folding markov chains: the origamcmc
TRANSCRIPT
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Folding Markov chains:the origaMCMC
Christian P. RobertUniversite Paris-Dauphine PSL and University of Warwick
Joint on-goin’ work with R. Douc and G. Roberts
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Outline
IntroductionMotivating exampleFolding the Markov chain
ConvergenceImproving the acceptance rateAsymptotic variance
Practicals [under development]
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Introduction
IntroductionMotivating exampleFolding the Markov chain
Convergence
Practicals [under development]
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motivating example
Consider the target
π(x) =1
(1 + x2)π
standard Cauchy distributionBasic Metropolis-Hastings algorithm with uniform proposalzt ∼ U(xt − ε, xt + ε) cannot be geometrically ergodic
[Mengersen and Tweedie (1996)]
−8 −6 −4 −2 0
020
0040
0060
0080
0010
000
x
t
Dynamics of a standard random-walkMetropolis–Hastings algorithm whentargeting a Cauchy distribution, based on104 iterations and a uniform scale of ε = .1.
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motivating example
Consider the target
π(x) =1
(1 + x2)π
standard Cauchy distributionBasic Metropolis-Hastings algorithm with uniform proposalzt ∼ U(xt − ε, xt + ε) cannot be geometrically ergodic
[Mengersen and Tweedie (1996)]
−8 −6 −4 −2 0
020
0040
0060
0080
0010
000
x
t
Dynamics of a standard random-walkMetropolis–Hastings algorithm whentargeting a Cauchy distribution, based on104 iterations and a uniform scale of ε = .1.
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new proposal
Metropolis-Hastings alternative:
1. the current value xt of the Markov chain is first inverted intoyt = 1/xt if found outside (−1, 1),
2. then moved by a random walk on (−1, 1) tozt ∼ U(yt − ε, yt + ε), which value is accepted or notaccording to the standard Metropolis-Hastings ratio,
3. and outcome inverted into xt+1 = 1/yt+1 with probability 1/2
simple version of the folding algorithm, with folding set the unitinterval (−1, 1)
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new proposal
Metropolis-Hastings alternative:
1. the current value xt of the Markov chain is first inverted intoyt = 1/xt if found outside (−1, 1),
2. then moved by a random walk on (−1, 1) tozt ∼ U(yt − ε, yt + ε), which value is accepted or notaccording to the standard Metropolis-Hastings ratio,
3. and outcome inverted into xt+1 = 1/yt+1 with probability 1/2
simple version of the folding algorithm, with folding set the unitinterval (−1, 1)
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new proposal
Metropolis-Hastings alternative:
1. the current value xt of the Markov chain is first inverted intoyt = 1/xt if found outside (−1, 1),
2. then moved by a random walk on (−1, 1) tozt ∼ U(yt − ε, yt + ε), which value is accepted or notaccording to the standard Metropolis-Hastings ratio,
3. and outcome inverted into xt+1 = 1/yt+1 with probability 1/2
simple version of the folding algorithm, with folding set the unitinterval (−1, 1)
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validation
simple version of the folding algorithm, with folding set the unitinterval (−1, 1)
I Cauchy target still stationary for this distribution
I probability 1/2 resulting from Jacobian rather than fromP(|X | < 1) = 1/2
I not-so-simple [but still-manageable] probabilty if chosingfolding interval (−2, 2) and inversion yt = 4/xt
I fundamental reason is that Cauchy is invariant by inversion
I resulting Markov chain is uniformly ergodic
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validation
simple version of the folding algorithm, with folding set the unitinterval (−1, 1)
I Cauchy target still stationary for this distribution
I probability 1/2 resulting from Jacobian rather than fromP(|X | < 1) = 1/2
I not-so-simple [but still-manageable] probabilty if chosingfolding interval (−2, 2) and inversion yt = 4/xt
I fundamental reason is that Cauchy is invariant by inversion
I resulting Markov chain is uniformly ergodic
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simulation outcome
−4 −2 0 2 4
020
0040
0060
0080
00
x
t
x
Den
sity
−20 −10 0 10 20
0.00
0.05
0.10
0.15
0.20
0.25
Figure : (Left) Folded Markov chain for Cauchy target with same scaleof the random walk. (Right) Empirical distribution of the Markov chainand fit to the Cauchy target
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folding the Markov chain
Consider target π on state space XLet A0,A1, . . . ,AM be a finite partition of the state space andcreate differentiable bijections g1, . . . , gM from A0 to A1, . . . ,AM ,respectively. Set X? = A0 as the folded spaceDefine the distribution
π?(x?) = π(x?) + π(g1x?) |dxg1 (x?)|+ . . .+ π(gMx?) |dxgM (x?)|
on X?
