RSS Read Paper by Mark Girolami

Manifold Monte Carlo Methods

Mark Girolami

Department of Statistical ScienceUniversity College London

Joint work with Ben Calderhead

Research Section Ordinary Meeting

The Royal Statistical SocietyOctober 13, 2010

Riemann manifold Langevin and Hamiltonian Monte Carlo Methods

I Riemann manifold Langevin and Hamiltonian Monte Carlo MethodsGirolami, M. & Calderhead, B., J.R.Statist. Soc. B (2011), 73, 2, 1 - 37.

I Advanced Monte Carlo methodology founded on geometric principles

Riemann manifold Langevin and Hamiltonian Monte Carlo Methods

I Riemann manifold Langevin and Hamiltonian Monte Carlo MethodsGirolami, M. & Calderhead, B., J.R.Statist. Soc. B (2011), 73, 2, 1 - 37.

I Advanced Monte Carlo methodology founded on geometric principles

Riemann manifold Langevin and Hamiltonian Monte Carlo Methods

I Riemann manifold Langevin and Hamiltonian Monte Carlo MethodsGirolami, M. & Calderhead, B., J.R.Statist. Soc. B (2011), 73, 2, 1 - 37.

I Advanced Monte Carlo methodology founded on geometric principles

Riemann manifold Langevin and Hamiltonian Monte Carlo Methods

I Riemann manifold Langevin and Hamiltonian Monte Carlo MethodsGirolami, M. & Calderhead, B., J.R.Statist. Soc. B (2011), 73, 2, 1 - 37.

I Advanced Monte Carlo methodology founded on geometric principles

Talk Outline

I Motivation to improve MCMC capability for challenging problems

I Stochastic diffusion as adaptive proposal process

I Exploring geometric concepts in MCMC methodology

I Diffusions across Riemann manifold as proposal mechanism

I Deterministic geodesic flows on manifold form basis of MCMC methods

I Illustrative Examples:-

I Warped Bivariate Gaussian

I Gaussian mixture model

I Log-Gaussian Cox process

I Conclusions

Talk Outline

I Motivation to improve MCMC capability for challenging problems

I Stochastic diffusion as adaptive proposal process

I Exploring geometric concepts in MCMC methodology

I Diffusions across Riemann manifold as proposal mechanism

I Deterministic geodesic flows on manifold form basis of MCMC methods

I Illustrative Examples:-

I Warped Bivariate Gaussian

I Gaussian mixture model

I Log-Gaussian Cox process

I Conclusions

Talk Outline

I Motivation to improve MCMC capability for challenging problems

I Stochastic diffusion as adaptive proposal process

I Exploring geometric concepts in MCMC methodology

I Diffusions across Riemann manifold as proposal mechanism

I Deterministic geodesic flows on manifold form basis of MCMC methods

I Illustrative Examples:-

I Warped Bivariate Gaussian

I Gaussian mixture model

I Log-Gaussian Cox process

I Conclusions

Talk Outline

I Motivation to improve MCMC capability for challenging problems

I Stochastic diffusion as adaptive proposal process

I Exploring geometric concepts in MCMC methodology

I Diffusions across Riemann manifold as proposal mechanism

I Deterministic geodesic flows on manifold form basis of MCMC methods

I Illustrative Examples:-

I Warped Bivariate Gaussian

I Gaussian mixture model

I Log-Gaussian Cox process

I Conclusions

Talk Outline

I Motivation to improve MCMC capability for challenging problems

I Stochastic diffusion as adaptive proposal process

I Exploring geometric concepts in MCMC methodology

I Diffusions across Riemann manifold as proposal mechanism

I Deterministic geodesic flows on manifold form basis of MCMC methods

I Illustrative Examples:-

I Warped Bivariate Gaussian

I Gaussian mixture model

I Log-Gaussian Cox process

I Conclusions

Talk Outline

I Motivation to improve MCMC capability for challenging problems

I Stochastic diffusion as adaptive proposal process

I Exploring geometric concepts in MCMC methodology

I Diffusions across Riemann manifold as proposal mechanism

I Deterministic geodesic flows on manifold form basis of MCMC methods

I Illustrative Examples:-

I Warped Bivariate Gaussian

I Gaussian mixture model

I Log-Gaussian Cox process

I Conclusions

Talk Outline

I Motivation to improve MCMC capability for challenging problems

I Stochastic diffusion as adaptive proposal process

I Exploring geometric concepts in MCMC methodology

I Diffusions across Riemann manifold as proposal mechanism

I Deterministic geodesic flows on manifold form basis of MCMC methods

I Illustrative Examples:-

I Warped Bivariate Gaussian

I Gaussian mixture model

I Log-Gaussian Cox process

I Conclusions

Talk Outline

I Motivation to improve MCMC capability for challenging problems

I Stochastic diffusion as adaptive proposal process

I Exploring geometric concepts in MCMC methodology

I Diffusions across Riemann manifold as proposal mechanism

I Deterministic geodesic flows on manifold form basis of MCMC methods

I Illustrative Examples:-

I Warped Bivariate Gaussian

I Gaussian mixture model

I Log-Gaussian Cox process

I Conclusions

Talk Outline

I Motivation to improve MCMC capability for challenging problems

I Stochastic diffusion as adaptive proposal process

I Exploring geometric concepts in MCMC methodology

I Diffusions across Riemann manifold as proposal mechanism

I Deterministic geodesic flows on manifold form basis of MCMC methods

I Illustrative Examples:-

I Warped Bivariate Gaussian

I Gaussian mixture model

I Log-Gaussian Cox process

I Conclusions

Motivation Simulation Based Inference

I Monte Carlo method employs samples from p(θ) to obtain estimateZφ(θ)p(θ)dθ =

1N

Xnφ(θn) +O(N−

12 )

I Draw θn from ergodic Markov process with stationary distribution p(θ)

I Construct process in two parts

I Propose a move θ → θ′ with probability pp(θ′|θ)

I accept or reject proposal with probability

pa(θ′|θ) = min

»1,

p(θ′)pp(θ|θ′)p(θ)pp(θ′|θ)

–

I Efficiency dependent on pp(θ′|θ) defining proposal mechanism

I Success of MCMC reliant upon appropriate proposal design

Motivation Simulation Based Inference

I Monte Carlo method employs samples from p(θ) to obtain estimateZφ(θ)p(θ)dθ =

1N

Xnφ(θn) +O(N−

12 )

I Draw θn from ergodic Markov process with stationary distribution p(θ)

I Construct process in two parts

I Propose a move θ → θ′ with probability pp(θ′|θ)

I accept or reject proposal with probability

pa(θ′|θ) = min

»1,

p(θ′)pp(θ|θ′)p(θ)pp(θ′|θ)

–

I Efficiency dependent on pp(θ′|θ) defining proposal mechanism

I Success of MCMC reliant upon appropriate proposal design

Motivation Simulation Based Inference

I Monte Carlo method employs samples from p(θ) to obtain estimateZφ(θ)p(θ)dθ =

1N

Xnφ(θn) +O(N−

12 )

I Draw θn from ergodic Markov process with stationary distribution p(θ)

I Construct process in two parts

I Propose a move θ → θ′ with probability pp(θ′|θ)

I accept or reject proposal with probability

pa(θ′|θ) = min

»1,

p(θ′)pp(θ|θ′)p(θ)pp(θ′|θ)

–

I Efficiency dependent on pp(θ′|θ) defining proposal mechanism

I Success of MCMC reliant upon appropriate proposal design

Motivation Simulation Based Inference

I Monte Carlo method employs samples from p(θ) to obtain estimateZφ(θ)p(θ)dθ =

1N

Xnφ(θn) +O(N−

12 )

I Draw θn from ergodic Markov process with stationary distribution p(θ)

I Construct process in two parts

I Propose a move θ → θ′ with probability pp(θ′|θ)

I accept or reject proposal with probability

pa(θ′|θ) = min

»1,

p(θ′)pp(θ|θ′)p(θ)pp(θ′|θ)

–

I Efficiency dependent on pp(θ′|θ) defining proposal mechanism

I Success of MCMC reliant upon appropriate proposal design

Motivation Simulation Based Inference

I Monte Carlo method employs samples from p(θ) to obtain estimateZφ(θ)p(θ)dθ =

1N

Xnφ(θn) +O(N−

12 )

I Draw θn from ergodic Markov process with stationary distribution p(θ)

I Construct process in two parts

I Propose a move θ → θ′ with probability pp(θ′|θ)

I accept or reject proposal with probability

pa(θ′|θ) = min

»1,

p(θ′)pp(θ|θ′)p(θ)pp(θ′|θ)

