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Numerical
analysis of the
Metropolis
algorithms
Peter Mathé
Numerical analysis of the Metropolis
Introduction
algorithms
Numerical
integration
Formal problem
formulation
Two approaches
Peter Mathé
from ergodic theory
Negative results
Weierstrass Institute, Berlin
Metropolis
method
Ball walk E-mail: mathe@wias-berlin.de
Uniform ergodicity
Ball walk with
Homepage: http://www.wias-berlin.de/people/mathe
Metropolis filter
Conclusions joint work with Erich Novak, Jena
References
Fleurance, June 2007
Outline
Numerical
analysis of the
Metropolis
algorithms 1 Introduction
Peter Mathé
Introduction
2 Numerical integration
Numerical
Formal problem formulation
integration
Formal problem
Two approaches from ergodic theory
formulation
Two approaches
Negative results
from ergodic theory
Negative results
Metropolis 3 Metropolis method
method
Ball walk Ball walk
Uniform ergodicity
Ball walk with Uniform ergodicity
Metropolis filter
Conclusions
Ball walk with Metropolis filter
References
4 Conclusions
Situation
Numerical
analysis of the
Metropolis
algorithms
We are given an integrable weight function : Ω → R+ on
Peter Mathé
some (Ω, µ).
Introduction Task
Numerical
integration
1 Sample according to
Formal problem
formulation
Two approaches
from ergodic theory µ (dx) ∝ (x) µ(dx)!
Negative results
Metropolis
method
Ball walk
Uniform ergodicity
2 Compute Ω (x) µ(dx)!
Ball walk with
Metropolis filter
Conclusions
Remark
References
Typically the second problem is as hard as the fitrst one!
Statistical Physics
Numerical
analysis of the
Metropolis
algorithms
Here
Peter Mathé Ω the collection of all states of a physical system.
Introduction E(x) the energy assigned to x ∈ Ω,
Numerical
integration
the probab. µE of a state is proportional to e−θE(x) .
Formal problem
formulation
Two approaches
from ergodic theory
Negative results
Metropolis
method
Ball walk
Uniform ergodicity
Ball walk with
Metropolis filter
Conclusions
References
Statistical Physics
Numerical
analysis of the
Metropolis
algorithms
Here
Peter Mathé Ω the collection of all states of a physical system.
Introduction E(x) the energy assigned to x ∈ Ω,
Numerical
integration
the probab. µE of a state is proportional to e−θE(x) .
Formal problem
formulation
Two approaches
from ergodic theory
Task
Negative results
Determine states of minimal energy (ground states).
Metropolis
method
Ball walk
Let A be an observable (function of the states).
Uniform ergodicity
Ball walk with
Determine
Metropolis filter
A(x)e−θE(x)
Conclusions A := A(x) µE (dx) = !
Ω e−θE(x)
References
Statistics
Numerical
analysis of the
Metropolis Let X1 , X2 , . . . be an i.i.d. sample according to some
algorithms
model (Pθ ) , θ ∈ Θ.
Peter Mathé
Let L(y |θ) be the likelihood function of the data, given
Introduction
the true parameter.
Numerical
integration
Formal problem
Bayes analysis provides us at any given prior p(θ)as
formulation
Two approaches
Posterior distribution
from ergodic theory
Negative results
Metropolis π(θ) ∝ L(y |θ)p(θ)
method
Ball walk
Uniform ergodicity
Ball walk with
Metropolis filter
Conclusions
References
Statistics
Numerical
analysis of the
Metropolis Let X1 , X2 , . . . be an i.i.d. sample according to some
algorithms
model (Pθ ) , θ ∈ Θ.
Peter Mathé
Let L(y |θ) be the likelihood function of the data, given
Introduction
the true parameter.
Numerical
integration
Formal problem
Bayes analysis provides us at any given prior p(θ)as
formulation
Two approaches
Posterior distribution
from ergodic theory
Negative results
Metropolis π(θ) ∝ L(y |θ)p(θ)
method
Ball walk
Uniform ergodicity
Ball walk with
Metropolis filter
Conclusions Task
References
Determine the posterior distribution!
Determine functionals f dπ!
