This document discusses multiple-try Metropolis-Hastings (MH) algorithms. It introduces the multiple-try Metropolis (MCTM) algorithm and multiple-try Metropolis with control (MTM-C) algorithms, which generate multiple candidate samples from a proposal distribution before selecting one to accept or reject. The document also discusses analyzing these algorithms through optimal scaling and optimizing their speedup over standard MH algorithms.
1. Scaling analysis of multiple-try MCMC
methods
Randal DOUC
randal.douc@it-sudparis.eu
Travail joint avec Mylène Bédard et Eric Moulines.
1 / 25
2. Themes
1 MCMC algorithms with multiple proposals: MCTM, MTM-C.
2 Analysis through optimal scaling (introduced by Roberts,
Gelman, Gilks, 1998)
3 Hit and Run algorithm.
2 / 25
3. Themes
1 MCMC algorithms with multiple proposals: MCTM, MTM-C.
2 Analysis through optimal scaling (introduced by Roberts,
Gelman, Gilks, 1998)
3 Hit and Run algorithm.
2 / 25
4. Themes
1 MCMC algorithms with multiple proposals: MCTM, MTM-C.
2 Analysis through optimal scaling (introduced by Roberts,
Gelman, Gilks, 1998)
3 Hit and Run algorithm.
2 / 25
5. Themes
1 MCMC algorithms with multiple proposals: MCTM, MTM-C.
2 Analysis through optimal scaling (introduced by Roberts,
Gelman, Gilks, 1998)
3 Hit and Run algorithm.
2 / 25
6. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
Plan de l’exposé
1 Introduction
2 MH algorithms with multiple proposals
Random Walk MH
MCTM algorithm
MTM-C algorithms
3 Optimal scaling
Main results
4 Optimising the speed up process
MCTM algorithm
MTM-C algorithms
5 Conclusion
3 / 25
7. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
Plan de l’exposé
1 Introduction
2 MH algorithms with multiple proposals
Random Walk MH
MCTM algorithm
MTM-C algorithms
3 Optimal scaling
Main results
4 Optimising the speed up process
MCTM algorithm
MTM-C algorithms
5 Conclusion
3 / 25
8. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
Plan de l’exposé
1 Introduction
2 MH algorithms with multiple proposals
Random Walk MH
MCTM algorithm
MTM-C algorithms
3 Optimal scaling
Main results
4 Optimising the speed up process
MCTM algorithm
MTM-C algorithms
5 Conclusion
3 / 25
9. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
Plan de l’exposé
1 Introduction
2 MH algorithms with multiple proposals
Random Walk MH
MCTM algorithm
MTM-C algorithms
3 Optimal scaling
Main results
4 Optimising the speed up process
MCTM algorithm
MTM-C algorithms
5 Conclusion
3 / 25
10. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
Plan de l’exposé
1 Introduction
2 MH algorithms with multiple proposals
Random Walk MH
MCTM algorithm
MTM-C algorithms
3 Optimal scaling
Main results
4 Optimising the speed up process
MCTM algorithm
MTM-C algorithms
5 Conclusion
3 / 25
11. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
Plan
1 Introduction
2 MH algorithms with multiple proposals
Random Walk MH
MCTM algorithm
MTM-C algorithms
3 Optimal scaling
Main results
4 Optimising the speed up process
MCTM algorithm
MTM-C algorithms
5 Conclusion
4 / 25
12. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
Metropolis Hastings (MH) algorithm
1 We wish to approximate
π(x )
I= h(x ) dx = h(x )¯ (x )dx
π
π(u)du
2 x → π(x ) is known but not π(u)du.
Approximate I with ˜ = n t=1 h(X [t]) where (X [t]) is a Markov
1 n
3 I
chain with limiting distribution π .
¯
4 In MH algorithm, the last condition is obtained from a detailed
balance condition
∀x , y , π(x )p(x , y ) = π(y )p(y , x )
5 Quality of the approximation are obtained from Law of Large
Numbers or CLT for Markov chains.
5 / 25
13. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
Metropolis Hastings (MH) algorithm
1 We wish to approximate
π(x )
I= h(x ) dx = h(x )¯ (x )dx
π
π(u)du
2 x → π(x ) is known but not π(u)du.
Approximate I with ˜ = n t=1 h(X [t]) where (X [t]) is a Markov
1 n
3 I
chain with limiting distribution π .
¯
4 In MH algorithm, the last condition is obtained from a detailed
balance condition
∀x , y , π(x )p(x , y ) = π(y )p(y , x )
5 Quality of the approximation are obtained from Law of Large
Numbers or CLT for Markov chains.
5 / 25
14. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
Metropolis Hastings (MH) algorithm
1 We wish to approximate
π(x )
I= h(x ) dx = h(x )¯ (x )dx
π
π(u)du
2 x → π(x ) is known but not π(u)du.
Approximate I with ˜ = n t=1 h(X [t]) where (X [t]) is a Markov
1 n
3 I
chain with limiting distribution π .
¯
4 In MH algorithm, the last condition is obtained from a detailed
balance condition
∀x , y , π(x )p(x , y ) = π(y )p(y , x )
5 Quality of the approximation are obtained from Law of Large
Numbers or CLT for Markov chains.
5 / 25
15. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
Metropolis Hastings (MH) algorithm
1 We wish to approximate
π(x )
I= h(x ) dx = h(x )¯ (x )dx
π
π(u)du
2 x → π(x ) is known but not π(u)du.
Approximate I with ˜ = n t=1 h(X [t]) where (X [t]) is a Markov
1 n
3 I
chain with limiting distribution π .