–

I Efficiency dependent on pp(θ′|θ) defining proposal mechanism

I Success of MCMC reliant upon appropriate proposal design

Motivation Simulation Based Inference

I Monte Carlo method employs samples from p(θ) to obtain estimateZφ(θ)p(θ)dθ =

1N

Xnφ(θn) +O(N−

12 )

I Draw θn from ergodic Markov process with stationary distribution p(θ)

I Construct process in two parts

I Propose a move θ → θ′ with probability pp(θ′|θ)

I accept or reject proposal with probability

pa(θ′|θ) = min

»1,

p(θ′)pp(θ|θ′)p(θ)pp(θ′|θ)

–

I Efficiency dependent on pp(θ′|θ) defining proposal mechanism

I Success of MCMC reliant upon appropriate proposal design

Motivation Simulation Based Inference

I Monte Carlo method employs samples from p(θ) to obtain estimateZφ(θ)p(θ)dθ =

1N

Xnφ(θn) +O(N−

12 )

I Draw θn from ergodic Markov process with stationary distribution p(θ)

I Construct process in two parts

I Propose a move θ → θ′ with probability pp(θ′|θ)

I accept or reject proposal with probability

pa(θ′|θ) = min

»1,

p(θ′)pp(θ|θ′)p(θ)pp(θ′|θ)

–

I Efficiency dependent on pp(θ′|θ) defining proposal mechanism

I Success of MCMC reliant upon appropriate proposal design

Adaptive Proposal Distributions - Exploit Discretised Diffusion

I For θ ∈ RD with density p(θ), L(θ) ≡ log p(θ), define Langevin diffusion

dθ(t) =12∇θL(θ(t))dt + db(t)

I First order Euler-Maruyama discrete integration of diffusion

θ(τ + ε) = θ(τ) +ε2

2∇θL(θ(τ)) + εz(τ)

I Proposalpp(θ′|θ) = N (θ′|µ(θ, ε), ε2I) with µ(θ, ε) = θ +

ε2

2∇θL(θ)

I Acceptance probability to correct for bias

pa(θ′|θ) = min»1,

p(θ′)pp(θ|θ′)p(θ)pp(θ′|θ)

–I Isotropic diffusion inefficient, employ pre-conditioning

θ′ = θ + ε2M∇θL(θ)/2 + ε√

Mz

I How to set M systematically? Tuning in transient & stationary phases

Adaptive Proposal Distributions - Exploit Discretised Diffusion

I For θ ∈ RD with density p(θ), L(θ) ≡ log p(θ), define Langevin diffusion

dθ(t) =12∇θL(θ(t))dt + db(t)

I First order Euler-Maruyama discrete integration of diffusion

θ(τ + ε) = θ(τ) +ε2

2∇θL(θ(τ)) + εz(τ)

I Proposalpp(θ′|θ) = N (θ′|µ(θ, ε), ε2I) with µ(θ, ε) = θ +

ε2

2∇θL(θ)

I Acceptance probability to correct for bias

pa(θ′|θ) = min»1,

p(θ′)pp(θ|θ′)p(θ)pp(θ′|θ)

–I Isotropic diffusion inefficient, employ pre-conditioning

θ′ = θ + ε2M∇θL(θ)/2 + ε√

Mz

I How to set M systematically? Tuning in transient & stationary phases

Adaptive Proposal Distributions - Exploit Discretised Diffusion

I For θ ∈ RD with density p(θ), L(θ) ≡ log p(θ), define Langevin diffusion

dθ(t) =12∇θL(θ(t))dt + db(t)

I First order Euler-Maruyama discrete integration of diffusion

θ(τ + ε) = θ(τ) +ε2

2∇θL(θ(τ)) + εz(τ)

I Proposalpp(θ′|θ) = N (θ′|µ(θ, ε), ε2I) with µ(θ, ε) = θ +

ε2

2∇θL(θ)

I Acceptance probability to correct for bias

pa(θ′|θ) = min»1,

p(θ′)pp(θ|θ′)p(θ)pp(θ′|θ)

–I Isotropic diffusion inefficient, employ pre-conditioning

θ′ = θ + ε2M∇θL(θ)/2 + ε√

Mz

I How to set M systematically? Tuning in transient & stationary phases

Adaptive Proposal Distributions - Exploit Discretised Diffusion

I For θ ∈ RD with density p(θ), L(θ) ≡ log p(θ), define Langevin diffusion

dθ(t) =12∇θL(θ(t))dt + db(t)

I First order Euler-Maruyama discrete integration of diffusion

θ(τ + ε) = θ(τ) +ε2

2∇θL(θ(τ)) + εz(τ)

I Proposalpp(θ′|θ) = N (θ′|µ(θ, ε), ε2I) with µ(θ, ε) = θ +

ε2

2∇θL(θ)

I Acceptance probability to correct for bias

pa(θ′|θ) = min»1,

p(θ′)pp(θ|θ′)p(θ)pp(θ′|θ)

–I Isotropic diffusion inefficient, employ pre-conditioning

θ′ = θ + ε2M∇θL(θ)/2 + ε√

Mz

I How to set M systematically? Tuning in transient & stationary phases

Adaptive Proposal Distributions - Exploit Discretised Diffusion

I For θ ∈ RD with density p(θ), L(θ) ≡ log p(θ), define Langevin diffusion

dθ(t) =12∇θL(θ(t))dt + db(t)

I First order Euler-Maruyama discrete integration of diffusion

θ(τ + ε) = θ(τ) +ε2

2∇θL(θ(τ)) + εz(τ)

I Proposalpp(θ′|θ) = N (θ′|µ(θ, ε), ε2I) with µ(θ, ε) = θ +

ε2

2∇θL(θ)

I Acceptance probability to correct for bias

pa(θ′|θ) = min»1,

p(θ′)pp(θ|θ′)p(θ)pp(θ′|θ)

–

I Isotropic diffusion inefficient, employ pre-conditioning

θ′ = θ + ε2M∇θL(θ)/2 + ε√

Mz

I How to set M systematically? Tuning in transient & stationary phases

Adaptive Proposal Distributions - Exploit Discretised Diffusion

I For θ ∈ RD with density p(θ), L(θ) ≡ log p(θ), define Langevin diffusion

dθ(t) =12∇θL(θ(t))dt + db(t)

I First order Euler-Maruyama discrete integration of diffusion

θ(τ + ε) = θ(τ) +ε2

2∇θL(θ(τ)) + εz(τ)

I Proposalpp(θ′|θ) = N (θ′|µ(θ, ε), ε2I) with µ(θ, ε) = θ +

ε2

2∇θL(θ)

I Acceptance probability to correct for bias

pa(θ′|θ) = min»1,

p(θ′)pp(θ|θ′)p(θ)pp(θ′|θ)

–I Isotropic diffusion inefficient, employ pre-conditioning

θ′ = θ + ε2M∇θL(θ)/2 + ε√

Mz

I How to set M systematically? Tuning in transient & stationary phases

Adaptive Proposal Distributions - Exploit Discretised Diffusion

I For θ ∈ RD with density p(θ), L(θ) ≡ log p(θ), define Langevin diffusion

dθ(t) =12∇θL(θ(t))dt + db(t)

I First order Euler-Maruyama discrete integration of diffusion

θ(τ + ε) = θ(τ) +ε2

2∇θL(θ(τ)) + εz(τ)

I Proposalpp(θ′|θ) = N (θ′|µ(θ, ε), ε2I) with µ(θ, ε) = θ +

ε2

2∇θL(θ)

I Acceptance probability to correct for bias

pa(θ′|θ) = min»1,

p(θ′)pp(θ|θ′)p(θ)pp(θ′|θ)

–I Isotropic diffusion inefficient, employ pre-conditioning

θ′ = θ + ε2M∇θL(θ)/2 + ε√

Mz

I How to set M systematically? Tuning in transient & stationary phases

Geometric Concepts in MCMC

I Rao, 1945; Jeffreys, 1948, to first orderZp(y; θ + δθ) log

p(y; θ + δθ)

p(y; θ)dθ ≈ δθTG(θ)δθ

where

G(θ) = Ey|θ

(∇θp(y; θ)

p(y; θ)

∇θp(y; θ)

p(y; θ)

T)

I Fisher Information G(θ) is p.d. metric defining a Riemann manifoldI Non-Euclidean geometry for probabilities - distances, metrics, invariants,

curvature, geodesicsI Asymptotic statistical analysis. Amari, 1981, 85, 90; Murray & Rice,

1993; Critchley et al, 1993; Kass, 1989; Efron, 1975; Dawid, 1975;Lauritsen, 1989

I Statistical shape analysis Kent et al, 1996; Dryden & Mardia, 1998

I Can geometric structure be employed in MCMC methodology?

