1. Manifold Monte Carlo Methods
Mark Girolami
Department of Statistical Science
University College London
Joint work with Ben Calderhead
Research Section Ordinary Meeting
The Royal Statistical Society
October 13, 2010
4. Riemann manifold Langevin and Hamiltonian Monte Carlo Methods
Riemann manifold Langevin and Hamiltonian Monte Carlo Methods
Girolami, M. & Calderhead, B., J.R.Statist. Soc. B (2011), 73, 2, 1 - 37.
5. Riemann manifold Langevin and Hamiltonian Monte Carlo Methods
Riemann manifold Langevin and Hamiltonian Monte Carlo Methods
Girolami, M. & Calderhead, B., J.R.Statist. Soc. B (2011), 73, 2, 1 - 37.
Advanced Monte Carlo methodology founded on geometric principles
6. Talk Outline
Motivation to improve MCMC capability for challenging problems
7. Talk Outline
Motivation to improve MCMC capability for challenging problems
Stochastic diffusion as adaptive proposal process
8. Talk Outline
Motivation to improve MCMC capability for challenging problems
Stochastic diffusion as adaptive proposal process
Exploring geometric concepts in MCMC methodology
9. Talk Outline
Motivation to improve MCMC capability for challenging problems
Stochastic diffusion as adaptive proposal process
Exploring geometric concepts in MCMC methodology
Diffusions across Riemann manifold as proposal mechanism
10. Talk Outline
Motivation to improve MCMC capability for challenging problems
Stochastic diffusion as adaptive proposal process
Exploring geometric concepts in MCMC methodology
Diffusions across Riemann manifold as proposal mechanism
Deterministic geodesic flows on manifold form basis of MCMC methods
11. Talk Outline
Motivation to improve MCMC capability for challenging problems
Stochastic diffusion as adaptive proposal process
Exploring geometric concepts in MCMC methodology
Diffusions across Riemann manifold as proposal mechanism
Deterministic geodesic flows on manifold form basis of MCMC methods
Illustrative Examples:-
Warped Bivariate Gaussian
12. Talk Outline
Motivation to improve MCMC capability for challenging problems
Stochastic diffusion as adaptive proposal process
Exploring geometric concepts in MCMC methodology
Diffusions across Riemann manifold as proposal mechanism
Deterministic geodesic flows on manifold form basis of MCMC methods
Illustrative Examples:-
Warped Bivariate Gaussian
Gaussian mixture model
13. Talk Outline
Motivation to improve MCMC capability for challenging problems
Stochastic diffusion as adaptive proposal process
Exploring geometric concepts in MCMC methodology
Diffusions across Riemann manifold as proposal mechanism
Deterministic geodesic flows on manifold form basis of MCMC methods
Illustrative Examples:-
Warped Bivariate Gaussian
Gaussian mixture model
Log-Gaussian Cox process
14. Talk Outline
Motivation to improve MCMC capability for challenging problems
Stochastic diffusion as adaptive proposal process
Exploring geometric concepts in MCMC methodology
Diffusions across Riemann manifold as proposal mechanism
Deterministic geodesic flows on manifold form basis of MCMC methods
Illustrative Examples:-
Warped Bivariate Gaussian
Gaussian mixture model
Log-Gaussian Cox process
Conclusions
15. Motivation Simulation Based Inference
Monte Carlo method employs samples from p(θ) to obtain estimate
Z
1 X 1
φ(θ)p(θ)dθ = φ(θ n ) + O(N − 2 )
N n
16. Motivation Simulation Based Inference
Monte Carlo method employs samples from p(θ) to obtain estimate
Z
1 X 1
φ(θ)p(θ)dθ = φ(θ n ) + O(N − 2 )
N n
Draw θ n from ergodic Markov process with stationary distribution p(θ)
17. Motivation Simulation Based Inference