c© π?(·) is a proper density on X?
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folding the Markov chain
Consider target π on state space XLet A0,A1, . . . ,AM be a finite partition of the state space andcreate differentiable bijections g1, . . . , gM from A0 to A1, . . . ,AM ,respectively. Set X? = A0 as the folded spaceDefine the distribution
π?(x?) = π(x?) + π(g1x?) |dxg1 (x?)|+ . . .+ π(gMx?) |dxgM (x?)|
on X?
c© π?(·) is a proper density on X?
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unfolding the folded Markov chain
Simulating from π? is equivalent to simulating from π:
LemmaIf x? ∼ π?, then
x =
x? with probability π(x?)/π?(x?)
g1x? with probability π(g1x?) |dxg1 (x?)| /π?(x?)
· · ·gMx? with probability π(gMx?) |dxgM (x?)| /π?(x?)
is distributed from the target π.
c© build MCMC sampler aiming at π?
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unfolding the folded Markov chain
Simulating from π? is equivalent to simulating from π:
LemmaIf x? ∼ π?, then
x =
x? with probability π(x?)/π?(x?)
g1x? with probability π(g1x?) |dxg1 (x?)| /π?(x?)
· · ·gMx? with probability π(gMx?) |dxgM (x?)| /π?(x?)
is distributed from the target π.
c© build MCMC sampler aiming at π?
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Cauchy example validated
For the Cauchy example:
I A0 = (−1, 1), A1 = (−1, 1)c, g1x? = 1/x?
I and
π?(x) = π(x?) + π(g1x?) |dxg1 (x?)|
=1
(1 + x2)π+
1
(1 + 1/x2)π
1
x2
=2
(1 + x2)π
I unfolding by x =
{x? w.p. 1/2
1/x? w.p. 1/2
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Cauchy example validated
For the alternative
I A0 = (−2, 2), A1 = (−2, 2)c, g1x? = 4/x?
I and
π?(x) = π(x?) + π(g1x?) |dxg1 (x?)|
=1
(1 + x2)π+
1
(1 + 4/x2)π
4
x2=
1
(1 + x2)π+
4
(4 + x2)π
I unfolding by x =
{x? w.p. π(x?)/π?(x?)
1/x? w.p. 4π(4/x?)/(x?)2π?(x?)
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Convergence
Introduction
ConvergenceImproving the acceptance rateAsymptotic variance
Practicals [under development]
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improving the acceptance rate
Define folded transition kernel, K ?(x?, dy?) as
k?(x?, y?) =M∑i=0
π(gix?) |dxgi (x?)|π?(x?)
M∑j=0
k(gix?, gjy
?) |dxgj (y?)|
Kernel considers allpossible images inoriginal space X andbrings them intothe folded space
X(1)k
X?,(1)k
X(1)k+1
X?,(1)k+1
Q Q
H〈K , π〉
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improving the acceptance rate
Define folded transition kernel, K ?(x?, dy?) as
k?(x?, y?) =M∑i=0
π(gix?) |dxgi (x?)|π?(x?)
M∑j=0
k(gix?, gjy
?) |dxgj (y?)|
Kernel considers allpossible images inoriginal space X andbrings them intothe folded space
X(0)k
X?,(0)k
X(0)k+1
X?,(0)k+1
Q? Q?