Geometric Concepts in MCMC

I Rao, 1945; Jeffreys, 1948, to first orderZp(y; θ + δθ) log

p(y; θ + δθ)

p(y; θ)dθ ≈ δθTG(θ)δθ

where

G(θ) = Ey|θ

(∇θp(y; θ)

p(y; θ)

∇θp(y; θ)

p(y; θ)

T)

I Fisher Information G(θ) is p.d. metric defining a Riemann manifoldI Non-Euclidean geometry for probabilities - distances, metrics, invariants,

curvature, geodesicsI Asymptotic statistical analysis. Amari, 1981, 85, 90; Murray & Rice,

1993; Critchley et al, 1993; Kass, 1989; Efron, 1975; Dawid, 1975;Lauritsen, 1989

I Statistical shape analysis Kent et al, 1996; Dryden & Mardia, 1998

I Can geometric structure be employed in MCMC methodology?

Geometric Concepts in MCMC

I Rao, 1945; Jeffreys, 1948, to first orderZp(y; θ + δθ) log

p(y; θ + δθ)

p(y; θ)dθ ≈ δθTG(θ)δθ

where

G(θ) = Ey|θ

(∇θp(y; θ)

p(y; θ)

∇θp(y; θ)

p(y; θ)

T)

I Fisher Information G(θ) is p.d. metric defining a Riemann manifoldI Non-Euclidean geometry for probabilities - distances, metrics, invariants,

curvature, geodesicsI Asymptotic statistical analysis. Amari, 1981, 85, 90; Murray & Rice,

1993; Critchley et al, 1993; Kass, 1989; Efron, 1975; Dawid, 1975;Lauritsen, 1989

I Statistical shape analysis Kent et al, 1996; Dryden & Mardia, 1998

I Can geometric structure be employed in MCMC methodology?

Geometric Concepts in MCMC

I Rao, 1945; Jeffreys, 1948, to first orderZp(y; θ + δθ) log

p(y; θ + δθ)

p(y; θ)dθ ≈ δθTG(θ)δθ

where

G(θ) = Ey|θ

(∇θp(y; θ)

p(y; θ)

∇θp(y; θ)

p(y; θ)

T)

I Fisher Information G(θ) is p.d. metric defining a Riemann manifold

I Non-Euclidean geometry for probabilities - distances, metrics, invariants,curvature, geodesics

I Asymptotic statistical analysis. Amari, 1981, 85, 90; Murray & Rice,1993; Critchley et al, 1993; Kass, 1989; Efron, 1975; Dawid, 1975;Lauritsen, 1989

I Statistical shape analysis Kent et al, 1996; Dryden & Mardia, 1998

I Can geometric structure be employed in MCMC methodology?

Geometric Concepts in MCMC

I Rao, 1945; Jeffreys, 1948, to first orderZp(y; θ + δθ) log

p(y; θ + δθ)

p(y; θ)dθ ≈ δθTG(θ)δθ

where

G(θ) = Ey|θ

(∇θp(y; θ)

p(y; θ)

∇θp(y; θ)

p(y; θ)

T)

I Fisher Information G(θ) is p.d. metric defining a Riemann manifoldI Non-Euclidean geometry for probabilities - distances, metrics, invariants,

curvature, geodesics

I Asymptotic statistical analysis. Amari, 1981, 85, 90; Murray & Rice,1993; Critchley et al, 1993; Kass, 1989; Efron, 1975; Dawid, 1975;Lauritsen, 1989

I Statistical shape analysis Kent et al, 1996; Dryden & Mardia, 1998

I Can geometric structure be employed in MCMC methodology?

Geometric Concepts in MCMC

I Rao, 1945; Jeffreys, 1948, to first orderZp(y; θ + δθ) log

p(y; θ + δθ)

p(y; θ)dθ ≈ δθTG(θ)δθ

where

G(θ) = Ey|θ

(∇θp(y; θ)

p(y; θ)

∇θp(y; θ)

p(y; θ)

T)

I Fisher Information G(θ) is p.d. metric defining a Riemann manifoldI Non-Euclidean geometry for probabilities - distances, metrics, invariants,

curvature, geodesicsI Asymptotic statistical analysis. Amari, 1981, 85, 90; Murray & Rice,

1993; Critchley et al, 1993; Kass, 1989; Efron, 1975; Dawid, 1975;Lauritsen, 1989

I Statistical shape analysis Kent et al, 1996; Dryden & Mardia, 1998

I Can geometric structure be employed in MCMC methodology?

Geometric Concepts in MCMC

I Rao, 1945; Jeffreys, 1948, to first orderZp(y; θ + δθ) log

p(y; θ + δθ)

p(y; θ)dθ ≈ δθTG(θ)δθ

where

G(θ) = Ey|θ

(∇θp(y; θ)

p(y; θ)

∇θp(y; θ)

p(y; θ)

T)

I Fisher Information G(θ) is p.d. metric defining a Riemann manifoldI Non-Euclidean geometry for probabilities - distances, metrics, invariants,

curvature, geodesicsI Asymptotic statistical analysis. Amari, 1981, 85, 90; Murray & Rice,

1993; Critchley et al, 1993; Kass, 1989; Efron, 1975; Dawid, 1975;Lauritsen, 1989

I Statistical shape analysis Kent et al, 1996; Dryden & Mardia, 1998

I Can geometric structure be employed in MCMC methodology?

Geometric Concepts in MCMC

I Rao, 1945; Jeffreys, 1948, to first orderZp(y; θ + δθ) log

p(y; θ + δθ)

p(y; θ)dθ ≈ δθTG(θ)δθ

where

G(θ) = Ey|θ

(∇θp(y; θ)

p(y; θ)

∇θp(y; θ)

p(y; θ)

T)

I Fisher Information G(θ) is p.d. metric defining a Riemann manifoldI Non-Euclidean geometry for probabilities - distances, metrics, invariants,

curvature, geodesicsI Asymptotic statistical analysis. Amari, 1981, 85, 90; Murray & Rice,

1993; Critchley et al, 1993; Kass, 1989; Efron, 1975; Dawid, 1975;Lauritsen, 1989

I Statistical shape analysis Kent et al, 1996; Dryden & Mardia, 1998

I Can geometric structure be employed in MCMC methodology?

Geometric Concepts in MCMC

I Tangent space - local metric defined by δθTG(θ)δθ =P

k,l gklδθkδθl

I Christoffel symbols - characterise connection on curved manifold

Γikl =

12

Xm

g im„∂gmk

∂θl +∂gml

∂θk −∂gkl

∂θm

«I Geodesics - shortest path between two points on manifold

d2θi

dt2 +Xk,l

Γikl

dθk

dtdθl

dt= 0

Geometric Concepts in MCMC

I Tangent space - local metric defined by δθTG(θ)δθ =P

k,l gklδθkδθl

I Christoffel symbols - characterise connection on curved manifold

Γikl =

12

Xm

g im„∂gmk

∂θl +∂gml

∂θk −∂gkl

∂θm

«I Geodesics - shortest path between two points on manifold

d2θi

dt2 +Xk,l

Γikl

dθk

dtdθl

dt= 0

Geometric Concepts in MCMC

I Tangent space - local metric defined by δθTG(θ)δθ =P

k,l gklδθkδθl

I Christoffel symbols - characterise connection on curved manifold

Γikl =

12

Xm

g im„∂gmk

∂θl +∂gml

∂θk −∂gkl

∂θm

«I Geodesics - shortest path between two points on manifold

d2θi

dt2 +Xk,l

Γikl

dθk

dtdθl

dt= 0

Geometric Concepts in MCMC

I Tangent space - local metric defined by δθTG(θ)δθ =P

k,l gklδθkδθl

I Christoffel symbols - characterise connection on curved manifold

Γikl =

12

Xm

g im„∂gmk

∂θl +∂gml

∂θk −∂gkl

∂θm

«

I Geodesics - shortest path between two points on manifold

d2θi

dt2 +Xk,l

Γikl

dθk

dtdθl

dt= 0

Geometric Concepts in MCMC

I Tangent space - local metric defined by δθTG(θ)δθ =P

k,l gklδθkδθl

I Christoffel symbols - characterise connection on curved manifold

Γikl =

12

Xm

g im„∂gmk

∂θl +∂gml

∂θk −∂gkl

∂θm

«I Geodesics - shortest path between two points on manifold

d2θi

dt2 +Xk,l

Γikl

dθk

dtdθl

dt= 0

Geometric Concepts in MCMC

I Tangent space - local metric defined by δθTG(θ)δθ =P

k,l gklδθkδθl

I Christoffel symbols - characterise connection on curved manifold

Γikl =

12

Xm

g im„∂gmk

∂θl +∂gml

∂θk −∂gkl

∂θm

«I Geodesics - shortest path between two points on manifold

d2θi

dt2 +Xk,l

Γikl

dθk

dtdθl

dt= 0

Illustration of Geometric Concepts

I Consider Normal density p(x |µ, σ) = Nx (µ, σ)