Monte Carlo method employs samples from p(θ) to obtain estimate
Z
1 X 1
φ(θ)p(θ)dθ = φ(θ n ) + O(N − 2 )
N n
Draw θ n from ergodic Markov process with stationary distribution p(θ)
Construct process in two parts
18. Motivation Simulation Based Inference
Monte Carlo method employs samples from p(θ) to obtain estimate
Z
1 X 1
φ(θ)p(θ)dθ = φ(θ n ) + O(N − 2 )
N n
Draw θ n from ergodic Markov process with stationary distribution p(θ)
Construct process in two parts
Propose a move θ → θ with probability pp (θ |θ)
19. Motivation Simulation Based Inference
Monte Carlo method employs samples from p(θ) to obtain estimate
Z
1 X 1
φ(θ)p(θ)dθ = φ(θ n ) + O(N − 2 )
N n
Draw θ n from ergodic Markov process with stationary distribution p(θ)
Construct process in two parts
Propose a move θ → θ with probability pp (θ |θ)
accept or reject proposal with probability
» –
p(θ )pp (θ|θ )
pa (θ |θ) = min 1,
p(θ)pp (θ |θ)
20. Motivation Simulation Based Inference
Monte Carlo method employs samples from p(θ) to obtain estimate
Z
1 X 1
φ(θ)p(θ)dθ = φ(θ n ) + O(N − 2 )
N n
Draw θ n from ergodic Markov process with stationary distribution p(θ)
Construct process in two parts
Propose a move θ → θ with probability pp (θ |θ)
accept or reject proposal with probability
» –
p(θ )pp (θ|θ )
pa (θ |θ) = min 1,
p(θ)pp (θ |θ)
Efficiency dependent on pp (θ |θ) defining proposal mechanism
21. Motivation Simulation Based Inference
Monte Carlo method employs samples from p(θ) to obtain estimate
Z
1 X 1
φ(θ)p(θ)dθ = φ(θ n ) + O(N − 2 )
N n
Draw θ n from ergodic Markov process with stationary distribution p(θ)
Construct process in two parts
Propose a move θ → θ with probability pp (θ |θ)
accept or reject proposal with probability
» –
p(θ )pp (θ|θ )
pa (θ |θ) = min 1,
p(θ)pp (θ |θ)
Efficiency dependent on pp (θ |θ) defining proposal mechanism
Success of MCMC reliant upon appropriate proposal design
38. Geometric Concepts in MCMC
Tangent space - local metric defined by δθ T G(θ)δθ =
P
k ,l gkl δθk δθl
39. Geometric Concepts in MCMC
Tangent space - local metric defined by δθ T G(θ)δθ = k ,l gkl δθk δθl
P
Christoffel symbols - characterise connection on curved manifold
40. Geometric Concepts in MCMC
Tangent space - local metric defined by δθ T G(θ)δθ = k ,l gkl δθk δθl
P
Christoffel symbols - characterise connection on curved manifold
„ «
1 X im ∂gmk ∂gml ∂gkl
Γikl = g + − m
2 m ∂θl ∂θk ∂θ
41. Geometric Concepts in MCMC
Tangent space - local metric defined by δθ T G(θ)δθ = k ,l gkl δθk δθl
P
Christoffel symbols - characterise connection on curved manifold
„ «
1 X im ∂gmk ∂gml ∂gkl
Γikl = g + − m
2 m ∂θl ∂θk ∂θ
Geodesics - shortest path between two points on manifold
42. Geometric Concepts in MCMC
Tangent space - local metric defined by δθ T G(θ)δθ = k ,l gkl δθk δθl
P
Christoffel symbols - characterise connection on curved manifold
„ «
1 X im ∂gmk ∂gml ∂gkl
Γikl = g + − m
2 m ∂θl ∂θk ∂θ
Geodesics - shortest path between two points on manifold
d 2 θi X i dθk dθl
+ Γkl =0
dt 2 dt dt
k,l
44. Illustration of Geometric Concepts
Consider Normal density p(x|µ, σ) = Nx (µ, σ)
Local inner product on tangent space defined by metric tensor, i.e.
δθ T G(θ)δθ, where θ = (µ, σ)T
45. Illustration of Geometric Concepts
Consider Normal density p(x|µ, σ) = Nx (µ, σ)
Local inner product on tangent space defined by metric tensor, i.e.
δθ T G(θ)δθ, where θ = (µ, σ)T
Metric is Fisher Information
σ −2
» –
0
G(µ, σ) =
0 2σ −2
46. Illustration of Geometric Concepts
Consider Normal density p(x|µ, σ) = Nx (µ, σ)
Local inner product on tangent space defined by metric tensor, i.e.
