1. A vanilla Rao–Blackwellisation of
Metropolis–Hastings algorithms
Randal DOUC and Christian ROBERT
Telecom SudParis, France
randal.douc@it-sudparis.eu
April 2009
1 / 24
2. Main themes
1 Rao–Blackwellisation on MCMC.
2 Can be performed in any Hastings Metropolis algorithm.
3 Asymptotically more efﬁcient to usual MCMC with a
controlled amount of calculations.
2 / 24
3. Main themes
1 Rao–Blackwellisation on MCMC.
2 Can be performed in any Hastings Metropolis algorithm.
3 Asymptotically more efﬁcient to usual MCMC with a
controlled amount of calculations.
2 / 24
4. Main themes
1 Rao–Blackwellisation on MCMC.
2 Can be performed in any Hastings Metropolis algorithm.
3 Asymptotically more efﬁcient to usual MCMC with a
controlled amount of calculations.
2 / 24
5. Main themes
1 Rao–Blackwellisation on MCMC.
2 Can be performed in any Hastings Metropolis algorithm.
3 Asymptotically more efﬁcient to usual MCMC with a
controlled amount of calculations.
2 / 24
6. Introduction Some properties of the HM algorithm Rao–Blackwellisation Illustrations Conclusion
Outline
1 Introduction
2 Some properties of the HM algorithm
3 Rao–Blackwellisation
Variance reduction
Asymptotic results
4 Illustrations
5 Conclusion
3 / 24
7. Introduction Some properties of the HM algorithm Rao–Blackwellisation Illustrations Conclusion
Outline
1 Introduction
2 Some properties of the HM algorithm
3 Rao–Blackwellisation
Variance reduction
Asymptotic results
4 Illustrations
5 Conclusion
3 / 24
8. Introduction Some properties of the HM algorithm Rao–Blackwellisation Illustrations Conclusion
Outline
1 Introduction
2 Some properties of the HM algorithm
3 Rao–Blackwellisation
Variance reduction
Asymptotic results
4 Illustrations
5 Conclusion
3 / 24
9. Introduction Some properties of the HM algorithm Rao–Blackwellisation Illustrations Conclusion
Outline
1 Introduction
2 Some properties of the HM algorithm
3 Rao–Blackwellisation
Variance reduction
Asymptotic results
4 Illustrations
5 Conclusion
3 / 24
10. Introduction Some properties of the HM algorithm Rao–Blackwellisation Illustrations Conclusion
Outline
1 Introduction
2 Some properties of the HM algorithm
3 Rao–Blackwellisation
Variance reduction
Asymptotic results
4 Illustrations
5 Conclusion
3 / 24
11. Introduction Some properties of the HM algorithm Rao–Blackwellisation Illustrations Conclusion
Outline
1 Introduction
2 Some properties of the HM algorithm
3 Rao–Blackwellisation
Variance reduction
Asymptotic results
4 Illustrations
5 Conclusion
4 / 24
12. Introduction Some properties of the HM algorithm Rao–Blackwellisation Illustrations Conclusion
Metropolis Hastings algorithm
1 We wish to approximate
h(x )π(x )dx
I= = h(x )¯ (x )dx
π
π(x )dx
2 x → π(x ) is known but not π(x )dx .
1 n
3 Approximate I with δ = n t=1 h(x (t) ) where (x (t) ) is a Markov
chain with limiting distribution π .
¯
4 Convergence obtained from Law of Large Numbers or CLT for
Markov chains.
5 / 24
13. Introduction Some properties of the HM algorithm Rao–Blackwellisation Illustrations Conclusion
Metropolis Hastings algorithm
1 We wish to approximate
h(x )π(x )dx
I= = h(x )¯ (x )dx
π
π(x )dx
2 x → π(x ) is known but not π(x )dx .
1 n
3 Approximate I with δ = n t=1 h(x (t) ) where (x (t) ) is a Markov
chain with limiting distribution π .
¯
4 Convergence obtained from Law of Large Numbers or CLT for
Markov chains.
5 / 24
14. Introduction Some properties of the HM algorithm Rao–Blackwellisation Illustrations Conclusion
Metropolis Hastings algorithm
1 We wish to approximate
h(x )π(x )dx
I= = h(x )¯ (x )dx
π
π(x )dx
2 x → π(x ) is known but not π(x )dx .