The problem
Numerical
analysis of the
Metropolis
algorithms Task
Peter Mathé Given any pair (f , ) of some integrand f and weight , find
Introduction
f (x) (x) µ(dx)
Numerical
integration
S(f , ) := f (x) µ (dx) = ,
Formal problem
(x) µ(dx)
formulation
Two approaches
from ergodic theory
Negative results
by using a finite number of (random) data
Metropolis (f (xi ), (xi )), i = 1, . . . , n !
method
Ball walk
Uniform ergodicity
Ball walk with
Metropolis filter
Conclusions
References
The problem
Numerical
analysis of the
Metropolis
algorithms Task
Peter Mathé Given any pair (f , ) of some integrand f and weight , find
Introduction
f (x) (x) µ(dx)
Numerical
integration
S(f , ) := f (x) µ (dx) = ,
Formal problem
(x) µ(dx)
formulation
Two approaches
from ergodic theory
Negative results
by using a finite number of (random) data
Metropolis (f (xi ), (xi )), i = 1, . . . , n !
method
Ball walk
Uniform ergodicity
Ball walk with Input Let (f , ) by any problem instance,
Metropolis filter
Conclusions method Let ϑ(f , ) any (Monte Carlo ) method,
References 1/2
error e(ϑ, (f , )) := E |S(f , ) − ϑ(f , )|2 (RMS).
Ergodic theorem
Numerical
analysis of the
Metropolis
algorithms Proposition (Simple Monte Carlo)
Peter Mathé
Let X1 , X2 , . . . be a sample from an ergodic Markov chain
Introduction with invariant distribution µ. For f , ∈ L1 (Ω, µ), > 0 we
Numerical
integration
have
Formal problem
n
formulation
i=1 f (Xi ) (Xi ) f (x) (x) µ(dx)
Two approaches
from ergodic theory ϑsimp (f , ) =
n n −→ = S(f , ).
Negative results
i=1 (Xi ) (x) µ(dx)
Metropolis
method
Ball walk
Uniform ergodicity
Remark
Ball walk with
Metropolis filter Need sample according to µ!
Conclusions
Simple Monte Carlo is non-adaptive! It does not use
References
for obtaining the sample.
Ergodic theorem
Numerical
analysis of the
Metropolis
algorithms
Proposition (General Monte Carlo)
Peter Mathé
Let X1 , X2 , . . . be a sample from an ergodic Markov chain
Introduction
with invariant distribution µ . For f ∈ L1 (Ω, µ ) we have
Numerical
integration
Formal problem
n
formulation
Two approaches ϑn (f , ) = f (Xi )−→ f (x) µ (dx) = S(f , ).
from ergodic theory
Negative results i=1
Metropolis
method
Ball walk
Uniform ergodicity
Remark
Ball walk with
Metropolis filter Need sample according to µ (asymptotically)!
Conclusions
It is adaptive with respect to , since it uses to obtain
References
the sample!
Basic questions
Numerical
analysis of the
Metropolis
algorithms
Peter Mathé
Introduction Both methods are analyzed in literature! Recent
Numerical study [Bassetti/Diaconis]!
integration
Formal problem
formulation
Two approaches
from ergodic theory
Negative results
Metropolis
method
Ball walk
Uniform ergodicity
Ball walk with
Metropolis filter
Conclusions
References
Basic questions
Numerical
analysis of the
Metropolis
algorithms
Peter Mathé
Introduction Both methods are analyzed in literature! Recent
Numerical study [Bassetti/Diaconis]!
integration
Formal problem The principle questions we want to address are
formulation
Two approaches
from ergodic theory
Negative results
Problem
Metropolis Which method to apply in specific application?
method
Ball walk
Uniform ergodicity
Ball walk with
Metropolis filter
Conclusions
References
Basic questions
Numerical
analysis of the
Metropolis
algorithms
Peter Mathé
Introduction Both methods are analyzed in literature! Recent
Numerical study [Bassetti/Diaconis]!
integration
Formal problem The principle questions we want to address are
formulation
Two approaches
from ergodic theory
Negative results
Problem
Metropolis Which method to apply in specific application?
method
Ball walk
Uniform ergodicity
Why and/or when to apply Metropolis algorithm?