¯
4 In MH algorithm, the last condition is obtained from a detailed
balance condition
∀x , y , π(x )p(x , y ) = π(y )p(y , x )
5 Quality of the approximation are obtained from Law of Large
Numbers or CLT for Markov chains.
5 / 25
16. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
Metropolis Hastings (MH) algorithm
1 We wish to approximate
π(x )
I= h(x ) dx = h(x )¯ (x )dx
π
π(u)du
2 x → π(x ) is known but not π(u)du.
Approximate I with ˜ = n t=1 h(X [t]) where (X [t]) is a Markov
1 n
3 I
chain with limiting distribution π .
¯
4 In MH algorithm, the last condition is obtained from a detailed
balance condition
∀x , y , π(x )p(x , y ) = π(y )p(y , x )
5 Quality of the approximation are obtained from Law of Large
Numbers or CLT for Markov chains.
5 / 25
17. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
Plan
1 Introduction
2 MH algorithms with multiple proposals
Random Walk MH
MCTM algorithm
MTM-C algorithms
3 Optimal scaling
Main results
4 Optimising the speed up process
MCTM algorithm
MTM-C algorithms
5 Conclusion
6 / 25
18. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
Random Walk MH
Notation w.p. = with probability
Algorithme (MCMC )
If X [t] = x , how is X [t + 1] simulated?
(a) Y ∼ q(x ; ·).
(b) Accept the proposal X [t + 1] = Y w.p. α(x , Y ) where
π(y )q(y ; x )
α(x , y ) = 1 ∧
π(x )q(x ; y )
(c) Otherwise X [t + 1] = x
7 / 25
19. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
Random Walk MH
Notation w.p. = with probability
Algorithme (MCMC )
If X [t] = x , how is X [t + 1] simulated?
(a) Y ∼ q(x ; ·).
(b) Accept the proposal X [t + 1] = Y w.p. α(x , Y ) where
π(y )q(y ; x )
α(x , y ) = 1 ∧
π(x )q(x ; y )
(c) Otherwise X [t + 1] = x
The chain is π-reversible since:
7 / 25
20. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
Random Walk MH
Notation w.p. = with probability
Algorithme (MCMC )
If X [t] = x , how is X [t + 1] simulated?
(a) Y ∼ q(x ; ·).
(b) Accept the proposal X [t + 1] = Y w.p. α(x , Y ) where
π(y )q(y ; x )
α(x , y ) = 1 ∧
π(x )q(x ; y )
(c) Otherwise X [t + 1] = x
The chain is π-reversible since:
π(x )α(x , y )q(x ; y ) = π(x )α(x , y ) ∧ π(y )α(y , x )
7 / 25
21. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
Random Walk MH
Notation w.p. = with probability
Algorithme (MCMC )
If X [t] = x , how is X [t + 1] simulated?
(a) Y ∼ q(x ; ·).
(b) Accept the proposal X [t + 1] = Y w.p. α(x , Y ) where
π(y )q(y ; x )
α(x , y ) = 1 ∧
π(x )q(x ; y )
(c) Otherwise X [t + 1] = x
The chain is π-reversible since:
π(x )α(x , y )q(x ; y ) = π(y )α(y , x )q(y ; x )
7 / 25
22. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
Random Walk MH
Assume that q(x ; y ) = q(y ; x ) ◮ the instrumental kernel is
symmetric. Typically Y = X + U where U has symm. distr.
Notation w.p. = with probability
Algorithme (MCMC with symmetric proposal)
If X [t] = x , how is X [t + 1] simulated?
(a) Y ∼ q(x ; ·).
(b) Accept the proposal X [t + 1] = Y w.p. α(x , Y ) where
π(y )q(y ; x )
α(x , y ) = 1 ∧
π(x )q(x ; y )
(c) Otherwise X [t + 1] = x
The chain is π-reversible since:
π(x )α(x , y )q(x ; y ) = π(y )α(y , x )q(y ; x )
7 / 25
23. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
Random Walk MH
Assume that q(x ; y ) = q(y ; x ) ◮ the instrumental kernel is
symmetric. Typically Y = X + U where U has symm. distr.
Notation w.p. = with probability
Algorithme (MCMC with symmetric proposal)
If X [t] = x , how is X [t + 1] simulated?
(a) Y ∼ q(x ; ·).
(b) Accept the proposal X [t + 1] = Y w.p. α(x , Y ) where
π(y )
α(x , y ) = 1 ∧
π(x )
(c) Otherwise X [t + 1] = x
The chain is π-reversible since:
π(x )α(x , y )q(x ; y ) = π(y )α(y , x )q(y ; x )
7 / 25
24. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
MCTM algorithm
Multiple proposal MCMC
1 Liu, Liang, Wong (2000) introduced the multiple proposal
MCMC. Generalized to multiple correlated proposals by Craiu
and Lemieux (2007).
2 A pool of candidates is drawn from (Y 1 , . . . , Y K ) X [t]
∼ q(X [t]; ·).
3 We select one candidate a priori according to some "informative"
criterium (with high values of π for example).
4 We accept the candidate with some well chosen probability.
◮ diversity of the candidates: some candidates are, other are far
away from the current state. Some additional notations:
Yj X [t]
∼ qj (X [t]; ·) (◮M ARGINAL DIST.) (1)
(Y i )i=j X [t],Y j
∼ qj (X [t], Y j ; ·) (◮S IM .
¯ OTHER CAND.) . (2)
8 / 25
25. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
MCTM algorithm
Multiple proposal MCMC
1 Liu, Liang, Wong (2000) introduced the multiple proposal
MCMC. Generalized to multiple correlated proposals by Craiu
and Lemieux (2007).
2 A pool of candidates is drawn from (Y 1 , . . . , Y K ) X [t]
∼ q(X [t]; ·).
