Bayesian Probabilistic Numerical Methods Numerical Disintegration - - PowerPoint PPT Presentation

Bayesian Probabilistic Numerical Methods

Numerical Disintegration and Pipelines

Jon Cockayne June 6, 2017

1

(Re)introduction

(Re)introduction

“Data” A(u) = a u ∼ µ Q#µa “Prior” “Information Equation” “Posterior”

2

Q1: How can we access µa?

2

(Re)introduction

Unless probabilistic numerical methods “agree” about what their uncertainty means, they cannot be composed coherently.

3

Modelling Electro-Mechanics in the Heart

4

Modelling Electro-Mechanics in the Heart

Ca Flux during Caffeine Ca transient Fit NCX Model Fit RyR Model Ca Flux during tail of Field Stimulation Ca Transient. Less Ca Flux through NCX (calculated) Fit SERCA Model ICaL Voltage clamp traces Fit ICaL Model Ca flux during start of Field Stimulation Ca

• Transient. Less Ca Flux through NCX,

SERCA and ICaL (calculated)

5

Q2: when is it “legal” to compose Bayesian PNM in pipelines?

5

Numerical Disintegration

Numerical Disintegration

Recall, the issue: X a = {u ∈ X : A(u) = a} µ(Xa) = 0

which means…

d

a

d

6

Numerical Disintegration

Recall, the issue: X a = {u ∈ X : A(u) = a} µ(Xa) = 0

which means…

∄ dµa dµ

6

Our Approach

Design an algorithm for approximately sampling µa. Two sources of error

• Intractability of

a (“Numerical Disintegration”)

• Intractability of non-Gaussian priors (“prior truncation”)

7

Our Approach

Design an algorithm for approximately sampling µa. Two sources of error

• Intractability of µa (“Numerical Disintegration”)
• Intractability of non-Gaussian priors (“prior truncation”)

7

Our Approach

Design an algorithm for approximately sampling µa. Two sources of error

• Intractability of µa (“Numerical Disintegration”)
• Intractability of non-Gaussian priors (“prior truncation”)

7

Three Considerations

Numerical Disintegration Prior Truncation

8

Three Considerations

Numerical Disintegration Prior Truncation Sampler Convergence

8

Three Considerations

Numerical Disintegration Prior Truncation Sampler Convergence

8

Numerical Disintegration

Introduce the δ-relaxed measure µa

δ…

dµa

δ

dµ ∝ ϕ (∥A(u) − a∥A δ ) a relaxation function chosen so that:

• 1
• r

0 as r .

9

Numerical Disintegration

Introduce the δ-relaxed measure µa

δ…

dµa

δ

dµ ∝ ϕ (∥A(u) − a∥A δ ) ϕ : R+ → R+ a relaxation function chosen so that:

• ϕ(0) = 1
• ϕ(r) → 0 as r → ∞.

9

Numerical Disintegration: Intuition

“Ideal” Radon–Nikodym derivative “dµa dµ ∝ I(u ∈ X a)”

10

Example Relaxation Functions

ϕ(r) = I(r < 1) ϕ(r) = exp(−r2)

11

a 1 a

Example Relaxation Functions

ϕ(r) = I(r < 1) Uniform noise over Bδ(a) ϕ(r) = exp(−r2) Gaussian noise with s.d. ∝ δ

11

a 1 a

Tempering for Sampling µa

δ

To sample µa

δ we take inspiration from rare event simulation and use tempering

schemes to sample the posterior. Set

1 N and consider a a

1

a

N

• a

0 is the prior and easy to sample.

• a

N has

N close to zero and is hard to sample.

• Intermediate distributions defjne a “ladder” which takes us from prior to posterior.

12

Tempering for Sampling µa

δ

To sample µa

δ we take inspiration from rare event simulation and use tempering

schemes to sample the posterior. Set δ0 > δ1 > . . . > δN and consider µa

δ0, µa δ1, . . . , µa δN

• a

0 is the prior and easy to sample.

• a

N has

N close to zero and is hard to sample.

• Intermediate distributions defjne a “ladder” which takes us from prior to posterior.

12

Tempering for Sampling µa

δ

To sample µa

δ we take inspiration from rare event simulation and use tempering

schemes to sample the posterior. Set δ0 > δ1 > . . . > δN and consider µa

δ0, µa δ1, . . . , µa δN

• µa

δ0 is the prior and easy to sample.

• µa

δN has δN close to zero and is hard to sample.

• Intermediate distributions defjne a “ladder” which takes us from prior to posterior.

12

Example: Poisson’s Equation

Consider − d2 dx2 u(x)= sin(2πx) x ∈ (0, 1) u(x)= 0 x = 0, x = 1

• Use a Gaussian prior on u x .
• Impose boundary conditions explicitly.
• Impose interior conditions at x

1 3, x 2 3.