δθ T G(θ)δθ, where θ = (µ, σ)T
Metric is Fisher Information
σ −2
» –
0
G(µ, σ) =
0 2σ −2
Inner-product σ −2 (δµ2 + 2δσ 2 ) so densities N (0, 1) & N (1, 1) further
apart than the densities N (0, 2) & N (1, 2) - distance non-Euclidean
47. Illustration of Geometric Concepts
Consider Normal density p(x|µ, σ) = Nx (µ, σ)
Local inner product on tangent space defined by metric tensor, i.e.
δθ T G(θ)δθ, where θ = (µ, σ)T
Metric is Fisher Information
σ −2
» –
0
G(µ, σ) =
0 2σ −2
Inner-product σ −2 (δµ2 + 2δσ 2 ) so densities N (0, 1) & N (1, 1) further
apart than the densities N (0, 2) & N (1, 2) - distance non-Euclidean
Metric tensor for univariate Normal defines a Hyperbolic Space
49. Langevin Diffusion on Riemannian manifold
Discretised Langevin diffusion on manifold defines proposal mechanism
2 “ ” D
X “p ”
θd = θd + G −1 (θ) θ L(θ) − 2
G(θ)−1 Γd +
ij ij G−1 (θ)z
2 d d
i,j
50. Langevin Diffusion on Riemannian manifold
Discretised Langevin diffusion on manifold defines proposal mechanism
2 “ ” D
X “p ”
θd = θd + G −1 (θ) θ L(θ) − 2
G(θ)−1 Γd +
ij ij G−1 (θ)z
2 d d
i,j
Manifold with constant curvature then proposal mechanism reduces to
2
G −1 (θ)
p
θ =θ+ θ L(θ) + G−1 (θ)z
2
51. Langevin Diffusion on Riemannian manifold
Discretised Langevin diffusion on manifold defines proposal mechanism
2 “ ” D
X “p ”
θd = θd + G −1 (θ) θ L(θ) − 2
G(θ)−1 Γd +
ij ij G−1 (θ)z
2 d d
i,j
Manifold with constant curvature then proposal mechanism reduces to
2
G −1 (θ) θ L(θ) +
p
θ =θ+ G−1 (θ)z
2
MALA proposal with preconditioning
2 √
θ =θ+ M θ L(θ) + Mz
2
52. Langevin Diffusion on Riemannian manifold
Discretised Langevin diffusion on manifold defines proposal mechanism
2 “ ” D
X “p ”
θd = θd + G −1 (θ) θ L(θ) − 2
G(θ)−1 Γd +
ij ij G−1 (θ)z
2 d d
i,j
Manifold with constant curvature then proposal mechanism reduces to
2
G −1 (θ) θ L(θ) +
p
θ =θ+ G−1 (θ)z
2
MALA proposal with preconditioning
2 √
θ =θ+ M θ L(θ) + Mz
2
Proposal and acceptance probability
pp (θ |θ) = N (θ |µ(θ, ), 2 G(θ))
» –
p(θ )pp (θ|θ )
pa (θ |θ) = min 1,
p(θ)pp (θ |θ)
53. Langevin Diffusion on Riemannian manifold
Discretised Langevin diffusion on manifold defines proposal mechanism
2 “ ” D
X “p ”
θd = θd + G −1 (θ) θ L(θ) − 2
G(θ)−1 Γd +
ij ij G−1 (θ)z
2 d d
i,j
Manifold with constant curvature then proposal mechanism reduces to
2
G −1 (θ) θ L(θ) +
p
θ =θ+ G−1 (θ)z
2
MALA proposal with preconditioning
2 √
θ =θ+ M θ L(θ) + Mz
2
Proposal and acceptance probability
pp (θ |θ) = N (θ |µ(θ, ), 2 G(θ))
» –
p(θ )pp (θ|θ )
pa (θ |θ) = min 1,
p(θ)pp (θ |θ)
Proposal mechanism diffuses approximately along the manifold
56. Geodesic flow as proposal mechanism
Desirable that proposals follow direct path on manifold - geodesics
57. Geodesic flow as proposal mechanism
Desirable that proposals follow direct path on manifold - geodesics
Define geodesic flow on manifold by solving
d 2 θi X i dθk dθl
+ Γkl =0
dt 2 dt dt
k,l
58. Geodesic flow as proposal mechanism
Desirable that proposals follow direct path on manifold - geodesics
Define geodesic flow on manifold by solving
d 2 θi X i dθk dθl
+ Γkl =0
dt 2 dt dt
k,l
How can this be exploited in the design of a transition operator?