Ball walk with
Metropolis filter
Conclusions
References
A large class (global assumptions on )
Numerical
analysis of the
We let
Metropolis
algorithms sup
FC (Ω) = {(f , ) | f ∞ ≤ 1, > 0, ≤ C}.
Peter Mathé inf
Introduction
Numerical
integration
Formal problem
formulation
Two approaches
from ergodic theory
Negative results
Metropolis
method
Ball walk
Uniform ergodicity
Ball walk with
Metropolis filter
Conclusions
References
A large class (global assumptions on )
Numerical
analysis of the
We let
Metropolis
algorithms sup
FC (Ω) = {(f , ) | f ∞ ≤ 1, > 0, ≤ C}.
Peter Mathé inf
Introduction
Numerical
Proposition
integration
Formal problem
formulation For each Monte Carlo method ϑn , using n values of f and ,
Two approaches
from ergodic theory we have the following bound:
Negative results
Metropolis
method 1 C
Ball walk e(ϑn , FC (Ω)) ≥ 2 , 2n ≥ C − 1.
Uniform ergodicity
Ball walk with
6 n
Metropolis filter
Conclusions “Simple Monte Carlo” is optimal.
References
A large class (global assumptions on )
Numerical
analysis of the
We let
Metropolis
algorithms sup
FC (Ω) = {(f , ) | f ∞ ≤ 1, > 0, ≤ C}.
Peter Mathé inf
Introduction
Numerical
Proposition
integration
Formal problem
formulation For each Monte Carlo method ϑn , using n values of f and ,
Two approaches
from ergodic theory we have the following bound:
Negative results
Metropolis
method 1 C
Ball walk e(ϑn , FC (Ω)) ≥ 2 , 2n ≥ C − 1.
Uniform ergodicity
Ball walk with
6 n
Metropolis filter
Conclusions “Simple Monte Carlo” is optimal.
References
Remark
Need structure!
Class with local structure on d-dim. ball
Numerical
analysis of the Let f and be defined on the unit ball B d ⊂ Rd and µΩ
Metropolis
algorithms the normalized Lebesgue measure.
Peter Mathé The weights > 0 on Rα (B d ) are log-concave, i.e.,
(λx + (1 − λ)y ) ≥ (x)λ · (y )1−λ ,
Introduction
Numerical
integration
Formal problem
formulation
The logarithm of is Lipschitz continuous, i.e.,
Two approaches
from ergodic theory
Negative results | log (x) − log (y )| ≤ α x − y 2.
Metropolis
method The class of input is
Ball walk
Uniform ergodicity
Ball walk with
Metropolis filter F α (B d ) = (f , ) | ∈ Rα (B d ), f 2, ≤1 .
Conclusions
References
Class with local structure on d-dim. ball
Numerical
analysis of the Let f and be defined on the unit ball B d ⊂ Rd and µΩ
Metropolis
algorithms the normalized Lebesgue measure.
Peter Mathé The weights > 0 on are log-concave, i.e.,
(λx + (1 − λ)y ) ≥ (x)λ · (y )1−λ ,
Introduction
Numerical
integration
Formal problem
formulation
The logarithm of is Lipschitz continuous, i.e.,
Two approaches
from ergodic theory
Negative results | log (x) − log (y )| ≤ α x − y 2.
Metropolis
method The class of input is
Ball walk
Uniform ergodicity
Ball walk with
Metropolis filter F α (B d ) = (f , ) | ∈ Rα (B d ), f 2, ≤1 .
Conclusions
References
Problem
Can we exploit this structure?
Lower bounds for non-adaptive methods
Numerical
analysis of the
Metropolis
Proposition
algorithms
For each non-adaptive method ϑn we have
Peter Mathé
1 α d/2
Introduction
e(ϑn , F α (B d )) ≥ √ · n−1/2 ,
Numerical
integration
2 d! 2
Formal problem
if 2n ≥ nd and 2n ≥ (α/log 4)d und α ≥ 2.
formulation
Two approaches
from ergodic theory
Negative results
Metropolis
method This bound is bad wrt. α!
