polychord: next generation nested sampling - sampling
TRANSCRIPT
![Page 1: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/1.jpg)
PolyChord: Next Generation Nested SamplingSampling, Parameter Estimation and Bayesian Model
Comparison
Will [email protected]
Supervisors: Anthony Lasenby & Mike HobsonAstrophysics DepartmentCavendish Laboratory
University of Cambridge
December 11, 2015
![Page 2: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/2.jpg)
Parameter estimation & model comparison
Metropolis Hastings
Nested Sampling
PolyChord
Applications
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Notation
I Data: D
I Model: M
I Parameters: Θ
I Likelihood: P(D|Θ,M) = L(Θ)
I Posterior: P(Θ|D,M) = P(Θ)
I Prior: P(Θ|M) = π(Θ)
I Evidence: P(D|M) = Z
![Page 4: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/4.jpg)
Notation
I Data: D
I Model: M
I Parameters: Θ
I Likelihood: P(D|Θ,M) = L(Θ)
I Posterior: P(Θ|D,M) = P(Θ)
I Prior: P(Θ|M) = π(Θ)
I Evidence: P(D|M) = Z
![Page 5: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/5.jpg)
Notation
I Data: D
I Model: M
I Parameters: Θ
I Likelihood: P(D|Θ,M) = L(Θ)
I Posterior: P(Θ|D,M) = P(Θ)
I Prior: P(Θ|M) = π(Θ)
I Evidence: P(D|M) = Z
![Page 6: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/6.jpg)
Notation
I Data: D
I Model: M
I Parameters: Θ
I Likelihood: P(D|Θ,M) = L(Θ)
I Posterior: P(Θ|D,M) = P(Θ)
I Prior: P(Θ|M) = π(Θ)
I Evidence: P(D|M) = Z
![Page 7: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/7.jpg)
Notation
I Data: D
I Model: M
I Parameters: Θ
I Likelihood: P(D|Θ,M) = L(Θ)
I Posterior: P(Θ|D,M) = P(Θ)
I Prior: P(Θ|M) = π(Θ)
I Evidence: P(D|M) = Z
![Page 8: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/8.jpg)
Notation
I Data: D
I Model: M
I Parameters: Θ
I Likelihood: P(D|Θ,M) = L(Θ)
I Posterior: P(Θ|D,M) = P(Θ)
I Prior: P(Θ|M) = π(Θ)
I Evidence: P(D|M) = Z
![Page 9: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/9.jpg)
Notation
I Data: D
I Model: M
I Parameters: Θ
I Likelihood: P(D|Θ,M) = L(Θ)
I Posterior: P(Θ|D,M) = P(Θ)
I Prior: P(Θ|M) = π(Θ)
I Evidence: P(D|M) = Z
![Page 10: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/10.jpg)
Notation
I Data: D
I Model: M
I Parameters: Θ
I Likelihood: P(D|Θ,M) = L(Θ)
I Posterior: P(Θ|D,M) = P(Θ)
I Prior: P(Θ|M) = π(Θ)
I Evidence: P(D|M) = Z
![Page 11: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/11.jpg)
Bayes’ theoremParameter estimation
What does the data tell us about the params Θ of our model M?
Objective: Update our prior information π(Θ) in light of data D.
π(Θ) = P(Θ|M)D−→ P(Θ|D,M) = P(Θ)
Solution: Use the likelihood L via Bayes’ theorem:
P(Θ|D,M) =P(D|Θ,M)P(Θ|M)
P(D|M)
Posterior =Likelihood× Prior
Evidence
![Page 12: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/12.jpg)
Bayes’ theoremParameter estimation
What does the data tell us about the params Θ of our model M?
Objective: Update our prior information π(Θ) in light of data D.
π(Θ) = P(Θ|M)D−→ P(Θ|D,M) = P(Θ)
Solution: Use the likelihood L via Bayes’ theorem:
P(Θ|D,M) =P(D|Θ,M)P(Θ|M)
P(D|M)
Posterior =Likelihood× Prior
Evidence
![Page 13: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/13.jpg)
Bayes’ theoremParameter estimation
What does the data tell us about the params Θ of our model M?
Objective: Update our prior information π(Θ) in light of data D.
π(Θ) = P(Θ|M)D−→ P(Θ|D,M) = P(Θ)
Solution: Use the likelihood L via Bayes’ theorem:
P(Θ|D,M) =P(D|Θ,M)P(Θ|M)
P(D|M)
Posterior =Likelihood× Prior
Evidence
![Page 14: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/14.jpg)
Bayes’ theoremParameter estimation
What does the data tell us about the params Θ of our model M?
Objective: Update our prior information π(Θ) in light of data D.