59. Geodesic flow as proposal mechanism
Desirable that proposals follow direct path on manifold - geodesics
Define geodesic flow on manifold by solving
d 2 θi X i dθk dθl
+ Γkl =0
dt 2 dt dt
k,l
How can this be exploited in the design of a transition operator?
Need slight detour
60. Geodesic flow as proposal mechanism
Desirable that proposals follow direct path on manifold - geodesics
Define geodesic flow on manifold by solving
d 2 θi X i dθk dθl
+ Γkl =0
dt 2 dt dt
k,l
How can this be exploited in the design of a transition operator?
Need slight detour - first define log-density under model as L(θ)
61. Geodesic flow as proposal mechanism
Desirable that proposals follow direct path on manifold - geodesics
Define geodesic flow on manifold by solving
d 2 θi X i dθk dθl
+ Γkl =0
dt 2 dt dt
k,l
How can this be exploited in the design of a transition operator?
Need slight detour - first define log-density under model as L(θ)
Introduce auxiliary variable p ∼ N (0, G(θ))
62. Geodesic flow as proposal mechanism
Desirable that proposals follow direct path on manifold - geodesics
Define geodesic flow on manifold by solving
d 2 θi X i dθk dθl
+ Γkl =0
dt 2 dt dt
k,l
How can this be exploited in the design of a transition operator?
Need slight detour - first define log-density under model as L(θ)
Introduce auxiliary variable p ∼ N (0, G(θ))
Negative joint log density is
1 1
H(θ, p) = −L(θ) + log 2π D |G(θ)| + pT G(θ)−1 p
2 2
63. Riemannian Hamiltonian Monte Carlo
Negative joint log-density ≡ Hamiltonian defined on Riemann manifold
1 1
H(θ, p) = −L(θ) + log 2π D |G(θ)| + pT G(θ)−1 p
| 2 {z } |2 {z }
Potential Energy Kinetic Energy
64. Riemannian Hamiltonian Monte Carlo
Negative joint log-density ≡ Hamiltonian defined on Riemann manifold
1 1
H(θ, p) = −L(θ) + log 2π D |G(θ)| + pT G(θ)−1 p
| 2 {z } |2 {z }
Potential Energy Kinetic Energy
Marginal density follows as required
Z ff
exp {L(θ)} 1
p(θ) ∝ p exp − pT G(θ)−1 p dp = exp {L(θ)}
2π D |G(θ)| 2
65. Riemannian Hamiltonian Monte Carlo
Negative joint log-density ≡ Hamiltonian defined on Riemann manifold
1 1
H(θ, p) = −L(θ) + log 2π D |G(θ)| + pT G(θ)−1 p
| 2 {z } |2 {z }
Potential Energy Kinetic Energy
Marginal density follows as required
Z ff
exp {L(θ)} 1
p(θ) ∝ p exp − pT G(θ)−1 p dp = exp {L(θ)}
2π D |G(θ)| 2
Obtain samples from marginal p(θ) using Gibbs sampler for p(θ, p)
pn+1 |θ n ∼ N (0, G(θ n ))
n+1 n+1
θ |p ∼ p(θ n+1 |pn+1 )
66. Riemannian Hamiltonian Monte Carlo
Negative joint log-density ≡ Hamiltonian defined on Riemann manifold
1 1
H(θ, p) = −L(θ) + log 2π D |G(θ)| + pT G(θ)−1 p
| 2 {z } |2 {z }
Potential Energy Kinetic Energy
Marginal density follows as required
Z ff
exp {L(θ)} 1
p(θ) ∝ p exp − pT G(θ)−1 p dp = exp {L(θ)}
2π D |G(θ)| 2
Obtain samples from marginal p(θ) using Gibbs sampler for p(θ, p)
pn+1 |θ n ∼ N (0, G(θ n ))
n+1 n+1
θ |p ∼ p(θ n+1 |pn+1 )
Employ Hamiltonian dynamics to propose samples for p(θ n+1 |pn+1 ),
Duane et al, 1987; Neal, 2010.