3 We select one candidate a priori according to some "informative"
criterium (with high values of π for example).
4 We accept the candidate with some well chosen probability.
◮ diversity of the candidates: some candidates are, other are far
away from the current state. Some additional notations:
Yj X [t]
∼ qj (X [t]; ·) (◮M ARGINAL DIST.) (1)
(Y i )i=j X [t],Y j
∼ qj (X [t], Y j ; ·) (◮S IM .
¯ OTHER CAND.) . (2)
8 / 25
26. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
MCTM algorithm
Multiple proposal MCMC
1 Liu, Liang, Wong (2000) introduced the multiple proposal
MCMC. Generalized to multiple correlated proposals by Craiu
and Lemieux (2007).
2 A pool of candidates is drawn from (Y 1 , . . . , Y K ) X [t]
∼ q(X [t]; ·).
3 We select one candidate a priori according to some "informative"
criterium (with high values of π for example).
4 We accept the candidate with some well chosen probability.
◮ diversity of the candidates: some candidates are, other are far
away from the current state. Some additional notations:
Yj X [t]
∼ qj (X [t]; ·) (◮M ARGINAL DIST.) (1)
(Y i )i=j X [t],Y j
∼ qj (X [t], Y j ; ·) (◮S IM .
¯ OTHER CAND.) . (2)
8 / 25
27. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
MCTM algorithm
Multiple proposal MCMC
1 Liu, Liang, Wong (2000) introduced the multiple proposal
MCMC. Generalized to multiple correlated proposals by Craiu
and Lemieux (2007).
2 A pool of candidates is drawn from (Y 1 , . . . , Y K ) X [t]
∼ q(X [t]; ·).
3 We select one candidate a priori according to some "informative"
criterium (with high values of π for example).
4 We accept the candidate with some well chosen probability.
◮ diversity of the candidates: some candidates are, other are far
away from the current state. Some additional notations:
Yj X [t]
∼ qj (X [t]; ·) (◮M ARGINAL DIST.) (1)
(Y i )i=j X [t],Y j
∼ qj (X [t], Y j ; ·) (◮S IM .
¯ OTHER CAND.) . (2)
8 / 25
28. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
MCTM algorithm
Multiple proposal MCMC
1 Liu, Liang, Wong (2000) introduced the multiple proposal
MCMC. Generalized to multiple correlated proposals by Craiu
and Lemieux (2007).
2 A pool of candidates is drawn from (Y 1 , . . . , Y K ) X [t]
∼ q(X [t]; ·).
3 We select one candidate a priori according to some "informative"
criterium (with high values of π for example).
4 We accept the candidate with some well chosen probability.
◮ diversity of the candidates: some candidates are, other are far
away from the current state. Some additional notations:
Yj X [t]
∼ qj (X [t]; ·) (◮M ARGINAL DIST.) (1)
(Y i )i=j X [t],Y j
∼ qj (X [t], Y j ; ·) (◮S IM .
¯ OTHER CAND.) . (2)
8 / 25
29. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
MCTM algorithm
Multiple proposal MCMC
1 Liu, Liang, Wong (2000) introduced the multiple proposal
MCMC. Generalized to multiple correlated proposals by Craiu
and Lemieux (2007).
2 A pool of candidates is drawn from (Y 1 , . . . , Y K ) X [t]
∼ q(X [t]; ·).
3 We select one candidate a priori according to some "informative"
criterium (with high values of π for example).
4 We accept the candidate with some well chosen probability.
◮ diversity of the candidates: some candidates are, other are far
away from the current state. Some additional notations:
Yj X [t]
∼ qj (X [t]; ·) (◮M ARGINAL DIST.) (1)
(Y i )i=j X [t],Y j
∼ qj (X [t], Y j ; ·) (◮S IM .
¯ OTHER CAND.) . (2)
8 / 25
30. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
MCTM algorithm
Assume that qj (x ; y ) = qj (y ; x ) .
Algorithme (MCTM: Multiple Correlated try Metropolis alg.)
If X [t] = x , how is X [t + 1] simulated?
(a) (Y 1 , . . . , Y K ) ∼ q(x ; ·). (◮POOL OF CAND.)
(b) Draw an index J ∈ {1, . . . , K }, with probability proportional to
[π(Y 1 ), . . . , π(Y K )] . (◮S ELECTION A PRIORI)
˜
(c) {Y J,i }i=J ∼ qJ (Y J , x ; ·).
¯ (◮AUXILIARY VARIABLES)
˜
(d) Accept the proposal X [t + 1] = Y J w.p. αJ (x , (Y i )K , (Y J,i )i=J )
i=1
where
i=j π(y i ) + π(y j )
αj (x , (y i )K , (y j,i )i=j ) = 1 ∧
i=1 ˜ . (3)
˜ j,i
i=j π(y ) + π(x )
(◮MH ACCEPTANCE PROBABILITY)
(e) Otherwise, X [t + 1] = X [t]
See MTM-C 9 / 25
31. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
MCTM algorithm
Assume that qj (x ; y ) = qj (y ; x ) .
Algorithme (MCTM: Multiple Correlated try Metropolis alg.)
If X [t] = x , how is X [t + 1] simulated?
(a) (Y 1 , . . . , Y K ) ∼ q(x ; ·). (◮POOL OF CAND.)
(b) Draw an index J ∈ {1, . . . , K }, with probability proportional to
[π(Y 1 ), . . . , π(Y K )] . (◮S ELECTION A PRIORI)
˜
(c) {Y J,i }i=J ∼ qJ (Y J , x ; ·).