1 n
3 Approximate I with δ = n t=1 h(x (t) ) where (x (t) ) is a Markov
chain with limiting distribution π .
¯
4 Convergence obtained from Law of Large Numbers or CLT for
Markov chains.
5 / 24
15. Introduction Some properties of the HM algorithm Rao–Blackwellisation Illustrations Conclusion
Metropolis Hastings algorithm
1 We wish to approximate
h(x )π(x )dx
I= = h(x )¯ (x )dx
π
π(x )dx
2 x → π(x ) is known but not π(x )dx .
1 n
3 Approximate I with δ = n t=1 h(x (t) ) where (x (t) ) is a Markov
chain with limiting distribution π .
¯
4 Convergence obtained from Law of Large Numbers or CLT for
Markov chains.
5 / 24
16. Introduction Some properties of the HM algorithm Rao–Blackwellisation Illustrations Conclusion
Metropolis Hasting Algorithm
Suppose that x (t) is drawn.
1 Simulate yt ∼ q(·|x (t) ).
2 Set x (t+1) = yt with probability
π(yt ) q(x (t) |yt )
α(x (t) , yt ) = min 1,
π(x (t) ) q(yt |x (t) )
Otherwise, set x (t+1) = x (t) .
3 α is such that the detailed balance equation is satisﬁed: ⊲ π is
¯
the stationary distribution of (x (t) ).
◮ The accepted candidates are simulated with the rejection
algorithm.
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17. Introduction Some properties of the HM algorithm Rao–Blackwellisation Illustrations Conclusion
Metropolis Hasting Algorithm
Suppose that x (t) is drawn.
1 Simulate yt ∼ q(·|x (t) ).
2 Set x (t+1) = yt with probability
π(yt ) q(x (t) |yt )
α(x (t) , yt ) = min 1,
π(x (t) ) q(yt |x (t) )
Otherwise, set x (t+1) = x (t) .
3 α is such that the detailed balance equation is satisﬁed: ⊲ π is
¯
the stationary distribution of (x (t) ).
◮ The accepted candidates are simulated with the rejection
algorithm.
6 / 24
18. Introduction Some properties of the HM algorithm Rao–Blackwellisation Illustrations Conclusion
Metropolis Hasting Algorithm
Suppose that x (t) is drawn.
1 Simulate yt ∼ q(·|x (t) ).
2 Set x (t+1) = yt with probability
π(yt ) q(x (t) |yt )
α(x (t) , yt ) = min 1,
π(x (t) ) q(yt |x (t) )
Otherwise, set x (t+1) = x (t) .
3 α is such that the detailed balance equation is satisﬁed:
π(x )q(y |x )α(x , y ) = π(y )q(x |y )α(y , x ).
⊲ π is the stationary distribution of (x (t) ).
¯
◮ The accepted candidates are simulated with the rejection
algorithm.
6 / 24
19. Introduction Some properties of the HM algorithm Rao–Blackwellisation Illustrations Conclusion
Metropolis Hasting Algorithm
Suppose that x (t) is drawn.
1 Simulate yt ∼ q(·|x (t) ).
2 Set x (t+1) = yt with probability
π(yt ) q(x (t) |yt )
α(x (t) , yt ) = min 1,
π(x (t) ) q(yt |x (t) )
Otherwise, set x (t+1) = x (t) .
3 α is such that the detailed balance equation is satisﬁed:
π(x )q(y |x )α(x , y ) = π(y )q(x |y )α(y , x ).
⊲ π is the stationary distribution of (x (t) ).
¯
◮ The accepted candidates are simulated with the rejection
algorithm.
6 / 24
20. Introduction Some properties of the HM algorithm Rao–Blackwellisation Illustrations Conclusion
Outline
1 Introduction
2 Some properties of the HM algorithm
3 Rao–Blackwellisation
Variance reduction
Asymptotic results
4 Illustrations
5 Conclusion
7 / 24
21. Introduction Some properties of the HM algorithm Rao–Blackwellisation Illustrations Conclusion
1 Alternative representation of the estimator δ is
n MN
1 (t) 1
δ= h(x ) = ni h(zi ) ,
n N
t=1 i=1
where
zi ’s are the accepted yj ’s,
MN is the number of accepted yj ’s till time N,
ni is the number of times zi appears in the sequence (x (t) )t .