67. Riemannian Hamiltonian Monte Carlo
Negative joint log-density ≡ Hamiltonian defined on Riemann manifold
1 1
H(θ, p) = −L(θ) + log 2π D |G(θ)| + pT G(θ)−1 p
| 2 {z } |2 {z }
Potential Energy Kinetic Energy
Marginal density follows as required
Z ff
exp {L(θ)} 1
p(θ) ∝ p exp − pT G(θ)−1 p dp = exp {L(θ)}
2π D |G(θ)| 2
Obtain samples from marginal p(θ) using Gibbs sampler for p(θ, p)
pn+1 |θ n ∼ N (0, G(θ n ))
n+1 n+1
θ |p ∼ p(θ n+1 |pn+1 )
Employ Hamiltonian dynamics to propose samples for p(θ n+1 |pn+1 ),
Duane et al, 1987; Neal, 2010.
dθ ∂ dp ∂
= H(θ, p) = − H(θ, p)
dt ∂p dt ∂θ
69. Riemannian Manifold Hamiltonian Monte Carlo
˜ ˜
Consider the Hamiltonian H(θ, p) = 1 pT G(θ)−1 p
2
Hamiltonians with only a quadratic kinetic energy term exactly describe
geodesic flow on the coordinate space θ with metric G ˜
70. Riemannian Manifold Hamiltonian Monte Carlo
˜ ˜
Consider the Hamiltonian H(θ, p) = 1 pT G(θ)−1 p
2
Hamiltonians with only a quadratic kinetic energy term exactly describe
geodesic flow on the coordinate space θ with metric G ˜
However our Hamiltonian is H(θ, p) = V (θ) + 1 pT G(θ)−1 p
2
71. Riemannian Manifold Hamiltonian Monte Carlo
˜ ˜
Consider the Hamiltonian H(θ, p) = 1 pT G(θ)−1 p
2
Hamiltonians with only a quadratic kinetic energy term exactly describe
geodesic flow on the coordinate space θ with metric G ˜
However our Hamiltonian is H(θ, p) = V (θ) + 1 pT G(θ)−1 p
2
˜
If we define G(θ) = G(θ) × (h − V (θ)), where h is a constant H(θ, p)
72. Riemannian Manifold Hamiltonian Monte Carlo
˜ ˜
Consider the Hamiltonian H(θ, p) = 1 pT G(θ)−1 p
2
Hamiltonians with only a quadratic kinetic energy term exactly describe
geodesic flow on the coordinate space θ with metric G ˜
However our Hamiltonian is H(θ, p) = V (θ) + 1 pT G(θ)−1 p
2
˜
If we define G(θ) = G(θ) × (h − V (θ)), where h is a constant H(θ, p)
Then the Maupertuis principle tells us that the Hamiltonian flow for
˜
H(θ, p) and H(θ, p) are equivalent along energy level h
73. Riemannian Manifold Hamiltonian Monte Carlo
˜ ˜
Consider the Hamiltonian H(θ, p) = 1 pT G(θ)−1 p
2
Hamiltonians with only a quadratic kinetic energy term exactly describe
geodesic flow on the coordinate space θ with metric G ˜
However our Hamiltonian is H(θ, p) = V (θ) + 1 pT G(θ)−1 p
2
˜
If we define G(θ) = G(θ) × (h − V (θ)), where h is a constant H(θ, p)
Then the Maupertuis principle tells us that the Hamiltonian flow for
˜
H(θ, p) and H(θ, p) are equivalent along energy level h
The solution of
dθ ∂ dp ∂
= H(θ, p) = − H(θ, p)
dt ∂p dt ∂θ
74. Riemannian Manifold Hamiltonian Monte Carlo
˜ ˜
Consider the Hamiltonian H(θ, p) = 1 pT G(θ)−1 p
2
Hamiltonians with only a quadratic kinetic energy term exactly describe
geodesic flow on the coordinate space θ with metric G ˜
However our Hamiltonian is H(θ, p) = V (θ) + 1 pT G(θ)−1 p
2
˜
If we define G(θ) = G(θ) × (h − V (θ)), where h is a constant H(θ, p)
Then the Maupertuis principle tells us that the Hamiltonian flow for