¯ (◮AUXILIARY VARIABLES)
˜
(d) Accept the proposal X [t + 1] = Y J w.p. αJ (x , (Y i )K , (Y J,i )i=J )
i=1
where
i=j π(y i ) + π(y j )
αj (x , (y i )K , (y j,i )i=j ) = 1 ∧
i=1 ˜ . (3)
˜ j,i
i=j π(y ) + π(x )
(◮MH ACCEPTANCE PROBABILITY)
(e) Otherwise, X [t + 1] = X [t]
See MTM-C 9 / 25
32. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
MCTM algorithm
Assume that qj (x ; y ) = qj (y ; x ) .
Algorithme (MCTM: Multiple Correlated try Metropolis alg.)
If X [t] = x , how is X [t + 1] simulated?
(a) (Y 1 , . . . , Y K ) ∼ q(x ; ·). (◮POOL OF CAND.)
(b) Draw an index J ∈ {1, . . . , K }, with probability proportional to
[π(Y 1 ), . . . , π(Y K )] . (◮S ELECTION A PRIORI)
˜
(c) {Y J,i }i=J ∼ qJ (Y J , x ; ·).
¯ (◮AUXILIARY VARIABLES)
˜
(d) Accept the proposal X [t + 1] = Y J w.p. αJ (x , (Y i )K , (Y J,i )i=J )
i=1
where
i=j π(y i ) + π(y j )
αj (x , (y i )K , (y j,i )i=j ) = 1 ∧
i=1 ˜ . (3)
˜ j,i
i=j π(y ) + π(x )
(◮MH ACCEPTANCE PROBABILITY)
(e) Otherwise, X [t + 1] = X [t]
See MTM-C 9 / 25
33. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
MCTM algorithm
Assume that qj (x ; y ) = qj (y ; x ) .
Algorithme (MCTM: Multiple Correlated try Metropolis alg.)
If X [t] = x , how is X [t + 1] simulated?
(a) (Y 1 , . . . , Y K ) ∼ q(x ; ·). (◮POOL OF CAND.)
(b) Draw an index J ∈ {1, . . . , K }, with probability proportional to
[π(Y 1 ), . . . , π(Y K )] . (◮S ELECTION A PRIORI)
˜
(c) {Y J,i }i=J ∼ qJ (Y J , x ; ·).
¯ (◮AUXILIARY VARIABLES)
˜
(d) Accept the proposal X [t + 1] = Y J w.p. αJ (x , (Y i )K , (Y J,i )i=J )
i=1
where
i=j π(y i ) + π(y j )
αj (x , (y i )K , (y j,i )i=j ) = 1 ∧
i=1 ˜ . (3)
˜ j,i
i=j π(y ) + π(x )
(◮MH ACCEPTANCE PROBABILITY)
(e) Otherwise, X [t + 1] = X [t]
See MTM-C 9 / 25
34. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
MCTM algorithm
1 It generalises the classical Random Walk Hasting Metropolis
algorithm (which is the case K = 1). RWMC
10 / 25
35. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
MCTM algorithm
1 It generalises the classical Random Walk Hasting Metropolis
algorithm (which is the case K = 1). RWMC
2 It satisfies the detailed balance condition wrt π:
K
π(x ) ··· ¯
qj (x ; y )Qj x , y ; ¯
d(y i ) Qj y , x ; d(y j,i )
j=1 i=j i=j
π(y ) π(y ) + i=j π(y i )
1∧
π(y ) + i=j π(y i ) π(x ) + i=j π(y j,i )
10 / 25
36. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
MCTM algorithm
1 It generalises the classical Random Walk Hasting Metropolis
algorithm (which is the case K = 1). RWMC
2 It satisfies the detailed balance condition wrt π:
K
π(x )π(y ) qj (x ; y ) ··· ¯
Qj x , y ; ¯
d(y i ) Qj y , x ; d(y j,i )
j=1 i=j i=j
1 1
i
∧
π(y ) + i=j π(y ) π(x ) + i=j π(y j,i )
◮ symmetric wrt (x , y )
10 / 25
37. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
MCTM algorithm
1 The MCTM uses the simulation of K random variables for the
pool of candidates and K − 1 auxiliary variables to compute the
MH acceptance ratio.
2 Can we reduce the number of simulated variables while keeping
the diversity of the pool?
3 Draw one random variable and use transformations to create the
pool of candidates and auxiliary variables.
11 / 25
38. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
MCTM algorithm
1 The MCTM uses the simulation of K random variables for the
pool of candidates and K − 1 auxiliary variables to compute the
MH acceptance ratio.
2 Can we reduce the number of simulated variables while keeping
the diversity of the pool?
3 Draw one random variable and use transformations to create the
pool of candidates and auxiliary variables.
11 / 25
39. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
MCTM algorithm
1 The MCTM uses the simulation of K random variables for the
pool of candidates and K − 1 auxiliary variables to compute the
MH acceptance ratio.
2 Can we reduce the number of simulated variables while keeping
the diversity of the pool?
3 Draw one random variable and use transformations to create the
pool of candidates and auxiliary variables.
11 / 25
40. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
MTM-C algorithms
Ψi : X ×[0, 1)r → X
Let .
Ψj,i : X × X → X
Assume that
1 For all j ∈ {1, . . . , K }, set
Y j = Ψj (x , V ) (◮C OMMON R . V.)
where V ∼ U([0, 1)r )
2 For any (i, j) ∈ {1, . . . , K }2 ,
Y i = Ψj,i (x , Y j ) . (◮R ECONSTRUCTION OF THE OTHER CAND.)
(4)
Example:
i
ψ i (x , v ) = x + σΦ−1 (< v i + v >) where v i =< K a >, a ∈ Rr , Φ
cumulative repartition function of the normal distribution. ◮
Korobov seq. + Cranley Patterson rot.