Ball walk
Uniform ergodicity Local structure cannot be exploited by non-adaptive
Ball walk with
Metropolis filter
methods! In particular this holds true for Simple Monte
Conclusions
Carlo.
References
Lower bounds for non-adaptive methods
Numerical
analysis of the
Metropolis
Proposition
algorithms
For each non-adaptive method ϑn we have
Peter Mathé
1 α d/2
Introduction
e(ϑn , F α (B d )) ≥ √ · n−1/2 ,
Numerical
integration
2 d! 2
Formal problem
if 2n ≥ nd and 2n ≥ (α/log 4)d und α ≥ 2.
formulation
Two approaches
from ergodic theory
Negative results
Metropolis
method This bound is bad wrt. α!
Ball walk
Uniform ergodicity Local structure cannot be exploited by non-adaptive
Ball walk with
Metropolis filter
methods! In particular this holds true for Simple Monte
Conclusions
Carlo.
References
Task
Design Metropolis algorithm!
The Metropolis step
Numerical
analysis of the
Metropolis Procedure Metropolis-step(x, )
algorithms
Input : current position x, function ;
Peter Mathé
Output : next position;
Introduction Propose: y := symmetric-proposal-step(x);
Numerical
integration
Accept:
Formal problem
formulation
if (y ) ≥ rand() · (x) then
Two approaches
from ergodic theory
return y
Negative results
else
Metropolis
method
return x
Ball walk
Uniform ergodicity
end
Ball walk with
Metropolis filter
Source [M(RT )2 ]: N. Metropolis, A. W. Rosenbluth, M. N.
Conclusions
Rosenbluth, A. H. Teller, and E. Teller. Equations of state
References
calculations by fast computing machines. J. Chem. Phys.,
21:1087–1092, 1953.
The Metropolis step
Numerical
analysis of the
Metropolis Procedure Metropolis-step(x, )
algorithms
Input : current position x, function ;
Peter Mathé
Output : next position;
Introduction Propose: y := symmetric-proposal-step(x);
Numerical
integration
Accept:
Formal problem
formulation
if (y ) ≥ rand() · (x) then
Two approaches
from ergodic theory
return y
Negative results
else
Metropolis
method
return x
Ball walk
Uniform ergodicity
end
Ball walk with
Metropolis filter
Source [M(RT )2 ]: N. Metropolis, A. W. Rosenbluth, M. N.
Conclusions
Rosenbluth, A. H. Teller, and E. Teller. Equations of state
References
calculations by fast computing machines. J. Chem. Phys.,
21:1087–1092, 1953.
Ball walk
Numerical
analysis of the We consider the following symmetric proposal step:
Metropolis
algorithms Procedure Ball-walk-step(x, δ)
Peter Mathé Input : current position x; δ > 0;
Introduction Output : next position;
Numerical Propose: Choose y ∈ B(x, δ) uniformly ;
integration
Formal problem Accept:
formulation
Two approaches
from ergodic theory
if y ∈ Ω then
Negative results return y ;
Metropolis else
method
Ball walk return x;
Uniform ergodicity
Ball walk with end
Metropolis filter
Conclusions
References Remark
The ball walk is often used. It is a major ingredient in volume
algorithms, see [Lovász/Simonovits].
Ball walk
Numerical
analysis of the We consider the following symmetric proposal step:
Metropolis
algorithms Procedure Ball-walk-step(x, δ)
Peter Mathé Input : current position x; δ > 0;
Introduction Output : next position;
Numerical Propose: Choose y ∈ B(x, δ) uniformly ;
integration
Formal problem Accept:
formulation
Two approaches
from ergodic theory
if y ∈ Ω then
Negative results return y ;
Metropolis else
method
Ball walk return x;
Uniform ergodicity
Ball walk with end
Metropolis filter
Conclusions
References Remark
The ball walk is often used. It is a major ingredient in volume
algorithms, see [Lovász/Simonovits].