˜
H(θ, p) and H(θ, p) are equivalent along energy level h
The solution of
dθ ∂ dp ∂
= H(θ, p) = − H(θ, p)
dt ∂p dt ∂θ
is therefore equivalent to the solution of
d 2 θi X i dθk dθl
˜
+ Γkl =0
dt 2 dt dt
k,l
75. Riemannian Manifold Hamiltonian Monte Carlo
˜ ˜
Consider the Hamiltonian H(θ, p) = 1 pT G(θ)−1 p
2
Hamiltonians with only a quadratic kinetic energy term exactly describe
geodesic flow on the coordinate space θ with metric G ˜
However our Hamiltonian is H(θ, p) = V (θ) + 1 pT G(θ)−1 p
2
˜
If we define G(θ) = G(θ) × (h − V (θ)), where h is a constant H(θ, p)
Then the Maupertuis principle tells us that the Hamiltonian flow for
˜
H(θ, p) and H(θ, p) are equivalent along energy level h
The solution of
dθ ∂ dp ∂
= H(θ, p) = − H(θ, p)
dt ∂p dt ∂θ
is therefore equivalent to the solution of
d 2 θi X i dθk dθl
˜
+ Γkl =0
dt 2 dt dt
k,l
RMHMC proposals are along the manifold geodesics
78. Gaussian Mixture Model
Univariate finite mixture model
K
X 2
p(xi |θ) = πk N (xi |µk , σk )
k=1
79. Gaussian Mixture Model
Univariate finite mixture model
K
X 2
p(xi |θ) = πk N (xi |µk , σk )
k=1
FI based metric tensor non-analytic - employ empirical FI
80. Gaussian Mixture Model
Univariate finite mixture model
K
X 2
p(xi |θ) = πk N (xi |µk , σk )
k=1
FI based metric tensor non-analytic - employ empirical FI
1 T 1
G(θ) = S S − 2 ssT
¯¯ −−→
−− cov ( θ L(θ)) = I(θ)
N N N→∞
!
∂S T ∂ sT
„ «
∂G(θ) 1 ∂S 1 ¯
∂s T ¯
= S + ST − ¯ ¯
s +s
∂θd N ∂θd ∂θd N2 ∂θd ∂θd
∂ log p(xi |θ) PN
with score matrix S with elements Si,d = ∂θd
¯
and s = i=1 ST
i,·
81. Gaussian Mixture Model
Univariate finite mixture model
p(x|µ, σ 2 ) = 0.7 × N (x|0, σ 2 ) + 0.3 × N (x|µ, σ 2 )
82. Gaussian Mixture Model
Univariate finite mixture model
p(x|µ, σ 2 ) = 0.7 × N (x|0, σ 2 ) + 0.3 × N (x|µ, σ 2 )
4
3
2
1
lnσ
0
−1
−2
−3
−4
−4 −3 −2 −1 0 1 2 3 4
µ
Figure: Arrows correspond to the gradients and ellipses to the inverse metric
tensor. Dashed lines are isocontours of the joint log density
83. Gaussian Mixture Model
MALA path mMALA path Simplified mMALA path
4 4 4
−1410 −1410 −1410
−1310 −1310 −1310
3 3 3
−1210 −1210 −1210
−1110 −1110 −1110
2 2 2
−1010 −1010 −1010
1 1 1
−910 −910 −910
lnσ
lnσ
lnσ
0 0 0
−1 −1 −1
−1410 −1410 −1410
−2 −2 −2
−2910 −2910 −2910
−3 −3 −3
−4 −4 −4
−4 −3 −2 −1 0 1 2 3 4 −4 −3 −2 −1 0 1 2 3 4 −4 −3 −2 −1 0 1 2 3 4
µ µ µ
MALA Sample Autocorrelation Function mMALA Sample Autocorrelation Function Simplified mMALA Sample Autocorrelation Function
0.8 0.8 0.8
Sample Autocorrelation
Sample Autocorrelation
Sample Autocorrelation
0.6 0.6 0.6
0.4 0.4 0.4
0.2 0.2 0.2
0 0 0
−0.2 −0.2 −0.2
0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20
Lag Lag Lag
Figure: Comparison of MALA (left), mMALA (middle) and simplified mMALA (right)
convergence paths and autocorrelation plots. Autocorrelation plots are from the
stationary chains, i.e. once the chains have converged to the stationary distribution.
