rss discussion of girolami and calderhead, october 13, 2010
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About discretising Hamiltonians
Christian P. Robert
Universite Paris-Dauphine and CREST
http://xianblog.wordpress.com
Royal Statistical Society, October 13, 2010
Christian P. Robert About discretising Hamiltonians
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Hamiltonian dynamics
Dynamic on the level sets of
H (θ,p) = −L(θ) +1
2log{(2π)D|G(θ)|} +
1
2pTG(θ)−1p ,
where p is an auxiliary vector of dimension D, is associated withHamilton’s pde’s
p =∂H
∂p(θ,p) , θ =
∂H
∂θ(θ,p)
which preserve the potential H (θ,p) and hence the targetdistribution at all times t
Christian P. Robert About discretising Hamiltonians
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Discretised Hamiltonian
Girolami and Calderhead reproduce Hamiltonian equations withinthe simulation domain by discretisation via the generalised leapfrog(!) generator,
[Subliminal French bashing?!]
Christian P. Robert About discretising Hamiltonians
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Discretised Hamiltonian
Girolami and Calderhead reproduce Hamiltonian equations withinthe simulation domain by discretisation via the generalised leapfrog(!) generator,but...
Christian P. Robert About discretising Hamiltonians
![Page 5: RSS discussion of Girolami and Calderhead, October 13, 2010](https://reader033.vdocuments.mx/reader033/viewer/2022051817/5482f18c5806b5f7048b472f/html5/thumbnails/5.jpg)
Discretised Hamiltonian
Girolami and Calderhead reproduce Hamiltonian equations withinthe simulation domain by discretisation via the generalised leapfrog(!) generator,but...invariance and stability properties of the [background] continuoustime process the method do not carry to the discretised version ofthe process [e.g., Langevin]
Christian P. Robert About discretising Hamiltonians
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Discretised Hamiltonian (2)
Is it useful to so painstakingly reproduce the continuousbehaviour?
Approximations (see R&R’s Langevin) can be corrected by aMetropolis-Hastings step, so why bother with a second levelof approximation?
Discretisation induces a calibration problem: how long is longenough?
Convergence issues (for the MCMC algorithm) should not beimpacted by inexact renderings of the continuous time processin discrete time: loss of efficiency?
Christian P. Robert About discretising Hamiltonians
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An illustration
Comparison of the fits of discretised Langevin diffusion sequencesto the target f(x) ∝ exp(−x4) when using a discretisation stepσ2 = .1 and σ2 = .0001, after the same number T = 107 of steps.
Den
sity
−1.5 −1.0 −0.5 0.0 0.5 1.0 1.5
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Christian P. Robert About discretising Hamiltonians
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An illustration
Comparison of the fits of discretised Langevin diffusion sequencesto the target f(x) ∝ exp(−x4) when using a discretisation stepσ2 = .1 and σ2 = .0001, after the same number T = 107 of steps.
Den
sity
−1.5 −1.0 −0.5 0.0 0.5 1.0 1.5
0.0
0.2
0.4
0.6
0.8
Christian P. Robert About discretising Hamiltonians
![Page 9: RSS discussion of Girolami and Calderhead, October 13, 2010](https://reader033.vdocuments.mx/reader033/viewer/2022051817/5482f18c5806b5f7048b472f/html5/thumbnails/9.jpg)
An illustration
Comparison of the fits of discretised Langevin diffusion sequencesto the target f(x) ∝ exp(−x4) when using a discretisation stepσ2 = .1 and σ2 = .0001, after the same number T = 107 of steps.
−2 −1 0 1 2
0e+
002e
+04
4e+
046e
+04
8e+
041e
+05
time
Christian P. Robert About discretising Hamiltonians
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Back on Langevin
For the Langevin diffusion, the corresponding Langevin(discretised) algorithm could as well use another scale η for thegradient, rather than the one τ used for the noise
Christian P. Robert About discretising Hamiltonians
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Back on Langevin
For the Langevin diffusion, the corresponding Langevin(discretised) algorithm could as well use another scale η for thegradient, rather than the one τ used for the noise
y = xt + η∇π(x) + τǫt
rather than a strict Euler discretisation
y = xt + τ2∇π(x)/2 + τǫt
Christian P. Robert About discretising Hamiltonians
![Page 12: RSS discussion of Girolami and Calderhead, October 13, 2010](https://reader033.vdocuments.mx/reader033/viewer/2022051817/5482f18c5806b5f7048b472f/html5/thumbnails/12.jpg)
Back on Langevin
For the Langevin diffusion, the corresponding Langevin(discretised) algorithm could as well use another scale η for thegradient, rather than the one τ used for the noise
y = xt + η∇π(x) + τǫt
rather than a strict Euler discretisation
y = xt + τ2∇π(x)/2 + τǫt
A few experiments run in Robert and Casella (1999, Chap. 6, §6.5)hinted that using a scale η 6= τ2/2 could actually lead toimprovements
Christian P. Robert About discretising Hamiltonians
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Back on Langevin
For the Langevin diffusion, the corresponding Langevin(discretised) algorithm could as well use another scale η for thegradient, rather than the one τ used for the noise
y = xt + η∇π(x) + τǫt
rather than a strict Euler discretisation
y = xt + τ2∇π(x)/2 + τǫt
A few experiments run in Robert and Casella (1999, Chap. 6, §6.5)hinted that using a scale η 6= τ2/2 could actually lead toimprovementsWhich [independent] framework should we adopt forassessing discretised diffusions?
Christian P. Robert About discretising Hamiltonians