Scaling analysis of multiple-try MCMC
              methods

               Randal DOUC
       randal.douc@it-sudparis.eu
...
Themes

   1   MCMC algorithms with multiple proposals: MCTM, MTM-C.
   2   Analysis through optimal scaling (introduced b...
Themes

   1   MCMC algorithms with multiple proposals: MCTM, MTM-C.
   2   Analysis through optimal scaling (introduced b...
Themes

   1   MCMC algorithms with multiple proposals: MCTM, MTM-C.
   2   Analysis through optimal scaling (introduced b...
Themes

   1   MCMC algorithms with multiple proposals: MCTM, MTM-C.
   2   Analysis through optimal scaling (introduced b...
Introduction     MH algorithms with multiple proposals   Optimal scaling   Optimising the speed up process   Conclusion


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Introduction     MH algorithms with multiple proposals   Optimal scaling   Optimising the speed up process   Conclusion


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Introduction     MH algorithms with multiple proposals   Optimal scaling   Optimising the speed up process   Conclusion


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Introduction     MH algorithms with multiple proposals   Optimal scaling   Optimising the speed up process   Conclusion


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Introduction     MH algorithms with multiple proposals   Optimal scaling   Optimising the speed up process   Conclusion


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Introduction     MH algorithms with multiple proposals   Optimal scaling   Optimising the speed up process   Conclusion


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Introduction    MH algorithms with multiple proposals    Optimal scaling   Optimising the speed up process   Conclusion


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Introduction    MH algorithms with multiple proposals    Optimal scaling   Optimising the speed up process   Conclusion


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Introduction    MH algorithms with multiple proposals    Optimal scaling   Optimising the speed up process   Conclusion


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Introduction    MH algorithms with multiple proposals    Optimal scaling   Optimising the speed up process   Conclusion


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Introduction    MH algorithms with multiple proposals    Optimal scaling   Optimising the speed up process   Conclusion


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Introduction     MH algorithms with multiple proposals   Optimal scaling   Optimising the speed up process   Conclusion


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Introduction    MH algorithms with multiple proposals   Optimal scaling   Optimising the speed up process   Conclusion


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Introduction    MH algorithms with multiple proposals   Optimal scaling   Optimising the speed up process   Conclusion


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Introduction    MH algorithms with multiple proposals   Optimal scaling   Optimising the speed up process   Conclusion


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Introduction    MH algorithms with multiple proposals   Optimal scaling   Optimising the speed up process   Conclusion


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Introduction    MH algorithms with multiple proposals   Optimal scaling   Optimising the speed up process   Conclusion


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Introduction    MH algorithms with multiple proposals   Optimal scaling      Optimising the speed up process   Conclusion
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Introduction     MH algorithms with multiple proposals     Optimal scaling     Optimising the speed up process   Conclusio...
Introduction     MH algorithms with multiple proposals     Optimal scaling     Optimising the speed up process   Conclusio...
Introduction     MH algorithms with multiple proposals     Optimal scaling     Optimising the speed up process   Conclusio...
Introduction     MH algorithms with multiple proposals     Optimal scaling     Optimising the speed up process   Conclusio...
Introduction     MH algorithms with multiple proposals     Optimal scaling     Optimising the speed up process   Conclusio...
Introduction     MH algorithms with multiple proposals     Optimal scaling     Optimising the speed up process   Conclusio...
Introduction     MH algorithms with multiple proposals   Optimal scaling     Optimising the speed up process   Conclusion
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Introduction     MH algorithms with multiple proposals   Optimal scaling     Optimising the speed up process   Conclusion
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Introduction     MH algorithms with multiple proposals   Optimal scaling     Optimising the speed up process   Conclusion
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Introduction     MH algorithms with multiple proposals   Optimal scaling     Optimising the speed up process   Conclusion
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Introduction     MH algorithms with multiple proposals   Optimal scaling   Optimising the speed up process   Conclusion


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Introduction     MH algorithms with multiple proposals   Optimal scaling     Optimising the speed up process      Conclusi...
Introduction     MH algorithms with multiple proposals    Optimal scaling      Optimising the speed up process        Conc...
Introduction     MH algorithms with multiple proposals   Optimal scaling   Optimising the speed up process   Conclusion