8 / 24
22. Introduction Some properties of the HM algorithm Rao–Blackwellisation Illustrations Conclusion
α(zi , ·) q(·|zi ) q(·|zi )
˜
q (·|zi ) = ≤ ,
p(zi ) p(zi )
where p(zi ) = ˜
α(zi , y ) q(y |zi )dy . To simulate according to q (·|zi ):
1 Propose a candidate y ∼ q(·|zi )
2 Accept with probability
q(y |zi )
˜
q (y |zi )/ = α(zi , y )
p(zi )
Otherwise, reject it and starts again.
3 ◮ this is the transition of the HM algorithm.
˜
The transition kernel q admits π as a stationary distribution:
˜
˜ ˜
π (x )q (y |x ) =
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23. Introduction Some properties of the HM algorithm Rao–Blackwellisation Illustrations Conclusion
α(zi , ·) q(·|zi ) q(·|zi )
˜
q (·|zi ) = ≤ ,
p(zi ) p(zi )
where p(zi ) = ˜
α(zi , y ) q(y |zi )dy . To simulate according to q (·|zi ):
1 Propose a candidate y ∼ q(·|zi )
2 Accept with probability
q(y |zi )
˜
q (y |zi )/ = α(zi , y )
p(zi )
Otherwise, reject it and starts again.
3 ◮ this is the transition of the HM algorithm.
˜
The transition kernel q admits π as a stationary distribution:
˜
˜ ˜
π (x )q (y |x ) =
9 / 24
24. Introduction Some properties of the HM algorithm Rao–Blackwellisation Illustrations Conclusion
α(zi , ·) q(·|zi ) q(·|zi )
˜
q (·|zi ) = ≤ ,
p(zi ) p(zi )
where p(zi ) = ˜
α(zi , y ) q(y |zi )dy . To simulate according to q (·|zi ):
1 Propose a candidate y ∼ q(·|zi )
2 Accept with probability
q(y |zi )
˜
q (y |zi )/ = α(zi , y )
p(zi )
Otherwise, reject it and starts again.
3 ◮ this is the transition of the HM algorithm.
˜
The transition kernel q admits π as a stationary distribution:
˜
˜ ˜
π (x )q (y |x ) =
9 / 24
25. Introduction Some properties of the HM algorithm Rao–Blackwellisation Illustrations Conclusion
α(zi , ·) q(·|zi ) q(·|zi )
˜
q (·|zi ) = ≤ ,
p(zi ) p(zi )
where p(zi ) = ˜
α(zi , y ) q(y |zi )dy . To simulate according to q (·|zi ):
1 Propose a candidate y ∼ q(·|zi )
2 Accept with probability
q(y |zi )
˜
q (y |zi )/ = α(zi , y )
p(zi )
Otherwise, reject it and starts again.
3 ◮ this is the transition of the HM algorithm.
˜
The transition kernel q admits π as a stationary distribution:
˜
π(x )p(x ) α(x , y )q(y |x )
˜ ˜
π (x )q (y |x ) =
π(u)p(u)du p(x )
π (x)
˜ ˜
q (y |x)
9 / 24
26. Introduction Some properties of the HM algorithm Rao–Blackwellisation Illustrations Conclusion
α(zi , ·) q(·|zi ) q(·|zi )
˜
q (·|zi ) = ≤ ,
p(zi ) p(zi )
where p(zi ) = ˜
α(zi , y ) q(y |zi )dy . To simulate according to q (·|zi ):
1 Propose a candidate y ∼ q(·|zi )
2 Accept with probability
q(y |zi )
˜
q (y |zi )/ = α(zi , y )
p(zi )
Otherwise, reject it and starts again.
3 ◮ this is the transition of the HM algorithm.
˜
The transition kernel q admits π as a stationary distribution:
˜
π(x )α(x , y )q(y |x )
˜ ˜
π (x )q (y |x ) =
π(u)p(u)du
9 / 24
27. Introduction Some properties of the HM algorithm Rao–Blackwellisation Illustrations Conclusion
α(zi , ·) q(·|zi ) q(·|zi )
˜
q (·|zi ) = ≤ ,
p(zi ) p(zi )
where p(zi ) = ˜
α(zi , y ) q(y |zi )dy . To simulate according to q (·|zi ):
1 Propose a candidate y ∼ q(·|zi )
2 Accept with probability
q(y |zi )
˜
q (y |zi )/ = α(zi , y )
p(zi )
Otherwise, reject it and starts again.