I Local inner product on tangent space defined by metric tensor, i.e.δθTG(θ)δθ, where θ = (µ, σ)T

I Metric is Fisher Information

G(µ, σ) =

»σ−2 0

0 2σ−2

–

I Inner-product σ−2(δµ2 + 2δσ2) so densities N (0, 1) & N (1, 1) furtherapart than the densities N (0, 2) & N (1, 2) - distance non-Euclidean

I Metric tensor for univariate Normal defines a Hyperbolic Space

Illustration of Geometric Concepts

I Consider Normal density p(x |µ, σ) = Nx (µ, σ)

I Local inner product on tangent space defined by metric tensor, i.e.δθTG(θ)δθ, where θ = (µ, σ)T

I Metric is Fisher Information

G(µ, σ) =

»σ−2 0

0 2σ−2

–

I Inner-product σ−2(δµ2 + 2δσ2) so densities N (0, 1) & N (1, 1) furtherapart than the densities N (0, 2) & N (1, 2) - distance non-Euclidean

I Metric tensor for univariate Normal defines a Hyperbolic Space

Illustration of Geometric Concepts

I Consider Normal density p(x |µ, σ) = Nx (µ, σ)

I Local inner product on tangent space defined by metric tensor, i.e.δθTG(θ)δθ, where θ = (µ, σ)T

I Metric is Fisher Information

G(µ, σ) =

»σ−2 0

0 2σ−2

–

I Inner-product σ−2(δµ2 + 2δσ2) so densities N (0, 1) & N (1, 1) furtherapart than the densities N (0, 2) & N (1, 2) - distance non-Euclidean

I Metric tensor for univariate Normal defines a Hyperbolic Space

Illustration of Geometric Concepts

I Consider Normal density p(x |µ, σ) = Nx (µ, σ)

I Local inner product on tangent space defined by metric tensor, i.e.δθTG(θ)δθ, where θ = (µ, σ)T

I Metric is Fisher Information

G(µ, σ) =

»σ−2 0

0 2σ−2

–

I Inner-product σ−2(δµ2 + 2δσ2) so densities N (0, 1) & N (1, 1) furtherapart than the densities N (0, 2) & N (1, 2) - distance non-Euclidean

I Metric tensor for univariate Normal defines a Hyperbolic Space

Illustration of Geometric Concepts

I Consider Normal density p(x |µ, σ) = Nx (µ, σ)

I Local inner product on tangent space defined by metric tensor, i.e.δθTG(θ)δθ, where θ = (µ, σ)T

I Metric is Fisher Information

G(µ, σ) =

»σ−2 0

0 2σ−2

–

I Inner-product σ−2(δµ2 + 2δσ2) so densities N (0, 1) & N (1, 1) furtherapart than the densities N (0, 2) & N (1, 2) - distance non-Euclidean

I Metric tensor for univariate Normal defines a Hyperbolic Space

M.C. Escher, Heaven and Hell, 1960

Langevin Diffusion on Riemannian manifold

I Discretised Langevin diffusion on manifold defines proposal mechanism

θ′d = θd +ε2

2

“G−1(θ)∇θL(θ)

”d− ε2

DXi,j

G(θ)−1ij Γd

ij + ε“p

G−1(θ)z”

d

I Manifold with constant curvature then proposal mechanism reduces to

θ′ = θ +ε2

2G−1(θ)∇θL(θ) + ε

pG−1(θ)z

I MALA proposal with preconditioning

θ′ = θ +ε2

2M∇θL(θ) + ε

√Mz

I Proposal and acceptance probability

pp(θ′|θ) = N (θ′|µ(θ, ε), ε2G(θ))

pa(θ′|θ) = min»1,

p(θ′)pp(θ|θ′)p(θ)pp(θ′|θ)

–I Proposal mechanism diffuses approximately along the manifold

Langevin Diffusion on Riemannian manifold

I Discretised Langevin diffusion on manifold defines proposal mechanism

θ′d = θd +ε2

2

“G−1(θ)∇θL(θ)

”d− ε2

DXi,j

G(θ)−1ij Γd

ij + ε“p

G−1(θ)z”

d

I Manifold with constant curvature then proposal mechanism reduces to

θ′ = θ +ε2

2G−1(θ)∇θL(θ) + ε

pG−1(θ)z

I MALA proposal with preconditioning

θ′ = θ +ε2

2M∇θL(θ) + ε

√Mz

I Proposal and acceptance probability

pp(θ′|θ) = N (θ′|µ(θ, ε), ε2G(θ))

pa(θ′|θ) = min»1,

p(θ′)pp(θ|θ′)p(θ)pp(θ′|θ)

–I Proposal mechanism diffuses approximately along the manifold

Langevin Diffusion on Riemannian manifold

I Discretised Langevin diffusion on manifold defines proposal mechanism

θ′d = θd +ε2

2

“G−1(θ)∇θL(θ)

”d− ε2

DXi,j

G(θ)−1ij Γd

ij + ε“p

G−1(θ)z”

d

I Manifold with constant curvature then proposal mechanism reduces to

θ′ = θ +ε2

2G−1(θ)∇θL(θ) + ε

pG−1(θ)z

I MALA proposal with preconditioning

θ′ = θ +ε2

2M∇θL(θ) + ε

√Mz

I Proposal and acceptance probability

pp(θ′|θ) = N (θ′|µ(θ, ε), ε2G(θ))

pa(θ′|θ) = min»1,

p(θ′)pp(θ|θ′)p(θ)pp(θ′|θ)

–I Proposal mechanism diffuses approximately along the manifold

Langevin Diffusion on Riemannian manifold

I Discretised Langevin diffusion on manifold defines proposal mechanism

θ′d = θd +ε2

2

“G−1(θ)∇θL(θ)

”d− ε2

DXi,j

G(θ)−1ij Γd

ij + ε“p

G−1(θ)z”

d

I Manifold with constant curvature then proposal mechanism reduces to

θ′ = θ +ε2

2G−1(θ)∇θL(θ) + ε

pG−1(θ)z

I MALA proposal with preconditioning

θ′ = θ +ε2

2M∇θL(θ) + ε

√Mz

I Proposal and acceptance probability

pp(θ′|θ) = N (θ′|µ(θ, ε), ε2G(θ))

pa(θ′|θ) = min»1,

p(θ′)pp(θ|θ′)p(θ)pp(θ′|θ)

–

I Proposal mechanism diffuses approximately along the manifold

Langevin Diffusion on Riemannian manifold

I Discretised Langevin diffusion on manifold defines proposal mechanism

θ′d = θd +ε2

2

“G−1(θ)∇θL(θ)

”d− ε2

DXi,j

G(θ)−1ij Γd

ij + ε“p

G−1(θ)z”

d

I Manifold with constant curvature then proposal mechanism reduces to

θ′ = θ +ε2

2G−1(θ)∇θL(θ) + ε

pG−1(θ)z

I MALA proposal with preconditioning

θ′ = θ +ε2

2M∇θL(θ) + ε

√Mz

I Proposal and acceptance probability

pp(θ′|θ) = N (θ′|µ(θ, ε), ε2G(θ))

pa(θ′|θ) = min»1,

p(θ′)pp(θ|θ′)p(θ)pp(θ′|θ)

–I Proposal mechanism diffuses approximately along the manifold

Langevin Diffusion on Riemannian manifold

−5 0 55

10

15

20

25

30

35

40

µ

!

−5 0 55

10

15

20

25

30

35

40

µ

!

Langevin Diffusion on Riemannian manifold

−10 0 10

5

10

15

20

25

µ

!

−10 0 10

5

10

15

20

25

µ

!

Geodesic flow as proposal mechanism

I Desirable that proposals follow direct path on manifold - geodesics

I Define geodesic flow on manifold by solving

d2θi

dt2 +Xk,l

Γikl

dθk

dtdθl

dt= 0

I How can this be exploited in the design of a transition operator?