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Introduction     MH algorithms with multiple proposals   Optimal scaling   Optimising the speed up process   Conclusion


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Introduction     MH algorithms with multiple proposals   Optimal scaling   Optimising the speed up process   Conclusion


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Introduction    MH algorithms with multiple proposals   Optimal scaling   Optimising the speed up process   Conclusion


M...
Introduction    MH algorithms with multiple proposals   Optimal scaling   Optimising the speed up process   Conclusion


M...
Introduction    MH algorithms with multiple proposals   Optimal scaling   Optimising the speed up process   Conclusion


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Introduction    MH algorithms with multiple proposals   Optimal scaling   Optimising the speed up process   Conclusion


M...
Introduction     MH algorithms with multiple proposals   Optimal scaling   Optimising the speed up process   Conclusion


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Introduction     MH algorithms with multiple proposals   Optimal scaling   Optimising the speed up process   Conclusion


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Introduction     MH algorithms with multiple proposals   Optimal scaling   Optimising the speed up process   Conclusion


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Introduction     MH algorithms with multiple proposals   Optimal scaling   Optimising the speed up process   Conclusion


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Introduction     MH algorithms with multiple proposals   Optimal scaling   Optimising the speed up process   Conclusion


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Introduction      MH algorithms with multiple proposals    Optimal scaling   Optimising the speed up process   Conclusion
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Introduction      MH algorithms with multiple proposals    Optimal scaling   Optimising the speed up process   Conclusion
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Introduction      MH algorithms with multiple proposals    Optimal scaling   Optimising the speed up process   Conclusion
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Introduction    MH algorithms with multiple proposals   Optimal scaling    Optimising the speed up process   Conclusion


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Introduction    MH algorithms with multiple proposals   Optimal scaling    Optimising the speed up process   Conclusion


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Introduction    MH algorithms with multiple proposals   Optimal scaling    Optimising the speed up process   Conclusion


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Introduction    MH algorithms with multiple proposals   Optimal scaling    Optimising the speed up process   Conclusion


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Introduction    MH algorithms with multiple proposals   Optimal scaling    Optimising the speed up process   Conclusion


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Introduction   MH algorithms with multiple proposals   Optimal scaling   Optimising the speed up process   Conclusion




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Introduction   MH algorithms with multiple proposals   Optimal scaling   Optimising the speed up process   Conclusion




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Introduction   MH algorithms with multiple proposals   Optimal scaling   Optimising the speed up process   Conclusion




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Introduction      MH algorithms with multiple proposals   Optimal scaling   Optimising the speed up process   Conclusion

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Introduction      MH algorithms with multiple proposals   Optimal scaling   Optimising the speed up process   Conclusion

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Introduction    MH algorithms with multiple proposals     Optimal scaling   Optimising the speed up process   Conclusion

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Introduction    MH algorithms with multiple proposals     Optimal scaling   Optimising the speed up process   Conclusion

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Introduction    MH algorithms with multiple proposals     Optimal scaling   Optimising the speed up process   Conclusion

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Introduction     MH algorithms with multiple proposals   Optimal scaling   Optimising the speed up process   Conclusion


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Introduction     MH algorithms with multiple proposals   Optimal scaling   Optimising the speed up process   Conclusion


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Introduction     MH algorithms with multiple proposals   Optimal scaling     Optimising the speed up process   Conclusion
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Introduction     MH algorithms with multiple proposals   Optimal scaling     Optimising the speed up process   Conclusion
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Introduction   MH algorithms with multiple proposals   Optimal scaling     Optimising the speed up process   Conclusion


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Introduction     MH algorithms with multiple proposals   Optimal scaling   Optimising the speed up process   Conclusion


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Introduction       MH algorithms with multiple proposals   Optimal scaling   Optimising the speed up process   Conclusion
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Introduction       MH algorithms with multiple proposals   Optimal scaling   Optimising the speed up process   Conclusion
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Introduction       MH algorithms with multiple proposals   Optimal scaling   Optimising the speed up process   Conclusion
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Introduction       MH algorithms with multiple proposals   Optimal scaling   Optimising the speed up process   Conclusion
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Introduction       MH algorithms with multiple proposals   Optimal scaling   Optimising the speed up process   Conclusion
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Optimal scaling in MTM algorithms

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Optimal scaling in MTM algorithms

  1. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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

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