3 ◮ this is the transition of the HM algorithm.
˜
The transition kernel q admits π as a stationary distribution:
˜
π(y )α(y , x )q(x |y )
˜ ˜
π (x )q (y |x ) =
π(u)p(u)du
9 / 24
28. Introduction Some properties of the HM algorithm Rao–Blackwellisation Illustrations Conclusion
α(zi , ·) q(·|zi ) q(·|zi )
˜
q (·|zi ) = ≤ ,
p(zi ) p(zi )
where p(zi ) = ˜
α(zi , y ) q(y |zi )dy . To simulate according to q (·|zi ):
1 Propose a candidate y ∼ q(·|zi )
2 Accept with probability
q(y |zi )
˜
q (y |zi )/ = α(zi , y )
p(zi )
Otherwise, reject it and starts again.
3 ◮ this is the transition of the HM algorithm.
˜
The transition kernel q admits π as a stationary distribution:
˜
˜ ˜ ˜ ˜
π (x )q (y |x ) = π (y )q (x |y ) ,
9 / 24
29. Introduction Some properties of the HM algorithm Rao–Blackwellisation Illustrations Conclusion
Lemme
The sequence (zi , ni ) satisﬁes
1 (zi , ni )i is a Markov chain;
2 zi+1 and ni are independent given zi ;
3 ni is distributed as a geometric random variable with probability
parameter
p(zi ) := α(zi , y ) q(y |zi ) dy ; (1)
4 (zi )i is a Markov chain with transition kernel
˜ ˜
Q(z, dy ) = q (y |z)dy and stationary distribution π such that
˜
˜
q (·|z) ∝ α(z, ·) q(·|z) and π (·) ∝ π(·)p(·) .
˜
10 / 24
30. Introduction Some properties of the HM algorithm Rao–Blackwellisation Illustrations Conclusion
Lemme
The sequence (zi , ni ) satisﬁes
1 (zi , ni )i is a Markov chain;
2 zi+1 and ni are independent given zi ;
3 ni is distributed as a geometric random variable with probability
parameter
p(zi ) := α(zi , y ) q(y |zi ) dy ; (1)
4 (zi )i is a Markov chain with transition kernel
˜ ˜
Q(z, dy ) = q (y |z)dy and stationary distribution π such that
˜
˜
q (·|z) ∝ α(z, ·) q(·|z) and π (·) ∝ π(·)p(·) .
˜
10 / 24
31. Introduction Some properties of the HM algorithm Rao–Blackwellisation Illustrations Conclusion
Lemme
The sequence (zi , ni ) satisﬁes
1 (zi , ni )i is a Markov chain;
2 zi+1 and ni are independent given zi ;
3 ni is distributed as a geometric random variable with probability
parameter
p(zi ) := α(zi , y ) q(y |zi ) dy ; (1)
4 (zi )i is a Markov chain with transition kernel
˜ ˜
Q(z, dy ) = q (y |z)dy and stationary distribution π such that
˜
˜
q (·|z) ∝ α(z, ·) q(·|z) and π (·) ∝ π(·)p(·) .
˜
10 / 24
32. Introduction Some properties of the HM algorithm Rao–Blackwellisation Illustrations Conclusion
Lemme
The sequence (zi , ni ) satisﬁes
1 (zi , ni )i is a Markov chain;
2 zi+1 and ni are independent given zi ;
3 ni is distributed as a geometric random variable with probability
parameter
p(zi ) := α(zi , y ) q(y |zi ) dy ; (1)
4 (zi )i is a Markov chain with transition kernel
˜ ˜
Q(z, dy ) = q (y |z)dy and stationary distribution π such that
˜
˜
q (·|z) ∝ α(z, ·) q(·|z) and π (·) ∝ π(·)p(·) .