• Construct the posterior using ND with

1 0 10

2 10 4 .

• Use

r exp r2 .

13

Example: Poisson’s Equation

Consider − d2 dx2 u(x)= sin(2πx) x ∈ (0, 1) u(x)= 0 x = 0, x = 1

• Use a Gaussian prior on u(x).
• Impose boundary conditions explicitly.
• Impose interior conditions at x

1 3, x 2 3.

• Construct the posterior using ND with

1 0 10

2 10 4 .

• Use

r exp r2 .

13

Example: Poisson’s Equation

Consider − d2 dx2 u(x)= sin(2πx) x ∈ (0, 1) u(x)= 0 x = 0, x = 1

• Use a Gaussian prior on u(x).
• Impose boundary conditions explicitly.
• Impose interior conditions at x = 1/3, x = 2/3.
• Construct the posterior using ND with

1 0 10

2 10 4 .

• Use

r exp r2 .

13

Example: Poisson’s Equation

Consider − d2 dx2 u(x)= sin(2πx) x ∈ (0, 1) u(x)= 0 x = 0, x = 1

• Use a Gaussian prior on u(x).
• Impose boundary conditions explicitly.
• Impose interior conditions at x = 1/3, x = 2/3.
• Construct the posterior using ND with δ ∈

{ 1.0, 10−2, 10−4} .

• Use ϕ(r) = exp(−r2).

13

Example: Poisson’s Equation

In what follows, on the left are samples from the posterior µa

δ in X-space.

On the right are contours of ϕ (∥A(u) − a∥A δ ) in A-space.

14

Example: Poisson’s Equation

0.0 0.2 0.4 0.6 0.8 1.0 0.100 0.075 0.050 0.025 0.000 0.025 0.050 0.075 0.100 0.4 0.6 0.8 1.0 1.2 1.2 1.0 0.8 0.6 0.4

= 1.0

15

Example: Poisson’s Equation

0.0 0.2 0.4 0.6 0.8 1.0 0.100 0.075 0.050 0.025 0.000 0.025 0.050 0.075 0.100 0.4 0.6 0.8 1.0 1.2 1.2 1.0 0.8 0.6 0.4

= 0.01

15

Example: Poisson’s Equation

0.0 0.2 0.4 0.6 0.8 1.0 0.100 0.075 0.050 0.025 0.000 0.025 0.050 0.075 0.100 0.4 0.6 0.8 1.0 1.2 1.2 1.0 0.8 0.6 0.4

= 0.0001

15

Three Considerations

Numerical Disintegration Prior Truncation Sampler Convergence

16

Prior Construction

Assume X has a countable basis {ϕi}, i = 0, . . . , ∞. Then for any u ∈ X u(x) =

∞

∑

i=0

uiϕi(x) Difgerent

i require difgerent

for almost-sure convergence…

• i IID Uniform,

1

• i IID Gaussian,

2

• i IID Cauchy,

2

For practical computation we truncate to N terms.

17

Prior Construction

Assume X has a countable basis {ϕi}, i = 0, . . . , ∞. Then for any u ∈ X u(x) =

∞

∑

i=0

γiξiϕi(x) Difgerent ξi require difgerent γ for almost-sure convergence…

• ξi IID Uniform, γ ∈ ℓ1
• ξi IID Gaussian, γ ∈ ℓ2
• ξi IID Cauchy, γ ∈ ℓ2

For practical computation we truncate to N terms.

17

Prior Construction

Assume X has a countable basis {ϕi}, i = 0, . . . , ∞. Then for any u ∈ X uN(x) =

N

∑

i=0

γiξiϕi(x) Difgerent ξi require difgerent γ for almost-sure convergence…

• ξi IID Uniform, γ ∈ ℓ1
• ξi IID Gaussian, γ ∈ ℓ2
• ξi IID Cauchy, γ ∈ ℓ2

For practical computation we truncate to N terms.

17

Three Considerations

Numerical Disintegration Prior Truncation Sampler Convergence

18

Convergence, but in what metric?

All results show weak convergence framed in terms of an abstract integral probability metric1: dF(ν, ν′) = sup

∥f∥F≤1

• ν(f) − ν′(f)
• Results are generic to A u ,

. Examples: Total Variation, Wasserstein

1Müller [1997]

19

Convergence, but in what metric?

All results show weak convergence framed in terms of an abstract integral probability metric1: dF(ν, ν′) = sup

∥f∥F≤1

• ν(f) − ν′(f)
• Results are generic to A(u), µ.

Examples: Total Variation, Wasserstein

1Müller [1997]

19

Convergence, but in what metric?

All results show weak convergence framed in terms of an abstract integral probability metric1: dF(ν, ν′) = sup

∥f∥F≤1

• ν(f) − ν′(f)
• Results are generic to A(u), µ.

