# Numerical analysis of the Metropolis algorithms

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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).

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
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

References
Determine the posterior distribution!
Determine functionals     f dπ!
The problem

Numerical
analysis of the
Metropolis
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
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?

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

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
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.
integration
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
inequality.
Metropolis
method
Trans. Amer. Math. Soc., 309(2):557–580, 1988.
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
algorithm.
References
Random Structures Algorithms, 4(4):359–412, 1993.
Peter Mathé.
Numerical           Numerical integration using Markov chains.
analysis of the
Metropolis          Monte Carlo Methods Appl., 5(4):325–343, 1999.
algorithms

Peter Mathé
Peter Mathé and E. Novak.
Simple Monte Carlo and the Metropolis algorithm.
Introduction
submitted, 2006.
Numerical
integration
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
method
machines.
Ball walk
Uniform ergodicity
J. Chem. Phys., 21:1087–1092, 1953.
Ball walk with
Metropolis ﬁlter
Gareth O. Roberts and Jeffrey S. Rosenthal.
Conclusions
General state space Markov chains and MCMC
References
algorithms.
Probab. Surv., 1:20–71 (electronic), 2004.
Numerical
analysis of the
A. Sokal.
Metropolis
algorithms
Monte Carlo methods in statistical mechanics:
Peter Mathé
foundations and new algorithms.
In Functional integration (Cargèse, 1996), pages
Introduction
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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