˜
10 / 24
33. Introduction Some properties of the HM algorithm Rao–Blackwellisation Illustrations Conclusion
zi−1
11 / 24
34. Introduction Some properties of the HM algorithm Rao–Blackwellisation Illustrations Conclusion
indep
zi−1 zi
indep
ni−1
11 / 24
35. Introduction Some properties of the HM algorithm Rao–Blackwellisation Illustrations Conclusion
indep indep
zi−1 zi zi+1
indep indep
ni−1 ni
11 / 24
36. Introduction Some properties of the HM algorithm Rao–Blackwellisation Illustrations Conclusion
indep indep
zi−1 zi zi+1
indep indep
ni−1 ni
n MN
1 1
δ= h(x (t) ) = ni h(zi ) .
n N
t=1 i=1
11 / 24
37. Introduction Some properties of the HM algorithm Rao–Blackwellisation Illustrations Conclusion
indep indep
zi−1 zi zi+1
indep indep
ni−1 ni
n MN
1 1
δ= h(x (t) ) = ni h(zi ) .
n N
t=1 i=1
11 / 24
38. Introduction Some properties of the HM algorithm Rao–Blackwellisation Illustrations Conclusion
Outline
1 Introduction
2 Some properties of the HM algorithm
3 Rao–Blackwellisation
Variance reduction
Asymptotic results
4 Illustrations
5 Conclusion
12 / 24
39. Introduction Some properties of the HM algorithm Rao–Blackwellisation Illustrations Conclusion
1 A natural idea:
MN
1 h(zi )
δ∗ = ,
N p(zi )
i=1
13 / 24
41. Introduction Some properties of the HM algorithm Rao–Blackwellisation Illustrations Conclusion
1 A natural idea:
MN h(zi ) MN π(zi )
i=1 i=1 h(zi )
∗ p(zi ) π (zi )
˜
δ ≃ = .
MN 1 MN π(zi )
i=1 i=1
p(zi ) π (zi )
˜
2 But p not available in closed form.
13 / 24
42. Introduction Some properties of the HM algorithm Rao–Blackwellisation Illustrations Conclusion
1 A natural idea:
MN h(zi ) MN π(zi )
i=1 i=1 h(zi )
∗ p(zi ) π (zi )
˜
δ ≃ = .
MN 1 MN π(zi )
i=1 i=1
p(zi ) π (zi )
˜
2 But p not available in closed form.
3 The geometric ni is the obvious solution that is used in the
original Metropolis–Hastings estimate.
13 / 24
43. Introduction Some properties of the HM algorithm Rao–Blackwellisation Illustrations Conclusion
1 A natural idea:
MN h(zi ) MN π(zi )
i=1 i=1 h(zi )
p(zi ) π (zi )
˜
δ∗ ≃ = .
MN 1 MN π(zi )
i=1 i=1
p(zi ) π (zi )
˜
2 But p not available in closed form.
3 The geometric ni is the obvious solution that is used in the
original Metropolis–Hastings estimate.
∞
ni = 1 + I {uℓ ≥ α(zi , yℓ )} ,
j=1 ℓ≤j
13 / 24
44. Introduction Some properties of the HM algorithm Rao–Blackwellisation Illustrations Conclusion
∞
ni = 1 + I {uℓ ≥ α(zi , yℓ )} ,
j=1 ℓ≤j
Lemma
If (yj )j is an iid sequence with distribution q(y |zi ), the quantity
∞
ˆ
ξi = 1 + {1 − α(zi , yℓ )}
j=1 ℓ≤j
is an unbiased estimator of 1/p(zi ) which variance, conditional on zi ,
is lower than the conditional variance of ni , {1 − p(zi )}/p2 (zi ).
13 / 24
45. Introduction Some properties of the HM algorithm Rao–Blackwellisation Illustrations Conclusion
∞
ˆ
ξi = 1 + {1 − α(zi , yℓ )}
j=1 ℓ≤j
1 Inﬁnite sum but sometimes ﬁnite:
π(yt ) q(x (t) |yt )
α(x (t) , yt ) = min 1,
π(x (t) ) q(yt |x (t) )
For example: take a symetric random walk as a proposal.
2 What if we wish to be sure that the sum is ﬁnite?
14 / 24
46. Introduction Some properties of the HM algorithm Rao–Blackwellisation Illustrations Conclusion
Variance reduction
Proposition
If (yj )j is an iid sequence with distribution q(y |zi ) and (uj )j is an iid
uniform sequence, for any k ≥ 0, the quantity
∞
ˆ
ξik = 1 + {1 − α(zi , yj )} I {uℓ ≥ α(zi , yℓ )} (2)
j=1 1≤ℓ≤k ∧j k +1≤ℓ≤j
is an unbiased estimator of 1/p(zi ) with an almost sure ﬁnite number
of terms.