Uniform ergodicity
Numerical
analysis of the
Metropolis Definition
algorithms
A Markov chain K with invariant distribution π is said to be
Peter Mathé
uniformly ergodic, if there are constants M < ∞ and
Introduction 0 < η < 1 such that
Numerical
integration sup K n (x, ·) − π(·) tv ≤ Mη n
Formal problem
formulation x∈Ω
Two approaches
from ergodic theory
Negative results
Metropolis
method
Ball walk
Uniform ergodicity
Ball walk with
Metropolis filter
Conclusions
References
Uniform ergodicity
Numerical
analysis of the
Metropolis Definition
algorithms
A Markov chain K with invariant distribution π is said to be
Peter Mathé
uniformly ergodic, if there are constants M < ∞ and
Introduction 0 < η < 1 such that
Numerical
integration sup K n (x, ·) − π(·) tv ≤ Mη n
Formal problem
formulation x∈Ω
Two approaches
from ergodic theory
Negative results
Metropolis
Theorem
method ([Mathé-Markov-chains], )
Ball walk
Uniform ergodicity
Ball walk with
Let K be a reversible uniformly ergodic markov chain with
Metropolis filter
spectral gap 1 − β1 (K ) > 0. Then
Conclusions
1 + β1 (K )
References lim sup e(ϑn , f )2 · n = sup .
n→∞ f∈ f 2 f ∈ f 2 1 − β1 (K )
Uniform ergodicity
Numerical
analysis of the
Metropolis Definition
algorithms
A Markov chain K with invariant distribution π is said to be
Peter Mathé
uniformly ergodic, if there are constants M < ∞ and
Introduction 0 < η < 1 such that
Numerical
integration sup K n (x, ·) − π(·) tv ≤ Mη n
Formal problem
formulation x∈Ω
Two approaches
from ergodic theory
Negative results
Metropolis
Theorem
method ( ,[Mathé/Novak-Metropolis])
Ball walk
Uniform ergodicity
Ball walk with
Let K be a reversible uniformly ergodic markov chain with
Metropolis filter
spectral gap 1 − β1 (K ) > 0. Then
Conclusions
1 + β1 (K ,δ )
References lim sup sup e(ϑn , f )2 · n = sup sup .
n→∞ ∈R (Ω) f ∈ f
α 2 ∈Rα (Ω) f ∈ f 2 1 − β1 (K ,δ )
Ergodicity of the ball walk
Numerical
analysis of the
Metropolis
algorithms Let Ω be a convex body in Rd .
Peter Mathé
Proposition
Introduction
Numerical
The ball walk is uniformly ergodic.
integration
Formal problem The uniform distribution µΩ on Ω is the invariant
formulation
Two approaches
from ergodic theory
distribution.
Negative results
The spectral gap depends on the geometry of Ω.
Metropolis
method
Ball walk
Uniform ergodicity
Ball walk with
Metropolis filter
Conclusions
References
Ergodicity of the ball walk
Numerical
analysis of the
Metropolis
algorithms Let Ω be a convex body in Rd .
Peter Mathé
Proposition
Introduction
Numerical
The ball walk is uniformly ergodic.
integration
Formal problem The uniform distribution µΩ on Ω is the invariant
formulation
Two approaches
from ergodic theory
distribution.
Negative results
The spectral gap depends on the geometry of Ω.
Metropolis
method
Ball walk
For Ω = B d we have for δ δ −1/2 that
Uniform ergodicity
1 − β1 ≥ c/d 2 .
Ball walk with
Metropolis filter
Conclusions
References
Ergodicity of the ball walk
Numerical
analysis of the
Metropolis
algorithms Let Ω be a convex body in Rd .
Peter Mathé
Proposition
Introduction
Numerical
The ball walk is uniformly ergodic.
integration
Formal problem The uniform distribution µΩ on Ω is the invariant
formulation
Two approaches
from ergodic theory
distribution.
Negative results
The spectral gap depends on the geometry of Ω.
Metropolis
method
Ball walk
For Ω = B d we have for δ δ −1/2 that
Uniform ergodicity
1 − β1 ≥ c/d 2 .