85. Log-Gaussian Cox Point Process with Latent Field
The joint density for Poisson counts and latent Gaussian field
Y64
p(y, x|µ, σ, β) ∝ exp{yi,j xi,j −m exp(xi,j )} exp(−(x−µ1)T Σ−1 (x−µ1)/2)
θ
i,j
86. Log-Gaussian Cox Point Process with Latent Field
The joint density for Poisson counts and latent Gaussian field
Y64
p(y, x|µ, σ, β) ∝ exp{yi,j xi,j −m exp(xi,j )} exp(−(x−µ1)T Σ−1 (x−µ1)/2)
θ
i,j
Metric tensors
„ «
1 ∂Σθ −1 ∂Σθ
G(θ)i,j = trace Σ−1
θ Σθ
2 ∂θi ∂θj
G(x) = Λ + Σ−1
θ
where Λ is diagonal with elements m exp(µ + (Σθ )i,i )
87. Log-Gaussian Cox Point Process with Latent Field
The joint density for Poisson counts and latent Gaussian field
Y64
p(y, x|µ, σ, β) ∝ exp{yi,j xi,j −m exp(xi,j )} exp(−(x−µ1)T Σ−1 (x−µ1)/2)
θ
i,j
Metric tensors
„ «
1 ∂Σθ −1 ∂Σθ
G(θ)i,j = trace Σ−1
θ Σθ
2 ∂θi ∂θj
G(x) = Λ + Σ−1
θ
where Λ is diagonal with elements m exp(µ + (Σθ )i,i )
Latent field metric tensor defining flat manifold is 4096 × 4096, O(N 3 )
obtained from parameter conditional
88. Log-Gaussian Cox Point Process with Latent Field
The joint density for Poisson counts and latent Gaussian field
Y64
p(y, x|µ, σ, β) ∝ exp{yi,j xi,j −m exp(xi,j )} exp(−(x−µ1)T Σ−1 (x−µ1)/2)
θ
i,j
Metric tensors
„ «
1 ∂Σθ −1 ∂Σθ
G(θ)i,j = trace Σ−1
θ Σθ
2 ∂θi ∂θj
G(x) = Λ + Σ−1
θ
where Λ is diagonal with elements m exp(µ + (Σθ )i,i )
Latent field metric tensor defining flat manifold is 4096 × 4096, O(N 3 )
obtained from parameter conditional
MALA requires transformation of latent field to sample - with separate
tuning in transient and stationary phases of Markov chain
Manifold methods requires no pilot tuning or additional transformations
89. RMHMC for Log-Gaussian Cox Point Processes
Table: Sampling the latent variables of a Log-Gaussian Cox Process - Comparison of
sampling methods
Method Time ESS (Min, Med, Max) s/Min ESS Rel. Speed
MALA (Transient) 31,577 (3, 8, 50) 10,605 ×1
MALA (Stationary) 31,118 (4, 16, 80) 7836 ×1.35
mMALA 634 (26, 84, 174) 24.1 ×440
RMHMC 2936 (1951, 4545, 5000) 1.5 ×7070
91. Conclusion and Discussion
Geometry of statistical models harnessed in Monte Carlo methods
Diffusions that respect structure and curvature of space - Manifold MALA
92. Conclusion and Discussion
Geometry of statistical models harnessed in Monte Carlo methods
Diffusions that respect structure and curvature of space - Manifold MALA
Geodesic flows on model manifold - RMHMC - generalisation of HMC
93. Conclusion and Discussion
Geometry of statistical models harnessed in Monte Carlo methods
Diffusions that respect structure and curvature of space - Manifold MALA
Geodesic flows on model manifold - RMHMC - generalisation of HMC
Assessed on correlated & high-dimensional latent variable models
94. Conclusion and Discussion
Geometry of statistical models harnessed in Monte Carlo methods
Diffusions that respect structure and curvature of space - Manifold MALA
Geodesic flows on model manifold - RMHMC - generalisation of HMC
Assessed on correlated & high-dimensional latent variable models
Promising capability of methodology
95. Conclusion and Discussion
Geometry of statistical models harnessed in Monte Carlo methods
Diffusions that respect structure and curvature of space - Manifold MALA
Geodesic flows on model manifold - RMHMC - generalisation of HMC
Assessed on correlated & high-dimensional latent variable models
Promising capability of methodology
Ongoing development
96. Conclusion and Discussion
Geometry of statistical models harnessed in Monte Carlo methods
Diffusions that respect structure and curvature of space - Manifold MALA
Geodesic flows on model manifold - RMHMC - generalisation of HMC