15 / 24
47. Introduction Some properties of the HM algorithm Rao–Blackwellisation Illustrations Conclusion
Variance reduction
Proposition
If (yj )j is an iid sequence with distribution q(y |zi ) and (uj )j is an iid
uniform sequence, for any k ≥ 0, the quantity
∞
ˆ
ξik = 1 + {1 − α(zi , yj )} I {uℓ ≥ α(zi , yℓ )} (2)
j=1 1≤ℓ≤k ∧j k +1≤ℓ≤j
is an unbiased estimator of 1/p(zi ) with an almost sure ﬁnite number
of terms. Moreover, for k ≥ 1,
ˆ 1 − p(zi ) 1 − (1 − 2p(zi ) + r (zi ))k 2 − p(zi )
V ξik zi = − (p(zi )−r (zi )) ,
p2 (zi ) 2p(zi ) − r (zi ) p2 (zi )
where p(zi ) := α(zi , y ) q(y |zi ) dy . and r (zi ) := α2 (zi , y ) q(y |zi ) dy .
15 / 24
48. Introduction Some properties of the HM algorithm Rao–Blackwellisation Illustrations Conclusion
Variance reduction
Proposition
If (yj )j is an iid sequence with distribution q(y |zi ) and (uj )j is an iid
uniform sequence, for any k ≥ 0, the quantity
∞
ˆ
ξik = 1 + {1 − α(zi , yj )} I {uℓ ≥ α(zi , yℓ )} (2)
j=1 1≤ℓ≤k ∧j k +1≤ℓ≤j
is an unbiased estimator of 1/p(zi ) with an almost sure ﬁnite number
of terms. Therefore, we have
ˆ ˆ ˆ
V ξi zi ≤ V ξik zi ≤ V ξi0 zi = V [ni | zi ] .
15 / 24
49. Introduction Some properties of the HM algorithm Rao–Blackwellisation Illustrations Conclusion
Variance reduction
zi−1
∞
ˆ
ξik = 1 + {1 − α(zi , yj )} I {uℓ ≥ α(zi , yℓ )} (3)
j=1 1≤ℓ≤k ∧j k +1≤ℓ≤j
16 / 24
50. Introduction Some properties of the HM algorithm Rao–Blackwellisation Illustrations Conclusion
Variance reduction
not indep
zi−1 zi
not indep
ˆk
ξi−1
∞
ˆ
ξik = 1 + {1 − α(zi , yj )} I {uℓ ≥ α(zi , yℓ )} (3)
j=1 1≤ℓ≤k ∧j k +1≤ℓ≤j
16 / 24
51. Introduction Some properties of the HM algorithm Rao–Blackwellisation Illustrations Conclusion
Variance reduction
not indep not indep
zi−1 zi zi+1
not indep not indep
ˆk
ξi−1 ˆ
ξik
∞
ˆ
ξik = 1 + {1 − α(zi , yj )} I {uℓ ≥ α(zi , yℓ )} (3)
j=1 1≤ℓ≤k ∧j k +1≤ℓ≤j
16 / 24
52. Introduction Some properties of the HM algorithm Rao–Blackwellisation Illustrations Conclusion
Variance reduction
not indep not indep
zi−1 zi zi+1
not indep not indep
ˆk
ξi−1 ˆ
ξik
M ˆk
k i=1 ξi h(zi )
δM = M ˆk
.
i=1 ξi
16 / 24
53. Introduction Some properties of the HM algorithm Rao–Blackwellisation Illustrations Conclusion
Variance reduction
not indep not indep
zi−1 zi zi+1
not indep not indep
ˆk
ξi−1 ˆ
ξik
M ˆk
k i=1 ξi h(zi )
δM = M ˆk
.
i=1 ξi
16 / 24
54. Introduction Some properties of the HM algorithm Rao–Blackwellisation Illustrations Conclusion
Asymptotic results
Let
M ˆk
k i=1 ξi h(zi )
δM = M ˆk
.
i=1 ξi
For any positive function ϕ, we denote Cϕ = {h; |h/ϕ|∞ < ∞}.
17 / 24
55. Introduction Some properties of the HM algorithm Rao–Blackwellisation Illustrations Conclusion
Asymptotic results
Let
M ˆk
k i=1 ξi h(zi )
δM = M ˆk
.
i=1 ξi
For any positive function ϕ, we denote Cϕ = {h; |h/ϕ|∞ < ∞}.