ψ i (x , v ) = x + γ i Φ−1 (v ) . ◮ Hit and Run algorithm. 12 / 25
41. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
MTM-C algorithms
Ψi : X ×[0, 1)r → X
Let .
Ψj,i : X × X → X
Assume that
1 For all j ∈ {1, . . . , K }, set
Y j = Ψj (x , V ) (◮C OMMON R . V.)
where V ∼ U([0, 1)r )
2 For any (i, j) ∈ {1, . . . , K }2 ,
Y i = Ψj,i (x , Y j ) . (◮R ECONSTRUCTION OF THE OTHER CAND.)
(4)
Example:
i
ψ i (x , v ) = x + σΦ−1 (< v i + v >) where v i =< K a >, a ∈ Rr , Φ
cumulative repartition function of the normal distribution. ◮
Korobov seq. + Cranley Patterson rot.
ψ i (x , v ) = x + γ i Φ−1 (v ) . ◮ Hit and Run algorithm. 12 / 25
42. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
MTM-C algorithms
Ψi : X ×[0, 1)r → X
Let .
Ψj,i : X × X → X
Assume that
1 For all j ∈ {1, . . . , K }, set
Y j = Ψj (x , V ) (◮C OMMON R . V.)
where V ∼ U([0, 1)r )
2 For any (i, j) ∈ {1, . . . , K }2 ,
Y i = Ψj,i (x , Y j ) . (◮R ECONSTRUCTION OF THE OTHER CAND.)
(4)
Example:
i
ψ i (x , v ) = x + σΦ−1 (< v i + v >) where v i =< K a >, a ∈ Rr , Φ
cumulative repartition function of the normal distribution. ◮
Korobov seq. + Cranley Patterson rot.
ψ i (x , v ) = x + γ i Φ−1 (v ) . ◮ Hit and Run algorithm. 12 / 25
43. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
MTM-C algorithms
Ψi : X ×[0, 1)r → X
Let .
Ψj,i : X × X → X
Assume that
1 For all j ∈ {1, . . . , K }, set
Y j = Ψj (x , V ) (◮C OMMON R . V.)
where V ∼ U([0, 1)r )
2 For any (i, j) ∈ {1, . . . , K }2 ,
Y i = Ψj,i (x , Y j ) . (◮R ECONSTRUCTION OF THE OTHER CAND.)
(4)
Example:
i
ψ i (x , v ) = x + σΦ−1 (< v i + v >) where v i =< K a >, a ∈ Rr , Φ
cumulative repartition function of the normal distribution. ◮
Korobov seq. + Cranley Patterson rot.
ψ i (x , v ) = x + γ i Φ−1 (v ) . ◮ Hit and Run algorithm. 12 / 25
44. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
MTM-C algorithms
Algorithme (MTM-C: Multiple Try Metropolis alg. with common
proposal)
(a) Draw V ∼ U([0, 1)r ) and set Y i = Ψi (x , V ) for i = 1, . . . , K .
(b) Draw an index J ∈ {1, . . . , K }, with probability proportional to
[π(Y 1 ), . . . , π(Y K )] .
(c) Accept X [t + 1] = Y J with probability αJ (x , Y ), where, for
¯
j ∈ {1, . . . , K },
αj (x , y j ) = αj x , {Ψj,i (x , y j )}K , {Ψj,i (y j , x )}i=j
¯ i=1 , (5)
with αj given in (3) and reject otherwise.
(d) Otherwise X [t + 1] = Y J .
See MCTM
13 / 25
45. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
MTM-C algorithms
Algorithme (MTM-C: Multiple Try Metropolis alg. with common
proposal)
(a) Draw V ∼ U([0, 1)r ) and set Y i = Ψi (x , V ) for i = 1, . . . , K .
(b) Draw an index J ∈ {1, . . . , K }, with probability proportional to
[π(Y 1 ), . . . , π(Y K )] .
(c) Accept X [t + 1] = Y J with probability αJ (x , Y ), where, for
¯
j ∈ {1, . . . , K },
αj (x , y j ) = αj x , {Ψj,i (x , y j )}K , {Ψj,i (y j , x )}i=j
¯ i=1 , (5)
with αj given in (3) and reject otherwise.
(d) Otherwise X [t + 1] = Y J .
See MCTM
13 / 25
46. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
MTM-C algorithms
Algorithme (MTM-C: Multiple Try Metropolis alg. with common
proposal)
(a) Draw V ∼ U([0, 1)r ) and set Y i = Ψi (x , V ) for i = 1, . . . , K .
(b) Draw an index J ∈ {1, . . . , K }, with probability proportional to
[π(Y 1 ), . . . , π(Y K )] .
(c) Accept X [t + 1] = Y J with probability αJ (x , Y ), where, for
¯
j ∈ {1, . . . , K },
αj (x , y j ) = αj x , {Ψj,i (x , y j )}K , {Ψj,i (y j , x )}i=j
¯ i=1 , (5)
with αj given in (3) and reject otherwise.
(d) Otherwise X [t + 1] = Y J .
See MCTM
13 / 25
47. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
MTM-C algorithms
Algorithme (MTM-C: Multiple Try Metropolis alg. with common
proposal)
(a) Draw V ∼ U([0, 1)r ) and set Y i = Ψi (x , V ) for i = 1, . . . , K .
(b) Draw an index J ∈ {1, . . . , K }, with probability proportional to
[π(Y 1 ), . . . , π(Y K )] .