H〈Q?KQ, π?〉
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improving the acceptance rate
Define folded transition kernel, K ?(x?, dy?) as
k?(x?, y?) =M∑i=0
π(gix?) |dxgi (x?)|π?(x?)
M∑j=0
k(gix?, gjy
?) |dxgj (y?)|
Proposition
If α(x , y), resp. α?(x?, y?), is Metropolis–Hasting acceptanceprobability for the original, resp. folded, proposal kernel K then
E[α?(X ?,Y ?)] ≥ E[α(X ,Y )]
when expectations computed under respective stationarydistributions, π?(x?)K ?(x?, y?) and π(x)K (x , y)
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asymptotic variance
Given (X,X ) and (X?,X ?),define the folding mappingϕ : X→ X? and write
Q(x , dx?) = δϕ(x)(dx?)
X(1)k
X?,(1)k
X(1)k+1
X?,(1)k+1
Q Q
H〈K , π〉
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asymptotic variance
Given (X,X ) and (X?,X ?),define the folding mappingϕ : X→ X? and write
Q(x , dx?) = δϕ(x)(dx?)
For π target probability on(X,X ), set
π? = π ◦ ϕ−1
and define kernel Q? on X? ×Xby
π(dx)Q(x , dx?) = π?(dx?)Q?(x?, dx)
X(1)k
X?,(1)k
X(1)k+1
X?,(1)k+1
Q Q
H〈K , π〉
X(0)k
X?,(0)k
X(0)k+1
X?,(0)k+1
Q? Q?
H〈Q?KQ, π?〉
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asymptotic variance
Given (X,X ) and (X?,X ?), define the folding mapping ϕ : X→ X?
and writeQ(x ,dx?) = δϕ(x)(dx?)
set π? = π ◦ ϕ−1 and define kernel Q? on X? ×X by
π(dx)Q(x , dx?) = π?(dx?)Q?(x?, dx)
Lemmafor all (f , g) ∈ L2(π)× L2(π?),
〈f ; Qg〉π = 〈Q?f ; g〉π? where 〈f ; g〉µ = µ(fg)
If K is π-reversible, then, Q?KQ is π?-reversible, since
〈f ; Q?KQg〉π? = 〈Qf ; KQg〉π = 〈KQf ; Qg〉π = 〈Q?KQf ; g〉π?
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comparing two π-reversible Markov chains
Let P0 and P1 be two π-reversible Markov kernels.easy-to-check conditions on P0 and P1 ensuring that for all f insome ”class of functions”,
v(f ,P0) ≥ v(f ,P1)
where we have defined, for a Markov chain (X ()k)k∈N with
π-reversible transition kernel P and initial distribution π,
v(f ,P) := limn→∞
1
nVar
n−1∑k=0
f (Xk) = limn→∞
√nVarπn(f )
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two notions
Definition
1. P1 dominates P0 on the off-diagonal, i.e. P1 � P0, if
∀(x ,A), P1(x ,A \ {x}) ≥ P0(x ,A \ {x}) .
2. P1 dominates P0 in the covariance ordering, i.e. P1 < P0, if
∀f ∈ L2(π), 〈f ; P1f 〉π ≤ 〈f ; P0f 〉π
where 〈f ; g〉π =∫π(dx)f (x)g(x).
TheoremP1 � P0 ⇒ P1 < P0 ⇒ v(f ,P0) ≥ v(f ,P1) ∀f ∈ L2(π) .
[Peskun (1973) and Tierney (1998)]
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two notions
Definition
1. P1 dominates P0 on the off-diagonal, i.e. P1 � P0, if
∀(x ,A), P1(x ,A \ {x}) ≥ P0(x ,A \ {x}) .
2. P1 dominates P0 in the covariance ordering, i.e. P1 < P0, if
∀f ∈ L2(π), 〈f ; P1f 〉π ≤ 〈f ; P0f 〉π
where 〈f ; g〉π =∫π(dx)f (x)g(x).
TheoremP1 � P0 ⇒ P1 < P0 ⇒ v(f ,P0) ≥ v(f ,P1) ∀f ∈ L2(π) .