Assume that there exist a positive function ϕ ≥ 1 such that
M
i=1 h(zi )/p(zi ) P
∀h ∈ Cϕ , M
−→ π(h) (3)
i=1 1/p(zi )
Theorem
Under the assumption that π(p) > 0, the following convergence
property holds:
i) If h is in Cϕ , then
k P
δM −→M→∞ π(h) (◮C ONSISTENCY)
17 / 24
56. Introduction Some properties of the HM algorithm Rao–Blackwellisation Illustrations Conclusion
Asymptotic results
Let
M ˆk
k i=1 ξi h(zi )
δM = M ˆk
.
i=1 ξi
For any positive function ϕ, we denote Cϕ = {h; |h/ϕ|∞ < ∞}.
Assume that there exist a positive function ψ such that
√ M
i=1 h(zi )/p(zi ) L
∀h ∈ Cψ , M M
− π(h) −→ N (0, Γ(h))
i=1 1/p(zi )
Theorem
Under the assumption that π(p) > 0, the following convergence
property holds:
ii) If, in addition, h2 /p ∈ Cϕ and h ∈ Cψ , then
√ k L
M(δM − π(h)) −→M→∞ N (0, Vk [h − π(h)]) , (◮C LT)
where Vk (h) := π(p) ˆ
π(dz)V ξik z h2 (z)p(z) + Γ(h) .
17 / 24
57. Introduction Some properties of the HM algorithm Rao–Blackwellisation Illustrations Conclusion
Asymptotic results
We will need some additional assumptions. Assume a maximal
inequality for the Markov chain (zi )i : there exists a measurable
function ζ such that for any starting point x ,
i
NCh (x )
∀h ∈ Cζ , Px sup [h(zi ) − π (h)] > ǫ ≤
˜
0≤i≤N ǫ2
j=0
Theorem
Assume that h is such that h/p ∈ Cζ and {Ch/p , h2 /p2 } ⊂ Cφ . Assume
moreover that
√ 0 L
M δM − π(h) −→ N (0, V0 [h − π(h)]) .
Then, for any starting point x ,
N
t=1 h(x (t) ) L
MN − π(h) −→N→∞ N (0, V0 [h − π(h)]) ,
N
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58. Introduction Some properties of the HM algorithm Rao–Blackwellisation Illustrations Conclusion
Asymptotic results
We will need some additional assumptions. Assume a maximal
inequality for the Markov chain (zi )i : there exists a measurable
function ζ such that for any starting point x ,
i
NCh (x )
∀h ∈ Cζ , Px sup [h(zi ) − π (h)] > ǫ ≤
˜
0≤i≤N ǫ2
j=0
Moreover, assume that ∃φ ≥ 1 such that for any starting point x ,
∀h ∈ Cφ , ˜ P
Q n (x , h) −→ π (h) = π(ph)/π(p) ,
˜
Theorem
Assume that h is such that h/p ∈ Cζ and {Ch/p , h2 /p2 } ⊂ Cφ . Assume
moreover that
√ 0 L
M δM − π(h) −→ N (0, V0 [h − π(h)]) .
Then, for any starting point x ,
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59. Introduction Some properties of the HM algorithm Rao–Blackwellisation Illustrations Conclusion
Asymptotic results
We will need some additional assumptions. Assume a maximal
inequality for the Markov chain (zi )i : there exists a measurable
function ζ such that for any starting point x ,
i
NCh (x )
∀h ∈ Cζ , Px sup [h(zi ) − π (h)] > ǫ ≤
˜
0≤i≤N ǫ2
j=0
Moreover, assume that ∃φ ≥ 1 such that for any starting point x ,
∀h ∈ Cφ , ˜ P
Q n (x , h) −→ π (h) = π(ph)/π(p) ,
˜
Theorem
Assume that h is such that h/p ∈ Cζ and {Ch/p , h2 /p2 } ⊂ Cφ . Assume
moreover that
√ 0 L
M δM − π(h) −→ N (0, V0 [h − π(h)]) .
Then, for any starting point x ,
18 / 24
60. Introduction Some properties of the HM algorithm Rao–Blackwellisation Illustrations Conclusion
Asymptotic results
i
NCh (x )
∀h ∈ Cζ , Px sup [h(zi ) − π (h)] > ǫ ≤
˜
0≤i≤N j=0 ǫ2
∀h ∈ Cφ , ˜ P
Q n (x , h) −→ π (h) = π(ph)/π(p) ,
˜
Theorem
Assume that h is such that h/p ∈ Cζ and {Ch/p , h2 /p2 } ⊂ Cφ . Assume
moreover that
√ 0 L
M δM − π(h) −→ N (0, V0 [h − π(h)]) .