(c) Accept X [t + 1] = Y J with probability αJ (x , Y ), where, for
¯
j ∈ {1, . . . , K },
αj (x , y j ) = αj x , {Ψj,i (x , y j )}K , {Ψj,i (y j , x )}i=j
¯ i=1 , (5)
with αj given in (3) and reject otherwise.
(d) Otherwise X [t + 1] = Y J .
See MCTM
13 / 25
48. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
Plan
1 Introduction
2 MH algorithms with multiple proposals
Random Walk MH
MCTM algorithm
MTM-C algorithms
3 Optimal scaling
Main results
4 Optimising the speed up process
MCTM algorithm
MTM-C algorithms
5 Conclusion
14 / 25
49. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
How to compare two MH algorithms
◮ P ESKUN- If P1 and P2 are two π-reversible kernels and
∀x , y p1 (x , y ) ≤ p2 (x , y )
then P2 is better than P1 in terms of the asymptotic variance of
N −1 N h(X1 ).
i=1
1 Off diagonal order: Not always easy to compare!
2 Moreover, one expression of the asymptotic variance is:
∞
V = Varπ (h) + 2 Covπ (h(X0 ), h(Xt ))
t=1
15 / 25
50. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
How to compare two MH algorithms
◮ P ESKUN- If P1 and P2 are two π-reversible kernels and
∀x , y p1 (x , y ) ≤ p2 (x , y )
then P2 is better than P1 in terms of the asymptotic variance of
N −1 N h(X1 ).
i=1
1 Off diagonal order: Not always easy to compare!
2 Moreover, one expression of the asymptotic variance is:
∞
V = Varπ (h) + 2 Covπ (h(X0 ), h(Xt ))
t=1
15 / 25
51. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
How to compare two MH algorithms
◮ P ESKUN- If P1 and P2 are two π-reversible kernels and
∀x , y p1 (x , y ) ≤ p2 (x , y )
then P2 is better than P1 in terms of the asymptotic variance of
N −1 N h(X1 ).
i=1
1 Off diagonal order: Not always easy to compare!
2 Moreover, one expression of the asymptotic variance is:
∞
V = Varπ (h) + 2 Covπ (h(X0 ), h(Xt ))
t=1
15 / 25
52. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
Original idea of optimal scaling
For the RW-MH algorithm:
1 Increase dimension T .
T
2 Target distribution πT (x0:T ) = t=0 f (xt ) .
3 Assume that XT [0] ∼ πT .
4 Take a random walk increasingly conservative: draw candidate
ℓ
YT = XT [t] + √T UT [t] where UT [t] centered standard normal.
5 What is the "best" ℓ?
16 / 25
53. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
Original idea of optimal scaling
For the RW-MH algorithm:
1 Increase dimension T .
T
2 Target distribution πT (x0:T ) = t=0 f (xt ) .
3 Assume that XT [0] ∼ πT .
4 Take a random walk increasingly conservative: draw candidate
ℓ
YT = XT [t] + √T UT [t] where UT [t] centered standard normal.
5 What is the "best" ℓ?
16 / 25
54. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
Original idea of optimal scaling
For the RW-MH algorithm:
1 Increase dimension T .
T
2 Target distribution πT (x0:T ) = t=0 f (xt ) .
3 Assume that XT [0] ∼ πT .
4 Take a random walk increasingly conservative: draw candidate
ℓ
YT = XT [t] + √T UT [t] where UT [t] centered standard normal.
5 What is the "best" ℓ?
16 / 25
55. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
Original idea of optimal scaling
For the RW-MH algorithm:
1 Increase dimension T .
T
2 Target distribution πT (x0:T ) = t=0 f (xt ) .
3 Assume that XT [0] ∼ πT .
4 Take a random walk increasingly conservative: draw candidate
ℓ
YT = XT [t] + √T UT [t] where UT [t] centered standard normal.
5 What is the "best" ℓ?
16 / 25
56. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
Original idea of optimal scaling
For the RW-MH algorithm:
1 Increase dimension T .
T
2 Target distribution πT (x0:T ) = t=0 f (xt ) .
3 Assume that XT [0] ∼ πT .
4 Take a random walk increasingly conservative: draw candidate
ℓ
YT = XT [t] + √T UT [t] where UT [t] centered standard normal.
5 What is the "best" ℓ?
16 / 25
57. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
Théorème
The first component of (XT [⌊Ts⌋])0≤s≤1 weakly converges in the
Skorokhod topology to the stationary solution (W [λℓ s], s ∈ R+ ) of the
Langevin SDE
1
dW [s] = dB[s] + [ln f ]′ (W [s])ds ,
2
In particular, the first component of
(XT [0], XT [α1 T ], . . . , XT [αp T ])
converges weakly to the distribution of
(W [0], W [λℓ α1 T ], . . . , W [λℓ αp T ])
17 / 25
58. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
Théorème
The first component of (XT [⌊Ts⌋])0≤s≤1 weakly converges in the
Skorokhod topology to the stationary solution (W [λℓ s], s ∈ R+ ) of the
Langevin SDE
1
dW [s] = dB[s] + [ln f ]′ (W [s])ds ,
2
In particular, the first component of
(XT [0], XT [α1 T ], . . . , XT [αp T ])
converges weakly to the distribution of
(W [0], W [λℓ α1 T ], . . . , W [λℓ αp T ])
ℓ
√
Then, ℓ is chosen to maximize λℓ = 2ℓ2 Φ − 2 I where
2
I= {[ln f ]′ (x )} f (x )dx .