Examples: Total Variation, Wasserstein

1Müller [1997]

19

Convergence of µa

δ

Theorem Assume that dF(µa, µa′) ≤ Cµ

• a − a′

α for some Cµ, α constant and A#µ-almost-all a, a′ ∈ A. Then, for small d

a a

C 1 C for A

• almost-all a

20

Convergence of µa

δ

Theorem Assume that dF(µa, µa′) ≤ Cµ

• a − a′

α for some Cµ, α constant and A#µ-almost-all a, a′ ∈ A. Then, for small δ dF(µa

δ, µa) ≤ Cµ (1 + Cφ) δα

for A#µ-almost-all a ∈ A

20

Total Error

Denote by µa

δ,N the posterior distribution given by

dµa

δ,N

dµ ∝ ϕ (∥A ◦ PN(u) − a∥A δ )

21

Total Error

Denote by µa

δ,N the posterior distribution given by

dµa

δ,N

dµ ∝ ϕ (∥A ◦ PN(u) − a∥A δ )

21

Total Error

Denote by µa

δ,N the posterior distribution given by

dµa

δ,N

dµ ∝ ϕ (∥A ◦ PN(u) − a∥A δ ) Assume that ∥A(u) − A ◦ PN(u)∥ ≤ exp(m ∥u∥X )Φ(N)

21

Total Error

Denote by µa

δ,N the posterior distribution given by

dµa

δ,N

dµ ∝ ϕ (∥A ◦ PN(u) − a∥A δ ) Assume that ∥A(u) − A ◦ PN(u)∥ ≤ exp(m ∥u∥X )Φ(N) Then under certain assumptions it can be shown2 that: dF(µa, µa

δ,N) ≤ Cµ(1 + Cφ)δα + CδΦ(N)

2Cockayne et al. [2017]

21

Total Error

Denote by µa

δ,N the posterior distribution given by

dµa

δ,N

dµ ∝ ϕ (∥A ◦ PN(u) − a∥A δ ) Assume that ∥A(u) − A ◦ PN(u)∥ ≤ exp(m ∥u∥X )Φ(N) Then under certain assumptions it can be shown2 that: dF(µa, µa

δ,N) ≤ Cµ(1 + Cφ)δα + CδΦ(N)

Thus, we have convergence with δ provided CδΦ(N) is controlled.

2Cockayne et al. [2017]

21

Numerical Disintegration

Numerical Example

Painlevé’s First Transcendental

u′′(x) − u(x)2 = −x u(0) = 0 u(x) → √x as x → ∞

22

Painlevé’s First Transcendental

u′′(x) − u(x)2 = −x u(0) = 0 u(10) = √ 10

2 4 6 8 10

t

3 2 1 1 2 3 4

x(t)

10 20 30 40 50 60 70 80

i

10-14 10-12 10-10 10-8 10-6 10-4 10-2 100

un

Positive Negative

22

Painlevé’s First Transcendental

u′′(x) − u(x)2 = −x u(0) = 0 u(10) = √ 10 We use ϕ(x) = exp(−x2), and defjne a schedule of 1600 δ from 10 to 10−4. Following results are based on equi-spaced ti, i = 1, . . . , 15, and generated with an SMC algorithm based upon a Cauchy prior.

22

Painlevé: Posterior Measures

23

Painlevé: Posterior Measures

23

Painlevé: Posterior Measures

23

Painlevé: Posterior Measures

23

Pipelines

Example: Split Integration

∫ 1 u(x)dx = ∫ 0.5 u(x)dx + ∫ 1

0.5

u(x)dx Observations {u(x1), . . . , u(x2m)}, where u1 = 0, um = 0.5, u2m = 1 u x1 u xm

1

u xm u xm

1

u x2m B1 B2

0 5

u x dt

1 0 5 u x dt

B3

1 0 u x dt

When is the output of the pipeline Bayesian?

24

Example: Split Integration

∫ 1 u(x)dx = ∫ 0.5 u(x)dx + ∫ 1

0.5

u(x)dx Observations {u(x1), . . . , u(x2m)}, where u1 = 0, um = 0.5, u2m = 1 u(x1), . . . , u(xm−1) u(xm) u(xm+1), . . . , u(x2m) B1(µ, ·) B2(µ, ·) ∫ 0.5 u(x)dt ∫ 1

0.5 u(x)dt

B3(µ, ·) ∫ 1

0 u(x)dt

When is the output of the pipeline Bayesian?

24

Example: Split Integration

∫ 1 u(x)dx = ∫ 0.5 u(x)dx + ∫ 1

0.5

u(x)dx Observations {u(x1), . . . , u(x2m)}, where u1 = 0, um = 0.5, u2m = 1 u(x1), . . . , u(xm−1) u(xm) u(xm+1), . . . , u(x2m) B1(µ, ·) B2(µ, ·) ∫ 0.5 u(x)dt ∫ 1

0.5 u(x)dt

B3(µ, ·) ∫ 1

0 u(x)dt

When is the output of the pipeline Bayesian?