Then, for any starting point x ,
N
t=1 h(x (t) ) L
MN − π(h) −→N→∞ N (0, V0 [h − π(h)]) ,
N
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61. Introduction Some properties of the HM algorithm Rao–Blackwellisation Illustrations Conclusion
Asymptotic results
Theorem
Assume that h is such that h/p ∈ Cζ and {Ch/p , h2 /p2 } ⊂ Cφ . Assume
moreover that
√ 0 L
M δM − π(h) −→ N (0, V0 [h − π(h)]) .
Then, for any starting point x ,
N
t=1 h(x (t) ) L
MN − π(h) −→N→∞ N (0, V0 [h − π(h)]) ,
N
where MN is deﬁned by
MN MN +1
ˆ
ξi0 ≤ N < ˆ
ξi0 . (3)
i=1 i=1
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62. Introduction Some properties of the HM algorithm Rao–Blackwellisation Illustrations Conclusion
Asymptotic results
Theorem
Assume that h is such that h/p ∈ Cζ and {Ch/p , h2 /p2 } ⊂ Cφ . Assume
moreover that
√ 0 L
M δM − π(h) −→ N (0, V0 [h − π(h)]) .
Then, for any starting point x ,
N
t=1 h(x (t) ) L
MN − π(h) −→N→∞ N (0, V0 [h − π(h)]) ,
N
where MN is deﬁned by
MN MN +1
ˆ
ξi0 ≤ N < ˆ
ξi0 . (3)
i=1 i=1
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63. Introduction Some properties of the HM algorithm Rao–Blackwellisation Illustrations Conclusion
Outline
1 Introduction
2 Some properties of the HM algorithm
3 Rao–Blackwellisation
Variance reduction
Asymptotic results
4 Illustrations
5 Conclusion
19 / 24
64. Introduction Some properties of the HM algorithm Rao–Blackwellisation Illustrations Conclusion
Figure: Overlay of the variations of 250 iid realisations of the
estimates δ (gold) and δ ∞ (grey) of E[X ] = 0 for 1000 iterations, along
with the 90% interquantile range for the estimates δ (brown) and δ ∞
(pink), in the setting of a random walk Gaussian proposal with scale
τ = 10.
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65. Introduction Some properties of the HM algorithm Rao–Blackwellisation Illustrations Conclusion
Figure: Overlay of the variations of 500 iid realisations of the
estimates δ (deep grey), δ ∞ (medium grey) and of the importance
sampling version (light grey) of E[X ] = 10 when X ∼ Exp(.1) for 100
iterations, along with the 90% interquantile ranges (same colour
code), in the setting of an independent exponential proposal with
scale µ = 0.02.
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66. Introduction Some properties of the HM algorithm Rao–Blackwellisation Illustrations Conclusion
I|x−y |=1 if x > 0 ,
π(x ) = β(1 − β)x and 2q(y |x ) =
I|y |≤1 if x = 0 .
For this problem,
p(x ) = 1 − β/2 and r (x ) = 1 − β + β 2 /2 .
We can therefore compute the gain in variance
p(x ) − r (x ) 2 − p(x ) β(1 − β)(2 + β)
=2
2p(x ) − r (x ) p2 (x ) (2 − β 2 )(2 − β)2
which is optimal for β = 0.174, leading to a gain of 0.578 while the
relative gain in variance is
p(x ) − r (x ) 2 − p(x ) (1 − β)(2 + β)
=
2p(x ) − r (x ) 1 − p(x ) (2 − β 2 )
which is decreasing in β.
22 / 24
67. Introduction Some properties of the HM algorithm Rao–Blackwellisation Illustrations Conclusion
Outline
1 Introduction
2 Some properties of the HM algorithm
3 Rao–Blackwellisation
Variance reduction
Asymptotic results
4 Illustrations
5 Conclusion
23 / 24
68. Introduction Some properties of the HM algorithm Rao–Blackwellisation Illustrations Conclusion
a) Rao Blackwellisation of any HM algorithm with a controled
amount of additional calculation.
b) Link with the importance sampling of Markov chains.
c) Analysis with asymptotic results on triangular arrays.
24 / 24
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