[Peskun (1973) and Tierney (1998)]
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two notions
Definition
1. P1 dominates P0 on the off-diagonal, i.e. P1 � P0, if
∀(x ,A), P1(x ,A \ {x}) ≥ P0(x ,A \ {x}) .
2. P1 dominates P0 in the covariance ordering, i.e. P1 < P0, if
∀f ∈ L2(π), 〈f ; P1f 〉π ≤ 〈f ; P0f 〉π
where 〈f ; g〉π =∫π(dx)f (x)g(x).
TheoremP1 � P0 ⇒ P1 < P0 ⇒ v(f ,P0) ≥ v(f ,P1) ∀f ∈ L2(π) .
[Peskun (1973) and Tierney (1998)]
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induced kernel
Given a proposition kernel K and the target distribution π, writeH〈K , π〉 the Metropolis-Hastings kernel defined by:
H〈K , π〉(x ,A \ {x}) =
∫K (x , dy)α(x , y)IA\{x}(y)
where α(x , y) = 1 ∧ r(x , y)
r(x , y) = dµdν (x , y)
µ(dxdy) = π(dy)K (y , dx)
ν(dxdy) = π(dx)K (x ,dy)
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two approximate expectations
1. Let {X (1)k } a Markov chain of transition kernel H〈K , π〉. The
Rao-Blackwellised approximation is defined by
π(1)n (h) =
1
n
n∑k=1
QQ?h(X(1)k ) (1)
2. Let {X ?,(0)k } a Markov chain of transition kernel H〈Q?KQ, π?〉
and consider the Rao-Blackwellised approximation
π(0)n (h) =
1
n
n∑k=1
Q?h(X?,(0)k ) (2)
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two approximate expectations
1. Let {X (1)k } a Markov chain of transition kernel H〈K , π〉. The
Rao-Blackwellised approximation is defined by
π(1)n (h) =
1
n
n∑k=1
QQ?h(X(1)k ) (1)
2. Let {X ?,(0)k } a Markov chain of transition kernel H〈Q?KQ, π?〉
and consider the Rao-Blackwellised approximation
π(0)n (h) =
1
n
n∑k=1
Q?h(X?,(0)k ) (2)
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comparison
TheoremFor h real-valued measurable function on (X,X ) such thatπh2 <∞ and
I {X (1)k , k ∈ N} Markov chain with kernel H〈K , π〉 starting
from π
I {X ?,(0)k , k ∈ N} Markov chain with kernel H〈Q?KQ, π?〉
starting from π?
Then,limn→∞
nVar(π(0)n (h)) ≤ lim
n→∞nVar(π
(1)n (h))
with π(0)n (h) and π
(1)n (h) defined in (2) and (1)
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comparison
TheoremFor h real-valued measurable function on (X,X ) such thatπh2 <∞ and
I {X (1)k , k ∈ N} Markov chain with kernel H〈K , π〉 starting
from π
I {X ?,(0)k , k ∈ N} Markov chain with kernel H〈Q?KQ, π?〉
starting from π?
Then,limn→∞
nVar(π(0)n (h)) ≤ lim
n→∞nVar(π
(1)n (h))
with π(0)n (h) and π
(1)n (h) defined in (2) and (1)
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Practicals
Introduction
Convergence
Practicals [under development]
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folding set
Unless target distribution simple enough for informed choice,natural choice for A0 is HPD region
Hα = {x ∈ X; π(x) ≥ α}
as
I π? [and hence π] lower bounded on Hα
I resulting Hα compact
I some transition kernels produce uniform ergodic chains
I partition of X into A0,Ac0 with natural stereoscopic projection
[provided A0 star-convex]
g1(x?) =%2
|x?|2x?