Ball walk with
Metropolis filter
Conclusions
References
The Metropolis filter
Numerical
analysis of the Procedure Metropolis-step K ,δ (x, , δ)
Metropolis
algorithms Input : current position x, δ > 0, function ;
Peter Mathé Output : next position;
Introduction Propose: y := Ball-walk-step(x, δ);
Numerical Accept:
integration
Formal problem if (y ) ≥ (x) then
formulation
Two approaches
return y
from ergodic theory
Negative results else if (y ) ≥ rand() · (x) then
Metropolis return y
method
Ball walk else
Uniform ergodicity
Ball walk with
return x
Metropolis filter
end
Conclusions
References
The Metropolis filter
Numerical
analysis of the Procedure Metropolis-step K ,δ (x, , δ)
Metropolis
algorithms Input : current position x, δ > 0, function ;
Peter Mathé Output : next position;
Introduction Propose: y := Ball-walk-step(x, δ);
Numerical Accept:
integration
Formal problem if (y ) ≥ (x) then
formulation
Two approaches
return y
from ergodic theory
Negative results else if (y ) ≥ rand() · (x) then
Metropolis return y
method
Ball walk else
Uniform ergodicity
Ball walk with
return x
Metropolis filter
end
Conclusions
References
The Metropolis filter
Numerical
analysis of the Procedure Metropolis-step K ,δ (x, , δ)
Metropolis
algorithms Input : current position x, δ > 0, function ;
Peter Mathé Output : next position;
Introduction Propose: y := Ball-walk-step(x, δ);
Numerical Accept:
integration
Formal problem if (y ) ≥ (x) then
formulation
Two approaches
return y
from ergodic theory
Negative results else if (y ) ≥ rand() · (x) then
Metropolis return y
method
Ball walk else
Uniform ergodicity
Ball walk with
return x
Metropolis filter
end
Conclusions
References
The Metropolis filter
Numerical
analysis of the Procedure Metropolis-step K ,δ (x, , δ)
Metropolis
algorithms Input : current position x, δ > 0, function ;
Peter Mathé Output : next position;
Introduction Propose: y := Ball-walk-step(x, δ);
Numerical Accept:
integration
Formal problem if (y ) ≥ (x) then
formulation
Two approaches
return y
from ergodic theory
Negative results else if (y ) ≥ rand() · (x) then
Metropolis return y
method
Ball walk else
Uniform ergodicity
Ball walk with
return x
Metropolis filter
end
Conclusions
References
Remark
The ball walk is a Metropolis chain with (x) = χΩ (x).
Ergodicity of the Metropolis chain
Numerical
analysis of the
We shall study the Metropolis chain as a perturbation of the
Metropolis
algorithms
ball walk!
Peter Mathé
Introduction
Numerical
integration
Formal problem
formulation
Two approaches
from ergodic theory
Negative results
Metropolis
method
Ball walk
Uniform ergodicity
Ball walk with
Metropolis filter
Conclusions
References
Ergodicity of the Metropolis chain
Numerical
analysis of the
We shall study the Metropolis chain as a perturbation of the
Metropolis
algorithms
ball walk!
Peter Mathé
Let Ω ⊂ Rd be a convex body.
Introduction Proposition
Numerical
integration For each ∈ Rα (Ω) the Metropolis chain K ,δ is
Formal problem
formulation uniformly ergodic.
Two approaches
from ergodic theory
Negative results
The invariant distribution is µ .
Metropolis Uniformly for ∈ Rα (Ω) there is a common factor η of
method
Ball walk uniform ergodicity.
Uniform ergodicity
Ball walk with
Metropolis filter
Conclusions
References
Ergodicity of the Metropolis chain
Numerical
analysis of the
We shall study the Metropolis chain as a perturbation of the
Metropolis
algorithms
ball walk!
Peter Mathé
Let Ω ⊂ Rd be a convex body.
Introduction Proposition
Numerical
integration For each ∈ Rα (Ω) the Metropolis chain K ,δ is
Formal problem
formulation uniformly ergodic.
Two approaches
from ergodic theory
Negative results
The invariant distribution is µ .
Metropolis Uniformly for ∈ Rα (Ω) there is a common factor η of
method
Ball walk uniform ergodicity.