24

Dependence Graphs

The abstract structure of the graph allows us to establish a coherence condition 1 2 3 1 2 1 2 1 2 Pipeline The dependency graph of a pipeline is obtained by deleting the method nodes and connecting their inputs directly to their outputs.

25

Dependence Graphs

The abstract structure of the graph allows us to establish a coherence condition 1 2 3 4 5 6 Dependency Graph The dependency graph of a pipeline is obtained by deleting the method nodes and connecting their inputs directly to their outputs.

25

Coherence

1 2 3 4 5 6 Defjnition A prior is coherent for the dependency graph if Yk is conditionally independent of Yi given Yj. Here i, j < k, i are non-parent nodes and j are parent nodes.

26

Bayesian Pipelines

Theorem A pipeline is Bayesian for its output QoI if:

• 1. The prior is coherent for the dependence graph.
• 2. The composite PNM are Bayesian.

27

Split Integration: Coherence

u(x1), . . . , u(xm−1) u(xm) u(xm+1), . . . , u(x2m) B1(µ, ·) B2(µ, ·) ∫ 0.5 u(x)dt ∫ 1

0.5 u(x)dt

B3(µ, ·) ∫ 1

0 u(x)dt

Is

1 0 5 u x dt independent of u x1

u xm

1 ?

Sometimes - e.g. a Wiener process prior. Sometimes not - e.g. if implies a Wiener process on u s x .

28

Split Integration: Coherence

u(x1), . . . , u(xm−1) u(xm) u(xm+1), . . . , u(x2m) B1(µ, ·) B2(µ, ·) ∫ 0.5 u(x)dt ∫ 1

0.5 u(x)dt

B3(µ, ·) ∫ 1

0 u(x)dt

Is ∫ 1

0.5 u(x) dt independent of u(x1), . . . , u(xm−1)?

Sometimes - e.g. a Wiener process prior. Sometimes not - e.g. if implies a Wiener process on u s x .

28

Split Integration: Coherence

u(x1), . . . , u(xm−1) u(xm) u(xm+1), . . . , u(x2m) B1(µ, ·) B2(µ, ·) ∫ 0.5 u(x)dt ∫ 1

0.5 u(x)dt

B3(µ, ·) ∫ 1

0 u(x)dt

Is ∫ 1

0.5 u(x) dt independent of u(x1), . . . , u(xm−1)?

Sometimes - e.g. a Wiener process prior. Sometimes not - e.g. if implies a Wiener process on u s x .

28

Split Integration: Coherence

u(x1), . . . , u(xm−1) u(xm) u(xm+1), . . . , u(x2m) B1(µ, ·) B2(µ, ·) ∫ 0.5 u(x)dt ∫ 1

0.5 u(x)dt

B3(µ, ·) ∫ 1

0 u(x)dt

Is ∫ 1

0.5 u(x) dt independent of u(x1), . . . , u(xm−1)?

Sometimes - e.g. a Wiener process prior. Sometimes not - e.g. if µ implies a Wiener process on u(s)(x).

28

Conclusions

Conclusions

We have seen…

• A method for approximately sampling from µa.
• Theoretical results proving asymptotic convergence of that sampler.
• Coherence conditions for composing Bayesian PNM into a Bayesian pipeline.

29

More to come

Numerical disintegration is highly ineffjcient compared to classical numerical methods (and even conjugate PNM). This is not intended to be a practical numerical method, but a framework for comparing other “approximate Bayesian” methods to the ideal. Next steps:

• Make the algorithm more effjcient?
• Explore more effjcient approximations to the posterior

30

More to come

Numerical disintegration is highly ineffjcient compared to classical numerical methods (and even conjugate PNM). This is not intended to be a practical numerical method, but a framework for comparing other “approximate Bayesian” methods to the ideal. Next steps:

• Make the algorithm more effjcient?
• Explore more effjcient approximations to the posterior

30

More to come

Numerical disintegration is highly ineffjcient compared to classical numerical methods (and even conjugate PNM). This is not intended to be a practical numerical method, but a framework for comparing other “approximate Bayesian” methods to the ideal. Next steps:

• Make the algorithm more effjcient?
• Explore more effjcient approximations to the posterior

30

Thanks!

30

References I

• J. Cockayne, C. Oates, T. Sullivan, and M. Girolami. Bayesian probabilistic numerical methods, 2017.
• A. Müller. Integral probability metrics and their generating classes of functions. Adv. in Appl. Probab., 29(2):429–443, 1997. ISSN 0001-8678. doi:

10.2307/1428011. URL http://dx.doi.org/10.2307/1428011.

31