17 / 25
59. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
Théorème
The first component of (XT [⌊Ts⌋])0≤s≤1 weakly converges in the
Skorokhod topology to the stationary solution (W [s], s ∈ R+ ) of the
Langevin SDE
λℓ ′
dW [s] = λℓ dB[s] + [ln f ] (W [s])ds ,
2
In particular, the first component of
(XT [0], XT [α1 T ], . . . , XT [αp T ])
converges weakly to the distribution of
(W [0], W [λℓ α1 T ], . . . , W [λℓ αp T ])
ℓ
√
Then, ℓ is chosen to maximize λℓ = 2ℓ2 Φ − 2 I where
2
I= {[ln f ]′ (x )} f (x )dx .
17 / 25
60. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
Main results
Optimal scaling for the MCTM algorithm
◮ The pool of candidates
YT ,t [n + 1] = XT ,t [n] + T −1/2 Uti [n + 1] ,
i
0 ≤ t ≤ T, 1 ≤ i ≤ K,
where for any t ∈ {0, . . . , T },
(Uti [n + 1])K ∼ N (0, Σ) , (◮MCTM)
i=1
Uti [n + 1] = ψ i (Vt ), and Vt ∼ U[0, 1], (◮MTM-C)
◮ The auxiliary variables
˜ j,i ˜
YT ,t [n + 1] = XT ,t [n] + T −1/2 Utj,i [n + 1] , i =j ,
18 / 25
61. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
Main results
Optimal scaling for the MCTM algorithm
◮ The pool of candidates
YT ,t [n + 1] = XT ,t [n] + T −1/2 Uti [n + 1] ,
i
0 ≤ t ≤ T, 1 ≤ i ≤ K,
where for any t ∈ {0, . . . , T },
(Uti [n + 1])K ∼ N (0, Σ) , (◮MCTM)
i=1
Uti [n + 1] = ψ i (Vt ), and Vt ∼ U[0, 1], (◮MTM-C)
◮ The auxiliary variables
˜ j,i ˜
YT ,t [n + 1] = XT ,t [n] + T −1/2 Utj,i [n + 1] , i =j ,
18 / 25
62. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
Main results
Théorème
Suppose that XT [0] is distributed according to the target density πT .
Then, the process (XT ,0 [sT ], s ∈ R+ ) weakly converges in the
Skorokhod topology to the stationary solution (W [s], s ∈ R+ ) of the
Langevin SDE
1 ′
dW [s] = λ1/2 dB[s] + λ [ln f ] (W [s])ds ,
2
with λ λ I, (Γj )K , where Γj , 1 ≤ j ≤ K denotes the covariance
j=1
j i ˜ j,i
matrix of the random vector (U0 , (U0 )i=j , (U0 )i=j ).
For the MCTM, Γj = Γj (Σ).
2K −1
α(Γ) = E A Gi − Var[Gi ]/2 i=1
, (6)
where A is bounded lip. and (Gi )2K −1 ∼ N (0, Γ).
i=1
K
λ I, (Γj )K
j=1 Γj1,1 × α IΓj , (7)
19 / 25
63. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
Main results
Théorème
Suppose that XT [0] is distributed according to the target density πT .
Then, the process (XT ,0 [sT ], s ∈ R+ ) weakly converges in the
Skorokhod topology to the stationary solution (W [s], s ∈ R+ ) of the
Langevin SDE
1 ′
dW [s] = λ1/2 dB[s] + λ [ln f ] (W [s])ds ,
2
with λ λ I, (Γj )K , where Γj , 1 ≤ j ≤ K denotes the covariance
j=1
j i ˜ j,i
matrix of the random vector (U0 , (U0 )i=j , (U0 )i=j ).
For the MCTM, Γj = Γj (Σ).
2K −1
α(Γ) = E A Gi − Var[Gi ]/2 i=1
, (6)
where A is bounded lip. and (Gi )2K −1 ∼ N (0, Γ).
i=1
K
λ I, (Γj )K
j=1 Γj1,1 × α IΓj , (7)
19 / 25
64. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
Main results
Théorème
Suppose that XT [0] is distributed according to the target density πT .
Then, the process (XT ,0 [sT ], s ∈ R+ ) weakly converges in the
Skorokhod topology to the stationary solution (W [s], s ∈ R+ ) of the
Langevin SDE
1 ′
dW [s] = λ1/2 dB[s] + λ [ln f ] (W [s])ds ,
2
with λ λ I, (Γj )K , where Γj , 1 ≤ j ≤ K denotes the covariance
j=1
j i ˜ j,i
matrix of the random vector (U0 , (U0 )i=j , (U0 )i=j ).
For the MCTM, Γj = Γj (Σ).
2K −1
α(Γ) = E A Gi − Var[Gi ]/2 i=1
, (6)
where A is bounded lip. and (Gi )2K −1 ∼ N (0, Γ).
i=1
K
λ I, (Γj )K
j=1 Γj1,1 × α IΓj , (7)
19 / 25
65. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
Plan
1 Introduction
2 MH algorithms with multiple proposals
Random Walk MH
MCTM algorithm
MTM-C algorithms
3 Optimal scaling
Main results
4 Optimising the speed up process
MCTM algorithm
MTM-C algorithms
5 Conclusion
20 / 25
66. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
MCTM algorithm
We optimize the speed λ λ(I, (Γj (Σ))K ) over a subset G
j=1
G = Σ = diag(ℓ2 , . . . , ℓ2 ), (ℓ1 , . . . , ℓK ) ∈ RK : the proposals
1 K
have different scales but are independent.
G = Σ = ℓ2 Σa , ℓ2 ∈ R , where Σa is the extreme antithetic
covariance matrix:
K 1
Σa IK − 1K 1T
K
K −1 K −1
with 1K = (1, . . . , 1)T .