I Need slight detour - first define log-density under model as L(θ)

I Introduce auxiliary variable p ∼ N (0,G(θ))

I Negative joint log density is

H(θ,p) = −L(θ) +12

log 2πD|G(θ)|+ 12

pTG(θ)−1p

Geodesic flow as proposal mechanism

I Desirable that proposals follow direct path on manifold - geodesics

I Define geodesic flow on manifold by solving

d2θi

dt2 +Xk,l

Γikl

dθk

dtdθl

dt= 0

I How can this be exploited in the design of a transition operator?

I Need slight detour - first define log-density under model as L(θ)

I Introduce auxiliary variable p ∼ N (0,G(θ))

I Negative joint log density is

H(θ,p) = −L(θ) +12

log 2πD|G(θ)|+ 12

pTG(θ)−1p

Geodesic flow as proposal mechanism

I Desirable that proposals follow direct path on manifold - geodesics

I Define geodesic flow on manifold by solving

d2θi

dt2 +Xk,l

Γikl

dθk

dtdθl

dt= 0

I How can this be exploited in the design of a transition operator?

I Need slight detour - first define log-density under model as L(θ)

I Introduce auxiliary variable p ∼ N (0,G(θ))

I Negative joint log density is

H(θ,p) = −L(θ) +12

log 2πD|G(θ)|+ 12

pTG(θ)−1p

Geodesic flow as proposal mechanism

I Desirable that proposals follow direct path on manifold - geodesics

I Define geodesic flow on manifold by solving

d2θi

dt2 +Xk,l

Γikl

dθk

dtdθl

dt= 0

I How can this be exploited in the design of a transition operator?

I Need slight detour

- first define log-density under model as L(θ)

I Introduce auxiliary variable p ∼ N (0,G(θ))

I Negative joint log density is

H(θ,p) = −L(θ) +12

log 2πD|G(θ)|+ 12

pTG(θ)−1p

Geodesic flow as proposal mechanism

I Desirable that proposals follow direct path on manifold - geodesics

I Define geodesic flow on manifold by solving

d2θi

dt2 +Xk,l

Γikl

dθk

dtdθl

dt= 0

I How can this be exploited in the design of a transition operator?

I Need slight detour - first define log-density under model as L(θ)

I Introduce auxiliary variable p ∼ N (0,G(θ))

I Negative joint log density is

H(θ,p) = −L(θ) +12

log 2πD|G(θ)|+ 12

pTG(θ)−1p

Geodesic flow as proposal mechanism

I Desirable that proposals follow direct path on manifold - geodesics

I Define geodesic flow on manifold by solving

d2θi

dt2 +Xk,l

Γikl

dθk

dtdθl

dt= 0

I How can this be exploited in the design of a transition operator?

I Need slight detour - first define log-density under model as L(θ)

I Introduce auxiliary variable p ∼ N (0,G(θ))

I Negative joint log density is

H(θ,p) = −L(θ) +12

log 2πD|G(θ)|+ 12

pTG(θ)−1p

Geodesic flow as proposal mechanism

I Desirable that proposals follow direct path on manifold - geodesics

I Define geodesic flow on manifold by solving

d2θi

dt2 +Xk,l

Γikl

dθk

dtdθl

dt= 0

I How can this be exploited in the design of a transition operator?

I Need slight detour - first define log-density under model as L(θ)

I Introduce auxiliary variable p ∼ N (0,G(θ))

I Negative joint log density is

H(θ,p) = −L(θ) +12

log 2πD|G(θ)|+ 12

pTG(θ)−1p

Riemannian Hamiltonian Monte CarloI Negative joint log-density ≡ Hamiltonian defined on Riemann manifold

H(θ,p) = −L(θ) +12

log 2πD|G(θ)|| {z }Potential Energy

+12

pTG(θ)−1p| {z }Kinetic Energy

I Marginal density follows as required

p(θ) ∝ exp {L(θ)}p2πD|G(θ)|

Zexp

−1

2pTG(θ)−1p

ffdp = exp {L(θ)}

I Obtain samples from marginal p(θ) using Gibbs sampler for p(θ,p)

pn+1|θn ∼ N (0,G(θn))

θn+1|pn+1 ∼ p(θn+1|pn+1)

I Employ Hamiltonian dynamics to propose samples for p(θn+1|pn+1),Duane et al, 1987; Neal, 2010.

dθ

dt=

∂

∂pH(θ,p)

dpdt

= − ∂

∂θH(θ,p)

Riemannian Hamiltonian Monte CarloI Negative joint log-density ≡ Hamiltonian defined on Riemann manifold

H(θ,p) = −L(θ) +12

log 2πD|G(θ)|| {z }Potential Energy

+12

pTG(θ)−1p| {z }Kinetic Energy

I Marginal density follows as required

p(θ) ∝ exp {L(θ)}p2πD|G(θ)|

Zexp

−1

2pTG(θ)−1p

ffdp = exp {L(θ)}

I Obtain samples from marginal p(θ) using Gibbs sampler for p(θ,p)

pn+1|θn ∼ N (0,G(θn))

θn+1|pn+1 ∼ p(θn+1|pn+1)

I Employ Hamiltonian dynamics to propose samples for p(θn+1|pn+1),Duane et al, 1987; Neal, 2010.

dθ

dt=

∂

∂pH(θ,p)

dpdt

= − ∂

∂θH(θ,p)

Riemannian Hamiltonian Monte CarloI Negative joint log-density ≡ Hamiltonian defined on Riemann manifold

H(θ,p) = −L(θ) +12

log 2πD|G(θ)|| {z }Potential Energy

+12

pTG(θ)−1p| {z }Kinetic Energy

I Marginal density follows as required

p(θ) ∝ exp {L(θ)}p2πD|G(θ)|

Zexp

−1

2pTG(θ)−1p

ffdp = exp {L(θ)}

I Obtain samples from marginal p(θ) using Gibbs sampler for p(θ,p)

pn+1|θn ∼ N (0,G(θn))

θn+1|pn+1 ∼ p(θn+1|pn+1)

I Employ Hamiltonian dynamics to propose samples for p(θn+1|pn+1),Duane et al, 1987; Neal, 2010.

dθ

dt=

∂

∂pH(θ,p)

dpdt

= − ∂

∂θH(θ,p)

Riemannian Hamiltonian Monte CarloI Negative joint log-density ≡ Hamiltonian defined on Riemann manifold

H(θ,p) = −L(θ) +12

log 2πD|G(θ)|| {z }Potential Energy

+12

pTG(θ)−1p| {z }Kinetic Energy

I Marginal density follows as required

p(θ) ∝ exp {L(θ)}p2πD|G(θ)|

Zexp

−1

2pTG(θ)−1p

ffdp = exp {L(θ)}

I Obtain samples from marginal p(θ) using Gibbs sampler for p(θ,p)

pn+1|θn ∼ N (0,G(θn))

θn+1|pn+1 ∼ p(θn+1|pn+1)

I Employ Hamiltonian dynamics to propose samples for p(θn+1|pn+1),Duane et al, 1987; Neal, 2010.

dθ

dt=

∂

∂pH(θ,p)

dpdt

= − ∂

∂θH(θ,p)

Riemannian Hamiltonian Monte CarloI Negative joint log-density ≡ Hamiltonian defined on Riemann manifold

H(θ,p) = −L(θ) +12

log 2πD|G(θ)|| {z }Potential Energy

+12

pTG(θ)−1p| {z }Kinetic Energy

I Marginal density follows as required

p(θ) ∝ exp {L(θ)}p2πD|G(θ)|

Zexp

−1

2pTG(θ)−1p

ffdp = exp {L(θ)}

I Obtain samples from marginal p(θ) using Gibbs sampler for p(θ,p)

pn+1|θn ∼ N (0,G(θn))

θn+1|pn+1 ∼ p(θn+1|pn+1)

I Employ Hamiltonian dynamics to propose samples for p(θn+1|pn+1),Duane et al, 1987; Neal, 2010.

dθ

dt=

∂

∂pH(θ,p)

dpdt

= − ∂

∂θH(θ,p)

Riemannian Manifold Hamiltonian Monte Carlo

I Consider the Hamiltonian H̃(θ,p) = 12 pTG̃(θ)−1p

I Hamiltonians with only a quadratic kinetic energy term exactly describegeodesic flow on the coordinate space θ with metric G̃

I However our Hamiltonian is H(θ,p) = V (θ) + 12 pTG(θ)−1p

I If we define G̃(θ) = G(θ)× (h − V (θ)), where h is a constant H(θ,p)