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practical implementation
While Hα usually unavailable, approximations can be found frompreliminary MCMC runs when π(x) or unnormalised version of itcan be computed
I preliminary run produces simulations with [relative] values ofπ, π(x1), . . . , π(xN)
I derivation of higher density values [and potential clustering]
I choice of an HPD approximation as ball and g1 as naturalprojection
I reevaluation of the folding set after further simulations
note: black box compatibility with MCMC code
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practical implementation
While Hα usually unavailable, approximations can be found frompreliminary MCMC runs when π(x) or unnormalised version of itcan be computed
I preliminary run produces simulations with [relative] values ofπ, π(x1), . . . , π(xN)
I derivation of higher density values [and potential clustering]
I choice of an HPD approximation as ball and g1 as naturalprojection
I reevaluation of the folding set after further simulations
note: black box compatibility with MCMC code
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Cauchy illustration
I preliminary run producessimulations with values ofπ(x)
I derivation of higher densityvalues and clustering
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−1.5 −1.0 −0.5 0.0
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0.25
0.30
xπ(
x)
22 / 25
![Page 39: folding Markov chains: the origaMCMC](https://reader031.vdocuments.site/reader031/viewer/2022030306/586f719f1a28ab10258b503b/html5/thumbnails/39.jpg)
Cauchy illustration
I preliminary run producessimulations with values ofπ(x)
I derivation of higher densityvalues and clustering
I choice of an HPDapproximation as ball and g1as natural projection
I potential reevaluation of thefolding set after furthersimulations
xD
ensi
ty
−0.6 −0.4 −0.2 0.0 0.2
0.0
0.5
1.0
1.5
2.0
22 / 25
![Page 40: folding Markov chains: the origaMCMC](https://reader031.vdocuments.site/reader031/viewer/2022030306/586f719f1a28ab10258b503b/html5/thumbnails/40.jpg)
Cauchy illustration
I preliminary run producessimulations with values ofπ(x)
I derivation of higher densityvalues and clustering
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1.0 1.5 2.0 2.5
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0.15
0.20
x
π(x)
22 / 25
![Page 41: folding Markov chains: the origaMCMC](https://reader031.vdocuments.site/reader031/viewer/2022030306/586f719f1a28ab10258b503b/html5/thumbnails/41.jpg)
Cauchy illustration
I preliminary run producessimulations with values ofπ(x)
I derivation of higher densityvalues and clustering
I choice of an HPDapproximation as ball and g1as natural projection
I potential reevaluation of thefolding set after furthersimulations
x
Den
sity
0.8 1.0 1.2 1.4 1.6
01
23
4
22 / 25
![Page 42: folding Markov chains: the origaMCMC](https://reader031.vdocuments.site/reader031/viewer/2022030306/586f719f1a28ab10258b503b/html5/thumbnails/42.jpg)
Cauchy illustration
I preliminary run producessimulations with values ofπ(x)
I derivation of higher densityvalues and clustering
I choice of an HPDapproximation as ball and g1as natural projection
I potential reevaluation of thefolding set after furthersimulations
x
Den
sity
1.3 1.4 1.5 1.6 1.7
02
46
810
qMC version using sobol(1e5,3)
22 / 25
![Page 43: folding Markov chains: the origaMCMC](https://reader031.vdocuments.site/reader031/viewer/2022030306/586f719f1a28ab10258b503b/html5/thumbnails/43.jpg)
Gaussian sugarloaf
Targetπ(x) ∝ ϕ(x ;µ,Σ)× exp{−α/||x − x0||2}
23 / 25
![Page 44: folding Markov chains: the origaMCMC](https://reader031.vdocuments.site/reader031/viewer/2022030306/586f719f1a28ab10258b503b/html5/thumbnails/44.jpg)
Gaussian sugarloaf
Targetπ(x) ∝ ϕ(x ;µ,Σ)× exp{−α/||x − x0||2}
I preliminary run producessimulations with values ofπ(x)
I derivation of higher densityvalues and clustering
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23 / 25
![Page 45: folding Markov chains: the origaMCMC](https://reader031.vdocuments.site/reader031/viewer/2022030306/586f719f1a28ab10258b503b/html5/thumbnails/45.jpg)
Gaussian sugarloaf
Targetπ(x) ∝ ϕ(x ;µ,Σ)× exp{−α/||x − x0||2}