Assessed on correlated & high-dimensional latent variable models
Promising capability of methodology
Ongoing development
Potential bottleneck at metric tensor and Christoffel symbols
97. Conclusion and Discussion
Geometry of statistical models harnessed in Monte Carlo methods
Diffusions that respect structure and curvature of space - Manifold MALA
Geodesic flows on model manifold - RMHMC - generalisation of HMC
Assessed on correlated & high-dimensional latent variable models
Promising capability of methodology
Ongoing development
Potential bottleneck at metric tensor and Christoffel symbols
Theoretical analysis of convergence
98. Conclusion and Discussion
Geometry of statistical models harnessed in Monte Carlo methods
Diffusions that respect structure and curvature of space - Manifold MALA
Geodesic flows on model manifold - RMHMC - generalisation of HMC
Assessed on correlated & high-dimensional latent variable models
Promising capability of methodology
Ongoing development
Potential bottleneck at metric tensor and Christoffel symbols
Theoretical analysis of convergence
Investigate alternative manifold structures
99. Conclusion and Discussion
Geometry of statistical models harnessed in Monte Carlo methods
Diffusions that respect structure and curvature of space - Manifold MALA
Geodesic flows on model manifold - RMHMC - generalisation of HMC
Assessed on correlated & high-dimensional latent variable models
Promising capability of methodology
Ongoing development
Potential bottleneck at metric tensor and Christoffel symbols
Theoretical analysis of convergence
Investigate alternative manifold structures
Design and effect of metric
100. Conclusion and Discussion
Geometry of statistical models harnessed in Monte Carlo methods
Diffusions that respect structure and curvature of space - Manifold MALA
Geodesic flows on model manifold - RMHMC - generalisation of HMC
Assessed on correlated & high-dimensional latent variable models
Promising capability of methodology
Ongoing development
Potential bottleneck at metric tensor and Christoffel symbols
Theoretical analysis of convergence
Investigate alternative manifold structures
Design and effect of metric
Optimality of Hamiltonian flows as local geodesics
101. Conclusion and Discussion
Geometry of statistical models harnessed in Monte Carlo methods
Diffusions that respect structure and curvature of space - Manifold MALA
Geodesic flows on model manifold - RMHMC - generalisation of HMC
Assessed on correlated & high-dimensional latent variable models
Promising capability of methodology
Ongoing development
Potential bottleneck at metric tensor and Christoffel symbols
Theoretical analysis of convergence
Investigate alternative manifold structures
Design and effect of metric
Optimality of Hamiltonian flows as local geodesics
Alternative transition kernels
102. Conclusion and Discussion
Geometry of statistical models harnessed in Monte Carlo methods
Diffusions that respect structure and curvature of space - Manifold MALA
Geodesic flows on model manifold - RMHMC - generalisation of HMC
Assessed on correlated & high-dimensional latent variable models
Promising capability of methodology
Ongoing development
Potential bottleneck at metric tensor and Christoffel symbols
Theoretical analysis of convergence
Investigate alternative manifold structures
Design and effect of metric
Optimality of Hamiltonian flows as local geodesics
Alternative transition kernels
No silver bullet or cure all - new powerful methodology for MC toolkit
103. Funding Acknowledgment
Girolami funded by EPSRC Advanced Research Fellowship and BBSRC
project grant
Calderhead supported by Microsoft Research PhD Scholarship