21 / 25
67. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
MCTM algorithm
MCTM algorithms
Table: Optimal scaling constants, value of the speed, and mean
acceptance rate for independent proposals
K 1 2 3 4 5
ℓ⋆ 2.38 2.64 2.82 2.99 3.12
λ⋆ 1.32 2.24 2.94 3.51 4.00
a⋆ 0.23 0.32 0.37 0.39 0.41
22 / 25
68. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
MCTM algorithm
MCTM algorithms
Table: Optimal scaling constants, value of the speed, and mean
acceptance rate for extreme antithetic proposals
K 1 2 3 4 5
ℓ⋆ 2.38 2.37 2.64 2.83 2.99
λ⋆ 1.32 2.64 3.66 4.37 4.91
a⋆ 0.23 0.46 0.52 0.54 0.55
Table: Optimal scaling constants, value of the speed, and mean
acceptance rate for the optimal covariance
K 1 2 3 4 5
ℓ⋆ 2.38 2.37 2.66 2.83 2.98
λ⋆ 1.32 2.64 3.70 4.40 4.93
a⋆ 0.23 0.46 0.52 0.55 0.56
22 / 25
69. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
MTM-C algorithms
MTM-C algorithms
Table: Optimal scaling constants, optimal value of the speed and the
mean acceptance rate for the RQMC MTM algorithm based on the
Korobov sequence and Cranley-Patterson rotations
K 1 2 3 4 5
σ⋆ 2.38 2.59 2.77 2.91 3.03
λ⋆ 1.32 2.43 3.31 4.01 4.56
a⋆ 0.23 0.36 0.42 0.47 0.50
Table: Optimal scaling constants, value of the speed, and mean
acceptance rate for the hit-and-run algorithm
K 1 2 4 6 8
ℓ⋆ 2.38 2.37 7.11 11.85 16.75
λ⋆ 1.32 2.64 2.65 2.65 2.65
a⋆ 0.23 0.46 0.46 0.46 0.46 23 / 25
70. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
Plan
1 Introduction
2 MH algorithms with multiple proposals
Random Walk MH
MCTM algorithm
MTM-C algorithms
3 Optimal scaling
Main results
4 Optimising the speed up process
MCTM algorithm
MTM-C algorithms
5 Conclusion
24 / 25
71. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
Conclusion
◮ MCTM algorithm:
1 Extreme antithetic proposals improves upon the MTM with
independent proposals.
2 Still, the improvement is not overly impressive and since the
introduction of correlation makes the computation of the
acceptance ratio more complex.
◮ MTM-C algorithm:
1 The advantage of the MTM-C algorithms: only one simulation
is required for obtaining the pool of proposals and auxiliary
variables.
2 The MTM-RQMC ∼ the extreme antithetic proposals.
3 Our preferred choice: the MTM-HR algorithm. In particular,
the case K = 2 induces a speed which is twice that of the
Metropolis algorithm whereas the computational cost is
almost the same in many scenarios.
25 / 25
72. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
Conclusion
◮ MCTM algorithm:
1 Extreme antithetic proposals improves upon the MTM with
independent proposals.
2 Still, the improvement is not overly impressive and since the
introduction of correlation makes the computation of the
acceptance ratio more complex.
◮ MTM-C algorithm:
1 The advantage of the MTM-C algorithms: only one simulation
is required for obtaining the pool of proposals and auxiliary
variables.
2 The MTM-RQMC ∼ the extreme antithetic proposals.
3 Our preferred choice: the MTM-HR algorithm. In particular,
the case K = 2 induces a speed which is twice that of the
Metropolis algorithm whereas the computational cost is
almost the same in many scenarios.
25 / 25
73. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
Conclusion
◮ MCTM algorithm:
1 Extreme antithetic proposals improves upon the MTM with
independent proposals.
2 Still, the improvement is not overly impressive and since the
introduction of correlation makes the computation of the
acceptance ratio more complex.
◮ MTM-C algorithm:
1 The advantage of the MTM-C algorithms: only one simulation
is required for obtaining the pool of proposals and auxiliary
variables.
2 The MTM-RQMC ∼ the extreme antithetic proposals.
3 Our preferred choice: the MTM-HR algorithm. In particular,
the case K = 2 induces a speed which is twice that of the
Metropolis algorithm whereas the computational cost is
almost the same in many scenarios.
25 / 25
74. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
Conclusion
◮ MCTM algorithm:
1 Extreme antithetic proposals improves upon the MTM with
independent proposals.
2 Still, the improvement is not overly impressive and since the
introduction of correlation makes the computation of the
acceptance ratio more complex.
◮ MTM-C algorithm:
1 The advantage of the MTM-C algorithms: only one simulation
is required for obtaining the pool of proposals and auxiliary
variables.
2 The MTM-RQMC ∼ the extreme antithetic proposals.
3 Our preferred choice: the MTM-HR algorithm. In particular,
the case K = 2 induces a speed which is twice that of the
Metropolis algorithm whereas the computational cost is
almost the same in many scenarios.
25 / 25
75. Introduction MH algorithms with multiple proposals Optimal scaling Optimising the speed up process Conclusion
Conclusion
◮ MCTM algorithm:
1 Extreme antithetic proposals improves upon the MTM with
independent proposals.
2 Still, the improvement is not overly impressive and since the
introduction of correlation makes the computation of the
acceptance ratio more complex.
◮ MTM-C algorithm:
1 The advantage of the MTM-C algorithms: only one simulation
is required for obtaining the pool of proposals and auxiliary
variables.
2 The MTM-RQMC ∼ the extreme antithetic proposals.
3 Our preferred choice: the MTM-HR algorithm. In particular,
the case K = 2 induces a speed which is twice that of the
Metropolis algorithm whereas the computational cost is
almost the same in many scenarios.
25 / 25