 1001, . . . , 2000) 3 7 74 31 22

5 6 72 32 22

10 5 69 31 23

34 / 52





0.00 0.25 0.50 0.75 1.00

100

200

300

400

500

600

700

800

R1t

0.00 0.25 0.50 0.75 1.00

100

200

300

400

500

600

700

800R2t

Simple Block (K=2)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

R1t

R2t

Simple Block (K=2)

Figure: The histograms (left) and scatter plots (right) of R1t and R2t

(left) (t = 1001, . . . , 2000) for the simple sampler (dotted blue) and the

sampler with block sampling with K = 2 and M = 100 (solid red).

35 / 52





Table: Summary statistics of R1t and R2t for the simple sampling

and the block sampling (M = 100,K = 2) for t = 1001, . . . , 2000

Method Mean Median Std. dev.

R1t Simple 0.862 0.924 0.145

Block 0.975 0.988 0.057

R2t Simple 0.161 0.127 0.139

Block 0.970 0.988 0.068

36 / 52





1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000

-1.2

-1.0

-0.8

µ

1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000

0.018

0.019

0.020

0.021

σ2

Figure: True posterior means and 95% credible intervals (dotted black)

with their estimates (solid red) for µ and σ2.

37 / 52





1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000-110

-100

-90

-80

log p(yt−999:t)

1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000

-0.75

-0.50

-0.25

0.00

0.25

0.50

0.75

1.00

log p(yt−999:t)−log p(yt−999:t)

Figure: Top: true log marginal likelihoods log p(yt−999:t) (dotted black)

and their estimates (solid red). Bottom: estimation errors

log p(yt−999:t)− log p(yt−999:t) for t = 1001, . . . , 2000.38 / 52




Example 2 (Realized stochastic volatility model)

Let y1,t : daily stock log return and y2,t : log realized volatility.

y1,t = exp(αt/2)ϵt , ϵt ∼ N (0, 1), t = 1, . . . ,T

y2,t = αt + ξ + ut , ut ∼ N (0, σ2u), t = 1, . . . ,T

αt+1 = µ+ ϕ(αt − µ) + ηt , ηt ∼ N (0, σ2η), t = 1, . . . ,T ,

α1 = µ+1√

1− ϕ2η0, η0 ∼ N (0, σ2

η),

where ϵtutηt

∼ N

0

0

0

,

1 0 ρση0 σ2

u 0

ρση 0 σ2η

.

(The correlation between ϵt and ηt is introduced to express the

leverage effect)39 / 52




Example 2 (Realized stochastic volatility model)

Data

▶ y1t and y2t : the daily log returns and log realized volatilities

of Standard and Poor’s (S&P) 500 index (obtained from

Oxford-Man Institute Realized Library)

▶ The initial estimation period: from January 1, 2000 (t = 1) to

December 31, 2007 (t = 1988) with L+ 1 = 1988.

▶ The rolling estimation started from sampling from the

posterior distribution using this initial sample period and

moved the window until December 30, 2016 (T = 4248).

▶ K = 10, M = 300 and N = 1000.

▶ We implement 10 MCMC iterations for the comparison below.

40 / 52




Example 2 (RSV)

2000 2500 3000 3500 4000

0.2

0.4

0.6

0.8

1.0

1.2 R2tR1t

2000 2500 3000 3500 4000

0.2

0.4

0.6

0.8

1.0

1.2

Figure: Traceplot of R1t (left) and R2t (right), (t = 1988, . . . , 4248) in

RSV model.

41 / 52




Example 2 (RSV)

K Mean Median Std. dev.

R1t 5 0.981 0.995 0.058

10 0.985 0.996 0.053

15 0.986 0.997 0.055

R2t 5 0.983 0.993 0.044

10 0.988 0.994 0.036

15 0.988 0.994 0.035

Table: Summary statistics for R1t and R2t (t = 1988, . . . , 4248) in RSV

model with leverage using K = 5, 10 and 15.

42 / 52




Example 2 (RSV)

K=5 K=10 K=15

2000 2200 2400 2600 2800 3000 3200 3400 3600 3800 4000 4200

50

75

100

125

150

175

200

225

K=5 K=10 K=15

Figure: Cumulative computation times (wall time, unit time =1000

seconds) using K = 5, 10 and 15 (t = 1988, . . . , 4248) in RSV model.

43 / 52




Example 2 (RSV)

Block (1 update) Block (10 updates)

Block (5 updates) MCMC

-0.8 -0.6 -0.4 -0.2 0.0

0.5

1.0µ Block (1 update)

Block (10 updates) Block (5 updates) MCMC

0.9 0.91 0.92 0.93 0.94 0.95 0.96 0.97

0.5

1.0φ

0.08 0.10 0.12 0.14 0.16

0.5

1.0σ2

η

-0.25 -0.20 -0.15 -0.10 -0.05 0.00

0.5

1.0ξ

0.20 0.22 0.24 0.26 0.28

0.5

1.0σ2

u

-0.65 -0.60 -0.55 -0.50 -0.45 -0.40 -0.35

0.5

1.0ρ

Figure: The estimated posterior distribution functions of θ for

t = 2261, . . . , 4248. MCMC and Particle rolling MCMC: 1, 5 and 10

iterations.44 / 52




Example 2 (RSV)

Block (5 updates) Block (10 updates)

2000 2500 3000 3500 4000

-0.50

-0.25

0.00

0.25 µBlock (5 updates) Block (10 updates)

2000 2500 3000 3500 40000.925

0.950

0.975

φ

2000 2500 3000 3500 4000

0.050

0.075

0.100

0.125 σ2η

2000 2500 3000 3500 4000

-0.2

-0.1ξ

2000 2500 3000 3500 4000

0.20

0.25σ2

u

2000 2500 3000 3500 4000

-0.7

-0.5

ρ

Figure: Traceplot of estimated posterior means and 95% credible

intervals for parameters using S&P500 return in RSV model (from

December 31, 2007 to December 30, 2016).45 / 52




Example 2 (RSV)

RSV with leverage RSV without leverage

2000 2500 3000 3500 4000

-4500

-4400

-4300

-4200

-4100

RSV with leverage RSV without leverage

2000 2500 3000 3500 4000

80

100

120

Figure: Left: Estimates of log p(yt−1987:t) (t = 1988, . . . , 4248) for S&P

500 index return in RSV model with leverage (solid red) and in RSV

model without leverage (dotted black). Right: Difference between two

log marginal likelihoods.