I preliminary run producessimulations with values ofπ(x)
I derivation of higher densityvalues and clustering
I choice of an HPDapproximation as ball and g1as natural projection
I potential reevaluation of thefolding set after furthersimulations
x
Den
sity
−4 −2 0 2 4
0.0
0.1
0.2
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x
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sity
−4 −2 0 2 4
0.0
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0.4
x
Den
sity
−4 −2 0 2 4
0.0
0.1
0.2
0.3
0.4
x
Den
sity
−4 −2 0 2 4
0.00
0.10
0.20
0.30
23 / 25
![Page 46: folding Markov chains: the origaMCMC](https://reader031.vdocuments.site/reader031/viewer/2022030306/586f719f1a28ab10258b503b/html5/thumbnails/46.jpg)
Gaussian sugarloaf
Targetπ(x) ∝ ϕ(x ;µ,Σ)× exp{−α/||x − x0||2}
I preliminary run producessimulations with values ofπ(x)
I derivation of higher densityvalues and clustering
I choice of an HPDapproximation as ball and g1as natural projection
I potential reevaluation of thefolding set after furthersimulations
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0 20 40 60
−10
010
2030
40
23 / 25
![Page 47: folding Markov chains: the origaMCMC](https://reader031.vdocuments.site/reader031/viewer/2022030306/586f719f1a28ab10258b503b/html5/thumbnails/47.jpg)
keep folding: the origaMCMC
When A0 shows too much variability of π?, it can be folded again:the procedure can be iterated or a more elaborated partition canbe constructed by clustering
I cost of unfolding possibly a deterent
I over-concentration not an issue with projected proposal
I plus other proposals may be included
I possible connection with Wang-Landau flat histogramalgorithm, although Jacobian may prevent flatness (andbecome a liability)
[Jacob & Ryder, 2014]
24 / 25
![Page 48: folding Markov chains: the origaMCMC](https://reader031.vdocuments.site/reader031/viewer/2022030306/586f719f1a28ab10258b503b/html5/thumbnails/48.jpg)
keep folding: the origaMCMC
When A0 shows too much variability of π?, it can be folded again:the procedure can be iterated or a more elaborated partition canbe constructed by clustering
I cost of unfolding possibly a deterent
I over-concentration not an issue with projected proposal
I plus other proposals may be included
I possible connection with Wang-Landau flat histogramalgorithm, although Jacobian may prevent flatness (andbecome a liability)
[Jacob & Ryder, 2014]
24 / 25
![Page 49: folding Markov chains: the origaMCMC](https://reader031.vdocuments.site/reader031/viewer/2022030306/586f719f1a28ab10258b503b/html5/thumbnails/49.jpg)
further questions
1. Folding increases the acceptance probabilities and improve theasymptotic covariance. What about achieving geometricergodicity?
2. Folding is only possible if folding and unfolding the Markovchains is not costly. What about a computing time criterion?
3. Domination results are obtained with the kernel induced onthe folded space [black box]. What about selecting moreappropriate [black/white box] folded kernels?
25 / 25
![Page 50: folding Markov chains: the origaMCMC](https://reader031.vdocuments.site/reader031/viewer/2022030306/586f719f1a28ab10258b503b/html5/thumbnails/50.jpg)
further questions
1. Folding increases the acceptance probabilities and improve theasymptotic covariance. What about achieving geometricergodicity?
2. Folding is only possible if folding and unfolding the Markovchains is not costly. What about a computing time criterion?
3. Domination results are obtained with the kernel induced onthe folded space [black box]. What about selecting moreappropriate [black/white box] folded kernels?
25 / 25
![Page 51: folding Markov chains: the origaMCMC](https://reader031.vdocuments.site/reader031/viewer/2022030306/586f719f1a28ab10258b503b/html5/thumbnails/51.jpg)
further questions
1. Folding increases the acceptance probabilities and improve theasymptotic covariance. What about achieving geometricergodicity?
2. Folding is only possible if folding and unfolding the Markovchains is not costly. What about a computing time criterion?
3. Domination results are obtained with the kernel induced onthe folded space [black box]. What about selecting moreappropriate [black/white box] folded kernels?
25 / 25