Uniform ergodicity √
Ball walk with
Metropolis filter If Ω = B d and δ = min 1/ d + 1, 1/α , then
Conclusions
References c 1 1
1 − β1 ≥ min , .
d d α
Convergence result
Numerical
analysis of the
Metropolis
algorithms
Theorem
Peter Mathé
Let X1 , X2 , . . . be a trajectory of the ball walk with Metropolis
Introduction filter and ϑn (f , ) = 1/n n f (Xi ). Let
i=1
Numerical δ ∗ = min (d + 1)−1/2 , α−1 . Then
integration
Formal problem
formulation
∗ √ √ √
Two approaches
from ergodic theory e(Sn , F α (B d )) ·
δ ˜
n≤C· d + 1 max d + 1, α .
Negative results
Metropolis
method
Ball walk
Uniform ergodicity
Ball walk with
Metropolis filter
Conclusions
References
Convergence result
Numerical
analysis of the
Metropolis
algorithms
Theorem
Peter Mathé
Let X1 , X2 , . . . be a trajectory of the ball walk with Metropolis
Introduction filter and ϑn (f , ) = 1/n n f (Xi ). Let
i=1
Numerical δ ∗ = min (d + 1)−1/2 , α−1 . Then
integration
Formal problem
formulation
∗ √ √ √
Two approaches
from ergodic theory e(Sn , F α (B d )) ·
δ ˜
n≤C· d + 1 max d + 1, α .
Negative results
Metropolis
method
Ball walk
Uniform ergodicity
Ball walk with
Metropolis filter The rate is linear in α, independent of the dimension,
Conclusions as compared to non-adaptive methods.
References
Convergence result
Numerical
analysis of the
Metropolis
algorithms
Theorem
Peter Mathé
Let X1 , X2 , . . . be a trajectory of the ball walk with Metropolis
Introduction filter and ϑn (f , ) = 1/n n f (Xi ). Let
i=1
Numerical δ ∗ = min (d + 1)−1/2 , α−1 . Then
integration
Formal problem
formulation
∗ √ √ √
Two approaches
from ergodic theory e(Sn , F α (B d )) ·
δ ˜
n≤C· d + 1 max d + 1, α .
Negative results
Metropolis
method
Ball walk
Uniform ergodicity
Ball walk with
Metropolis filter The rate is linear in α, independent of the dimension,
Conclusions as compared to non-adaptive methods.
References
The local Lipschitz continuity can be exploited, if we
adjust the step size δ to the problem!
Conclusions
Numerical
analysis of the
Metropolis
algorithms
Peter Mathé
We analyzed the use of Markov chains for weighted
Introduction numerical integration.
Numerical
integration
Without structural assumptions no Monte Carlo method
Formal problem
formulation can be efficient, even Metropolis does not help!
Two approaches
from ergodic theory
Negative results
Under certain structure Metropolis can be superior to
Metropolis other methods.
method √
Ball walk Convergence rate typically is 1/ n, and the constant
Uniform ergodicity
Ball walk with
Metropolis filter
depends on the spectral gap.
Conclusions The geometry of the domain plays a crucial role.
References
Numerical
analysis of the
F. Bassetti and P. Diaconis.
Metropolis
algorithms
Examples comparing importance sampling and the
Peter Mathé
Metropolis algorithm.
Illinois J. of Math., 50(1):67–91, 2006.
Introduction
Numerical Gregory F. Lawler and Alan D. Sokal.
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Formal problem Bounds on the L2 spectrum for Markov chains and
formulation
Two approaches
from ergodic theory
Markov processes: a generalization of Cheeger’s
Negative results
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Metropolis
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Ball walk
Uniform ergodicity
Ball walk with
L. Lovász and M. Simonovits.
Metropolis filter
Random walks in a convex body and an improved volume
Conclusions
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Numerical
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Formal problem N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H.
formulation
Two approaches
from ergodic theory
Teller, and E. Teller.
Negative results
Equations of state calculations by fast computing
Metropolis
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Ball walk
Uniform ergodicity
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131–192. Plenum, New York, 1997.
Numerical
integration
Formal problem
formulation
Two approaches
from ergodic theory
Negative results
Metropolis
method
Ball walk
Uniform ergodicity
Ball walk with
Metropolis filter
Conclusions
References
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