I Then the Maupertuis principle tells us that the Hamiltonian flow forH(θ,p) and H̃(θ,p) are equivalent along energy level h

I The solution of

dθ

dt=

∂

∂pH(θ,p)

dpdt

= − ∂

∂θH(θ,p)

is therefore equivalent to the solution of

d2θi

dt2 +Xk,l

Γ̃ikl

dθk

dtdθl

dt= 0

I RMHMC proposals are along the manifold geodesics

Riemannian Manifold Hamiltonian Monte Carlo

I Consider the Hamiltonian H̃(θ,p) = 12 pTG̃(θ)−1p

I Hamiltonians with only a quadratic kinetic energy term exactly describegeodesic flow on the coordinate space θ with metric G̃

I However our Hamiltonian is H(θ,p) = V (θ) + 12 pTG(θ)−1p

I If we define G̃(θ) = G(θ)× (h − V (θ)), where h is a constant H(θ,p)

I Then the Maupertuis principle tells us that the Hamiltonian flow forH(θ,p) and H̃(θ,p) are equivalent along energy level h

I The solution of

dθ

dt=

∂

∂pH(θ,p)

dpdt

= − ∂

∂θH(θ,p)

is therefore equivalent to the solution of

d2θi

dt2 +Xk,l

Γ̃ikl

dθk

dtdθl

dt= 0

I RMHMC proposals are along the manifold geodesics

Riemannian Manifold Hamiltonian Monte Carlo

I Consider the Hamiltonian H̃(θ,p) = 12 pTG̃(θ)−1p

I Hamiltonians with only a quadratic kinetic energy term exactly describegeodesic flow on the coordinate space θ with metric G̃

I However our Hamiltonian is H(θ,p) = V (θ) + 12 pTG(θ)−1p

I If we define G̃(θ) = G(θ)× (h − V (θ)), where h is a constant H(θ,p)

I Then the Maupertuis principle tells us that the Hamiltonian flow forH(θ,p) and H̃(θ,p) are equivalent along energy level h

I The solution of

dθ

dt=

∂

∂pH(θ,p)

dpdt

= − ∂

∂θH(θ,p)

is therefore equivalent to the solution of

d2θi

dt2 +Xk,l

Γ̃ikl

dθk

dtdθl

dt= 0

I RMHMC proposals are along the manifold geodesics

Riemannian Manifold Hamiltonian Monte Carlo

I Consider the Hamiltonian H̃(θ,p) = 12 pTG̃(θ)−1p

I Hamiltonians with only a quadratic kinetic energy term exactly describegeodesic flow on the coordinate space θ with metric G̃

I However our Hamiltonian is H(θ,p) = V (θ) + 12 pTG(θ)−1p

I If we define G̃(θ) = G(θ)× (h − V (θ)), where h is a constant H(θ,p)

I Then the Maupertuis principle tells us that the Hamiltonian flow forH(θ,p) and H̃(θ,p) are equivalent along energy level h

I The solution of

dθ

dt=

∂

∂pH(θ,p)

dpdt

= − ∂

∂θH(θ,p)

is therefore equivalent to the solution of

d2θi

dt2 +Xk,l

Γ̃ikl

dθk

dtdθl

dt= 0

I RMHMC proposals are along the manifold geodesics

Riemannian Manifold Hamiltonian Monte Carlo

I Consider the Hamiltonian H̃(θ,p) = 12 pTG̃(θ)−1p

I Hamiltonians with only a quadratic kinetic energy term exactly describegeodesic flow on the coordinate space θ with metric G̃

I However our Hamiltonian is H(θ,p) = V (θ) + 12 pTG(θ)−1p

I If we define G̃(θ) = G(θ)× (h − V (θ)), where h is a constant H(θ,p)

I Then the Maupertuis principle tells us that the Hamiltonian flow forH(θ,p) and H̃(θ,p) are equivalent along energy level h

I The solution of

dθ

dt=

∂

∂pH(θ,p)

dpdt

= − ∂

∂θH(θ,p)

is therefore equivalent to the solution of

d2θi

dt2 +Xk,l

Γ̃ikl

dθk

dtdθl

dt= 0

I RMHMC proposals are along the manifold geodesics

Riemannian Manifold Hamiltonian Monte Carlo

I Consider the Hamiltonian H̃(θ,p) = 12 pTG̃(θ)−1p

I Hamiltonians with only a quadratic kinetic energy term exactly describegeodesic flow on the coordinate space θ with metric G̃

I However our Hamiltonian is H(θ,p) = V (θ) + 12 pTG(θ)−1p

I If we define G̃(θ) = G(θ)× (h − V (θ)), where h is a constant H(θ,p)

I Then the Maupertuis principle tells us that the Hamiltonian flow forH(θ,p) and H̃(θ,p) are equivalent along energy level h

I The solution of

dθ

dt=

∂

∂pH(θ,p)

dpdt

= − ∂

∂θH(θ,p)

is therefore equivalent to the solution of

d2θi

dt2 +Xk,l

Γ̃ikl

dθk

dtdθl

dt= 0

I RMHMC proposals are along the manifold geodesics

Riemannian Manifold Hamiltonian Monte Carlo

I Consider the Hamiltonian H̃(θ,p) = 12 pTG̃(θ)−1p

I Hamiltonians with only a quadratic kinetic energy term exactly describegeodesic flow on the coordinate space θ with metric G̃

I However our Hamiltonian is H(θ,p) = V (θ) + 12 pTG(θ)−1p

I If we define G̃(θ) = G(θ)× (h − V (θ)), where h is a constant H(θ,p)

I Then the Maupertuis principle tells us that the Hamiltonian flow forH(θ,p) and H̃(θ,p) are equivalent along energy level h

I The solution of

dθ

dt=

∂

∂pH(θ,p)

dpdt

= − ∂

∂θH(θ,p)

is therefore equivalent to the solution of

d2θi

dt2 +Xk,l

Γ̃ikl

dθk

dtdθl

dt= 0

I RMHMC proposals are along the manifold geodesics

Riemannian Manifold Hamiltonian Monte Carlo

I Consider the Hamiltonian H̃(θ,p) = 12 pTG̃(θ)−1p

I Hamiltonians with only a quadratic kinetic energy term exactly describegeodesic flow on the coordinate space θ with metric G̃

I However our Hamiltonian is H(θ,p) = V (θ) + 12 pTG(θ)−1p

I If we define G̃(θ) = G(θ)× (h − V (θ)), where h is a constant H(θ,p)

I Then the Maupertuis principle tells us that the Hamiltonian flow forH(θ,p) and H̃(θ,p) are equivalent along energy level h

I The solution of

dθ

dt=

∂

∂pH(θ,p)

dpdt

= − ∂

∂θH(θ,p)

is therefore equivalent to the solution of

d2θi

dt2 +Xk,l

Γ̃ikl

dθk

dtdθl

dt= 0

I RMHMC proposals are along the manifold geodesics

Warped Bivariate GaussianI p(w1,w2|y, σx , σy ) ∝

QNn=1N (yn|w1 + w2

2 , σ2y )N (w1,w2|0, σ2

x I)

-4 -3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3

-4 -3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3

w! w!

w" w"

HMC RMHMC

Warped Bivariate GaussianI p(w1,w2|y, σx , σy ) ∝

QNn=1N (yn|w1 + w2

2 , σ2y )N (w1,w2|0, σ2

x I)

-4 -3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3

-4 -3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3

w! w!

w" w"

HMC RMHMC

Gaussian Mixture Model

I Univariate finite mixture model

p(xi |θ) =KX

k=1

πkN (xi |µk , σ2k )

I FI based metric tensor non-analytic - employ empirical FI

G(θ) =1N

ST S − 1N2 s̄s̄T −−−−→

N→∞cov (∇θL(θ)) = I(θ)

∂G(θ)

∂θd=

1N

∂ST

∂θdS + ST ∂S

∂θd

!− 1

N2

„∂s̄∂θd

s̄T + s̄∂s̄T

∂θd

«with score matrix S with elements Si,d = ∂ log p(xi |θ)

∂θdand s̄ =

PNi=1 ST

i,·

Gaussian Mixture Model

I Univariate finite mixture model

p(xi |θ) =KX

k=1

πkN (xi |µk , σ2k )

I FI based metric tensor non-analytic - employ empirical FI

G(θ) =1N

ST S − 1N2 s̄s̄T −−−−→

N→∞cov (∇θL(θ)) = I(θ)

∂G(θ)