46 / 52




Comparison with SMC2 (without rolling window)

Figure: Cumulative computation times (wall time, unit time =1000

seconds) over 10 runs with SMC2 with c = 0.3 (left), 0.5 (middle) and

with the PRMCMC-based algorithm (right).

In the particle MH update steps of SMC2, the parameter θ is generated

using the random walk MH algorithm using the normal proposal

N (θ∗, cΣ) where θ∗ is a current value of the parameter θ, c is a tuning

parameter (0.3 or 0.5), and Σ is the estimated covariance matrix.47 / 52





-0.5 0.0

0.5

1.0

µ (c=0.3)

0.96 0.98

0.5

1.0

φ (c=0.3)

0.03 0.04 0.05

0.5

1.0

σ2η (c=0.3)

-0.3 -0.2 -0.1

0.5

1.0

ξ (c=0.3)

0.175 0.200 0.225

0.5

1.0

σ2u (c=0.3)

-0.7 -0.5

0.5

1.0

ρ (c=0.3)

-0.5 0.0

0.5

1.0

µ (c=0.5)

0.96 0.97 0.98 0.99

0.5

1.0

φ (c=0.5)

0.03 0.04 0.05

0.5

1.0

σ2η (c=0.5)

-0.3 -0.2 -0.1

0.5

1.0

ξ (c=0.5)

0.175 0.200 0.225

0.5

1.0

σ2u (c=0.5)

-0.8 -0.7 -0.6

0.5

1.0

ρ (c=0.5)

Figure: Estimated posterior cumulative distribution functions of θ given

y1:1988 over 10 runs with SMC2 with c = 0.3, 0.5 (red and blue,

respectively) compared to the result of MCMC (gray).48 / 52





-0.6 -0.4 -0.2 0.0 0.2

0.5

1.0µ

0.955 0.96 0.965 0.97 0.975 0.98 0.985

0.5

1.0φ

0.025 0.03 0.035 0.04 0.045 0.05 0.055

0.5

1.0σ2

η

-0.35 -0.30 -0.25 -0.20 -0.15 -0.10

0.5

1.0ξ

0.17 0.18 0.19 0.20 0.21 0.22 0.23

0.5

1.0σ2

u

-0.85 -0.80 -0.75 -0.70 -0.65 -0.60

0.5

1.0ρ

Figure: Estimated posterior cumulative distribution functions of θ given

y1:1988 over 10 runs with the PRMCMC-based method (red) compared to

the result of MCMC (gray).49 / 52




Conclusion

This paper considers the state space model for economic times

series and

▶ proposes the efficient rolling estimation method for both

parameter θ and states αs:t .

▶ gives the theoretical justification of the algorithm by proving

the marginal density of parameters and states.

▶ the algorithm is based on Particle Gibbs

▶ proposes, as a special case, the sequential Bayesian estimation

method (which is alternative to SMC2 based on Particle MH).

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Refereneces

▶ Andrieu, C., Doucet, A., & Holenstein, R. (2010). Particle markov

chain monte carlo methods. Journal of the Royal Statistical Society:

Series B (Statistical Methodology), 72(3), 269-342.

▶ Chopin, N., Jacob, P. E., & Papaspiliopoulos, O. (2013). SMC2: an

efficient algorithm for sequential analysis of state space models.

Journal of the Royal Statistical Society: Series B (Statistical

Methodology), 75(3), 397-426.

▶ Del Moral, P., Doucet, A., & Jasra, A. (2006). Sequential monte

carlo samplers. Journal of the Royal Statistical Society: Series B

(Statistical Methodology), 68(3), 411-436.

▶ Doucet, A., Briers, M., & Senecal, S. (2006). Efficient block

sampling strategies for sequential Monte Carlo methods. Journal of

Computational and Graphical Statistics, 15(3), 693-711.

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Refereneces

▶ Fulop, A., & Li, J. (2013). Efficient learning via simulation: A

marginalized resample-move approach. Journal of Econometrics,

176(2), 146-161.

▶ Polson, N. G., Stroud, J. R., & Muller, P. (2008). Practical filtering

with sequential parameter learning. Journal of the Royal Statistical

Society: Series B (Statistical Methodology), 70(2), 413-428.

▶ Takahashi, M., Omori, Y., & Watanabe, T. (2009). Estimating

stochastic volatility models using daily returns and realized volatility

simultaneously. Computational Statistics & Data Analysis, 53(6),

2404-2426.

▶ Whiteley, N., Andrieu, C., & Doucet, A. (2010). Efficient Bayesian

inference for switching state-space models using discrete particle

Markov chain Monte Carlo methods. arXiv preprint

arXiv:1011.2437.

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Documents

Particle rolling MCMC · 2018-08-12 · Introduction Particle rolling MCMC Theoretical justi cation of PRMCMC Numerical examples Motivation State space model Rolling parameter and