∂θd=

1N

∂ST

∂θdS + ST ∂S

∂θd

!− 1

N2

„∂s̄∂θd

s̄T + s̄∂s̄T

∂θd

«with score matrix S with elements Si,d = ∂ log p(xi |θ)

∂θdand s̄ =

PNi=1 ST

i,·

Gaussian Mixture Model

I Univariate finite mixture model

p(xi |θ) =KX

k=1

πkN (xi |µk , σ2k )

I FI based metric tensor non-analytic - employ empirical FI

G(θ) =1N

ST S − 1N2 s̄s̄T −−−−→

N→∞cov (∇θL(θ)) = I(θ)

∂G(θ)

∂θd=

1N

∂ST

∂θdS + ST ∂S

∂θd

!− 1

N2

„∂s̄∂θd

s̄T + s̄∂s̄T

∂θd

«with score matrix S with elements Si,d = ∂ log p(xi |θ)

∂θdand s̄ =

PNi=1 ST

i,·

Gaussian Mixture ModelI Univariate finite mixture model

p(x |µ, σ2) = 0.7×N (x |0, σ2) + 0.3×N (x |µ, σ2)

µ

lnσ

−4 −3 −2 −1 0 1 2 3 4−4

−3

−2

−1

0

1

2

3

4

Figure: Arrows correspond to the gradients and ellipses to the inverse metrictensor. Dashed lines are isocontours of the joint log density

Gaussian Mixture ModelI Univariate finite mixture model

p(x |µ, σ2) = 0.7×N (x |0, σ2) + 0.3×N (x |µ, σ2)

µ

lnσ

−4 −3 −2 −1 0 1 2 3 4−4

−3

−2

−1

0

1

2

3

4

Figure: Arrows correspond to the gradients and ellipses to the inverse metrictensor. Dashed lines are isocontours of the joint log density

Gaussian Mixture Model

µ

lnσ

MALA path

−910

−1010

−1110

−1210

−1310

−1410

−1410

−2910

−4 −3 −2 −1 0 1 2 3 4−4

−3

−2

−1

0

1

2

3

4

µ

lnσ

mMALA path

−910

−1010

−1110

−1210

−1310

−1410

−1410

−2910

−4 −3 −2 −1 0 1 2 3 4−4

−3

−2

−1

0

1

2

3

4

µ

lnσ

Simplified mMALA path

−910

−1010

−1110

−1210

−1310

−1410

−1410

−2910

−4 −3 −2 −1 0 1 2 3 4−4

−3

−2

−1

0

1

2

3

4

0 2 4 6 8 10 12 14 16 18 20−0.2

0

0.2

0.4

0.6

0.8

Lag

Sam

ple

Auto

corr

ela

tion

MALA Sample Autocorrelation Function

0 2 4 6 8 10 12 14 16 18 20−0.2

0

0.2

0.4

0.6

0.8

Lag

Sam

ple

Auto

corr

ela

tion

mMALA Sample Autocorrelation Function

0 2 4 6 8 10 12 14 16 18 20−0.2

0

0.2

0.4

0.6

0.8

Lag

Sam

ple

Auto

corr

ela

tion

Simplified mMALA Sample Autocorrelation Function

Figure: Comparison of MALA (left), mMALA (middle) and simplified mMALA (right)convergence paths and autocorrelation plots. Autocorrelation plots are from thestationary chains, i.e. once the chains have converged to the stationary distribution.

Gaussian Mixture Model

µ

lnσ

HMC path

−910

−1010

−1110

−1210

−1310

−1410

−1410

−2910

−4 −3 −2 −1 0 1 2 3 4−4

−3

−2

−1

0

1

2

3

4

µ

lnσ

RMHMC path

−910

−1010

−1110

−1210

−1310

−1410

−1410

−2910

−4 −3 −2 −1 0 1 2 3 4−4

−3

−2

−1

0

1

2

3

4

µ

lnσ

Gibbs path

−910

−1010

−1110

−1210

−1310

−1410

−1410

−2910

−4 −3 −2 −1 0 1 2 3 4−4

−3

−2

−1

0

1

2

3

4

0 2 4 6 8 10 12 14 16 18 20−0.2

0

0.2

0.4

0.6

0.8

Lag

Sam

ple

Auto

corr

ela

tion

HMC Sample Autocorrelation Function

0 2 4 6 8 10 12 14 16 18 20−0.2

0

0.2

0.4

0.6

0.8

Lag

Sam

ple

Auto

corr

ela

tion

RMHMC Sample Autocorrelation Function

0 2 4 6 8 10 12 14 16 18 20−0.2

0

0.2

0.4

0.6

0.8

Lag

Sam

ple

Auto

corr

ela

tion

Gibbs Sample Autocorrelation Function

Figure: Comparison of HMC (left), RMHMC (middle) and GIBBS (right) convergencepaths and autocorrelation plots. Autocorrelation plots are from the stationary chains,i.e. once the chains have converged to the stationary distribution.

Log-Gaussian Cox Point Process with Latent Field

I The joint density for Poisson counts and latent Gaussian field

p(y, x|µ, σ, β) ∝Y64

i,jexp{yi,jxi,j−m exp(xi,j )} exp(−(x−µ1)TΣ−1

θ (x−µ1)/2)

I Metric tensors

G(θ)i,j =12

trace„

Σ−1θ

∂Σθ

∂θiΣ−1

θ

∂Σθ

∂θj

«G(x) = Λ + Σ−1

θ

where Λ is diagonal with elements m exp(µ+ (Σθ)i,i )

I Latent field metric tensor defining flat manifold is 4096× 4096, O(N3)obtained from parameter conditional

I MALA requires transformation of latent field to sample - with separatetuning in transient and stationary phases of Markov chain

I Manifold methods requires no pilot tuning or additional transformations

Log-Gaussian Cox Point Process with Latent Field

I The joint density for Poisson counts and latent Gaussian field

p(y, x|µ, σ, β) ∝Y64

i,jexp{yi,jxi,j−m exp(xi,j )} exp(−(x−µ1)TΣ−1

θ (x−µ1)/2)

I Metric tensors

G(θ)i,j =12

trace„

Σ−1θ

∂Σθ

∂θiΣ−1

θ

∂Σθ

∂θj

«G(x) = Λ + Σ−1

θ

where Λ is diagonal with elements m exp(µ+ (Σθ)i,i )

I Latent field metric tensor defining flat manifold is 4096× 4096, O(N3)obtained from parameter conditional

I MALA requires transformation of latent field to sample - with separatetuning in transient and stationary phases of Markov chain

I Manifold methods requires no pilot tuning or additional transformations

Log-Gaussian Cox Point Process with Latent Field

I The joint density for Poisson counts and latent Gaussian field

p(y, x|µ, σ, β) ∝Y64

i,jexp{yi,jxi,j−m exp(xi,j )} exp(−(x−µ1)TΣ−1

θ (x−µ1)/2)

I Metric tensors

G(θ)i,j =12

trace„

Σ−1θ

∂Σθ

∂θiΣ−1

θ

∂Σθ

∂θj

«G(x) = Λ + Σ−1

θ

where Λ is diagonal with elements m exp(µ+ (Σθ)i,i )

I Latent field metric tensor defining flat manifold is 4096× 4096, O(N3)obtained from parameter conditional

I MALA requires transformation of latent field to sample - with separatetuning in transient and stationary phases of Markov chain

I Manifold methods requires no pilot tuning or additional transformations

Log-Gaussian Cox Point Process with Latent Field

I The joint density for Poisson counts and latent Gaussian field

p(y, x|µ, σ, β) ∝Y64

i,jexp{yi,jxi,j−m exp(xi,j )} exp(−(x−µ1)TΣ−1

θ (x−µ1)/2)

I Metric tensors

G(θ)i,j =12

trace„

Σ−1θ

∂Σθ

∂θiΣ−1

θ

∂Σθ

∂θj

«G(x) = Λ + Σ−1

θ

where Λ is diagonal with elements m exp(µ+ (Σθ)i,i )

I Latent field metric tensor defining flat manifold is 4096× 4096, O(N3)obtained from parameter conditional

I MALA requires transformation of latent field to sample - with separatetuning in transient and stationary phases of Markov chain

I Manifold methods requires no pilot tuning or additional transformations

RMHMC for Log-Gaussian Cox Point Processes

Table: Sampling the latent variables of a Log-Gaussian Cox Process - Comparison ofsampling methods

Method Time ESS (Min, Med, Max) s/Min ESS Rel. SpeedMALA (Transient) 31,577 (3, 8, 50) 10,605 ×1MALA (Stationary) 31,118 (4, 16, 80) 7836 ×1.35

mMALA 634 (26, 84, 174) 24.1 ×440RMHMC 2936 (1951, 4545, 5000) 1.5 ×7070

Conclusion and Discussion

I Geometry of statistical models harnessed in Monte Carlo methods

I Diffusions that respect structure and curvature of space - Manifold MALA

I Geodesic flows on model manifold - RMHMC - generalisation of HMC

I Assessed on correlated & high-dimensional latent variable models

I Promising capability of methodology

I Ongoing development

I Potential bottleneck at metric tensor and Christoffel symbols

I Theoretical analysis of convergence

I Investigate alternative manifold structures

I Design and effect of metric

I Optimality of Hamiltonian flows as local geodesics

I Alternative transition kernels

I No silver bullet or cure all - new powerful methodology for MC toolkit

Conclusion and Discussion

I Geometry of statistical models harnessed in Monte Carlo methods

I Diffusions that respect structure and curvature of space - Manifold MALA

I Geodesic flows on model manifold - RMHMC - generalisation of HMC

I Assessed on correlated & high-dimensional latent variable models

I Promising capability of methodology

I Ongoing development

I Potential bottleneck at metric tensor and Christoffel symbols

I Theoretical analysis of convergence

I Investigate alternative manifold structures

I Design and effect of metric

I Optimality of Hamiltonian flows as local geodesics

I Alternative transition kernels

I No silver bullet or cure all - new powerful methodology for MC toolkit

Conclusion and Discussion

I Geometry of statistical models harnessed in Monte Carlo methods

I Diffusions that respect structure and curvature of space - Manifold MALA

I Geodesic flows on model manifold - RMHMC - generalisation of HMC

I Assessed on correlated & high-dimensional latent variable models

I Promising capability of methodology

I Ongoing development

I Potential bottleneck at metric tensor and Christoffel symbols

I Theoretical analysis of convergence

I Investigate alternative manifold structures

I Design and effect of metric

I Optimality of Hamiltonian flows as local geodesics

I Alternative transition kernels

I No silver bullet or cure all - new powerful methodology for MC toolkit

Conclusion and Discussion

I Geometry of statistical models harnessed in Monte Carlo methods

I Diffusions that respect structure and curvature of space - Manifold MALA

I Geodesic flows on model manifold - RMHMC - generalisation of HMC

I Assessed on correlated & high-dimensional latent variable models

I Promising capability of methodology

I Ongoing development

I Potential bottleneck at metric tensor and Christoffel symbols

I Theoretical analysis of convergence

I Investigate alternative manifold structures

I Design and effect of metric

I Optimality of Hamiltonian flows as local geodesics

I Alternative transition kernels

I No silver bullet or cure all - new powerful methodology for MC toolkit

Conclusion and Discussion

I Geometry of statistical models harnessed in Monte Carlo methods

I Diffusions that respect structure and curvature of space - Manifold MALA

I Geodesic flows on model manifold - RMHMC - generalisation of HMC

I Assessed on correlated & high-dimensional latent variable models

I Promising capability of methodology

I Ongoing development

I Potential bottleneck at metric tensor and Christoffel symbols

I Theoretical analysis of convergence

I Investigate alternative manifold structures

I Design and effect of metric

I Optimality of Hamiltonian flows as local geodesics

I Alternative transition kernels

I No silver bullet or cure all - new powerful methodology for MC toolkit

Conclusion and Discussion

I Geometry of statistical models harnessed in Monte Carlo methods

I Diffusions that respect structure and curvature of space - Manifold MALA

I Geodesic flows on model manifold - RMHMC - generalisation of HMC

I Assessed on correlated & high-dimensional latent variable models

I Promising capability of methodology

I Ongoing development

I Potential bottleneck at metric tensor and Christoffel symbols

I Theoretical analysis of convergence

I Investigate alternative manifold structures

I Design and effect of metric

I Optimality of Hamiltonian flows as local geodesics

I Alternative transition kernels

I No silver bullet or cure all - new powerful methodology for MC toolkit

Conclusion and Discussion

I Geometry of statistical models harnessed in Monte Carlo methods

I Diffusions that respect structure and curvature of space - Manifold MALA

I Geodesic flows on model manifold - RMHMC - generalisation of HMC

I Assessed on correlated & high-dimensional latent variable models

I Promising capability of methodology

I Ongoing development

I Potential bottleneck at metric tensor and Christoffel symbols

I Theoretical analysis of convergence

I Investigate alternative manifold structures

I Design and effect of metric

I Optimality of Hamiltonian flows as local geodesics

I Alternative transition kernels

I No silver bullet or cure all - new powerful methodology for MC toolkit

Conclusion and Discussion

I Geometry of statistical models harnessed in Monte Carlo methods

I Diffusions that respect structure and curvature of space - Manifold MALA

I Geodesic flows on model manifold - RMHMC - generalisation of HMC

I Assessed on correlated & high-dimensional latent variable models

I Promising capability of methodology

I Ongoing development

I Potential bottleneck at metric tensor and Christoffel symbols

I Theoretical analysis of convergence

I Investigate alternative manifold structures

I Design and effect of metric

I Optimality of Hamiltonian flows as local geodesics

I Alternative transition kernels

I No silver bullet or cure all - new powerful methodology for MC toolkit

Conclusion and Discussion

I Geometry of statistical models harnessed in Monte Carlo methods

I Diffusions that respect structure and curvature of space - Manifold MALA

I Geodesic flows on model manifold - RMHMC - generalisation of HMC

I Assessed on correlated & high-dimensional latent variable models

I Promising capability of methodology

I Ongoing development

I Potential bottleneck at metric tensor and Christoffel symbols

I Theoretical analysis of convergence

I Investigate alternative manifold structures

I Design and effect of metric

I Optimality of Hamiltonian flows as local geodesics

I Alternative transition kernels

I No silver bullet or cure all - new powerful methodology for MC toolkit

Conclusion and Discussion

I Geometry of statistical models harnessed in Monte Carlo methods

I Diffusions that respect structure and curvature of space - Manifold MALA

I Geodesic flows on model manifold - RMHMC - generalisation of HMC

I Assessed on correlated & high-dimensional latent variable models

I Promising capability of methodology

I Ongoing development

I Potential bottleneck at metric tensor and Christoffel symbols

I Theoretical analysis of convergence

I Investigate alternative manifold structures

I Design and effect of metric

I Optimality of Hamiltonian flows as local geodesics

I Alternative transition kernels

I No silver bullet or cure all - new powerful methodology for MC toolkit

Conclusion and Discussion

I Geometry of statistical models harnessed in Monte Carlo methods

I Diffusions that respect structure and curvature of space - Manifold MALA

I Geodesic flows on model manifold - RMHMC - generalisation of HMC

I Assessed on correlated & high-dimensional latent variable models

I Promising capability of methodology

I Ongoing development

I Potential bottleneck at metric tensor and Christoffel symbols

I Theoretical analysis of convergence

I Investigate alternative manifold structures

I Design and effect of metric

I Optimality of Hamiltonian flows as local geodesics

I Alternative transition kernels

I No silver bullet or cure all - new powerful methodology for MC toolkit

Conclusion and Discussion

I Geometry of statistical models harnessed in Monte Carlo methods

I Diffusions that respect structure and curvature of space - Manifold MALA

I Geodesic flows on model manifold - RMHMC - generalisation of HMC

I Assessed on correlated & high-dimensional latent variable models

I Promising capability of methodology

I Ongoing development

I Potential bottleneck at metric tensor and Christoffel symbols

I Theoretical analysis of convergence

I Investigate alternative manifold structures

I Design and effect of metric

I Optimality of Hamiltonian flows as local geodesics

I Alternative transition kernels

I No silver bullet or cure all - new powerful methodology for MC toolkit

Conclusion and Discussion

I Geometry of statistical models harnessed in Monte Carlo methods

I Diffusions that respect structure and curvature of space - Manifold MALA

I Geodesic flows on model manifold - RMHMC - generalisation of HMC

I Assessed on correlated & high-dimensional latent variable models

I Promising capability of methodology

I Ongoing development

I Potential bottleneck at metric tensor and Christoffel symbols

I Theoretical analysis of convergence

I Investigate alternative manifold structures

I Design and effect of metric

I Optimality of Hamiltonian flows as local geodesics

I Alternative transition kernels

I No silver bullet or cure all - new powerful methodology for MC toolkit

Funding Acknowledgment

I Girolami funded by EPSRC Advanced Research Fellowship and BBSRCproject grant

I Calderhead supported by Microsoft Research PhD Scholarship