π(Θ) = P(Θ|M)D−→ P(Θ|D,M) = P(Θ)
Solution: Use the likelihood L via Bayes’ theorem:
P(Θ|D,M) =P(D|Θ,M)P(Θ|M)
P(D|M)
Posterior =Likelihood× Prior
Evidence
![Page 15: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/15.jpg)
Bayes’ theoremParameter estimation
What does the data tell us about the params Θ of our model M?
Objective: Update our prior information π(Θ) in light of data D.
π(Θ) = P(Θ|M)D−→ P(Θ|D,M) = P(Θ)
Solution: Use the likelihood L via Bayes’ theorem:
P(Θ|D,M) =P(D|Θ,M)P(Θ|M)
P(D|M)
Posterior =Likelihood× Prior
Evidence
![Page 16: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/16.jpg)
Bayes’ theoremParameter estimation
What does the data tell us about the params Θ of our model M?
Objective: Update our prior information π(Θ) in light of data D.
π(Θ) = P(Θ|M)D−→ P(Θ|D,M) = P(Θ)
Solution: Use the likelihood L via Bayes’ theorem:
P(Θ|D,M) =P(D|Θ,M)P(Θ|M)
P(D|M)
Posterior =Likelihood× Prior
Evidence
![Page 17: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/17.jpg)
Bayes’ theoremParameter estimation
What does the data tell us about the params Θ of our model M?
Objective: Update our prior information π(Θ) in light of data D.
π(Θ) = P(Θ|M)D−→ P(Θ|D,M) = P(Θ)
Solution: Use the likelihood L via Bayes’ theorem:
P(Θ|D,M) =P(D|Θ,M)P(Θ|M)
P(D|M)
Posterior =Likelihood× Prior
Evidence
![Page 18: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/18.jpg)
Bayes’ theoremModel comparison
What does the data tell us about our model Mi in relation to othermodels {M1,M2, · · · }?
P(Mi )D−→ P(Mi |D)
P(Mi |D) =P(D|Mi )P(Mi )
P(D)
P(D|Mi ) = Zi = Evidence of Mi
![Page 19: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/19.jpg)
Bayes’ theoremModel comparison
What does the data tell us about our model Mi in relation to othermodels {M1,M2, · · · }?
P(Mi )D−→ P(Mi |D)
P(Mi |D) =P(D|Mi )P(Mi )
P(D)
P(D|Mi ) = Zi = Evidence of Mi
![Page 20: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/20.jpg)
Bayes’ theoremModel comparison
What does the data tell us about our model Mi in relation to othermodels {M1,M2, · · · }?
P(Mi )D−→ P(Mi |D)
P(Mi |D) =P(D|Mi )P(Mi )
P(D)
P(D|Mi ) = Zi = Evidence of Mi
![Page 21: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/21.jpg)
Bayes’ theoremModel comparison
What does the data tell us about our model Mi in relation to othermodels {M1,M2, · · · }?
P(Mi )D−→ P(Mi |D)
P(Mi |D) =P(D|Mi )P(Mi )
P(D)
P(D|Mi ) = Zi = Evidence of Mi
![Page 22: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/22.jpg)
Bayes’ theoremModel comparison
What does the data tell us about our model Mi in relation to othermodels {M1,M2, · · · }?
P(Mi )D−→ P(Mi |D)
P(Mi |D) =P(D|Mi )P(Mi )
P(D)
P(D|Mi ) = Zi = Evidence of Mi
![Page 23: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/23.jpg)
Parameter estimation & model comparisonThe challenge
Parameter estimation: what does the data tell us about a model?(Computing posteriors)
Model comparison: what does the data tell us about all models?(Computing evidences)
Both of these are challenging things to compute.
I Markov-Chain Monte-Carlo (MCMC) can solve the first ofthese (kind of)
I Nested sampling (NS) promises to solve both simultaneously.
![Page 24: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/24.jpg)
Parameter estimation & model comparisonThe challenge
Parameter estimation: what does the data tell us about a model?(Computing posteriors)
Model comparison: what does the data tell us about all models?(Computing evidences)
Both of these are challenging things to compute.
I Markov-Chain Monte-Carlo (MCMC) can solve the first ofthese (kind of)
I Nested sampling (NS) promises to solve both simultaneously.
![Page 25: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/25.jpg)
Parameter estimation & model comparisonThe challenge
Parameter estimation: what does the data tell us about a model?(Computing posteriors)
Model comparison: what does the data tell us about all models?(Computing evidences)
Both of these are challenging things to compute.
I Markov-Chain Monte-Carlo (MCMC) can solve the first ofthese (kind of)
I Nested sampling (NS) promises to solve both simultaneously.
![Page 26: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/26.jpg)
Parameter estimation & model comparisonThe challenge
Parameter estimation: what does the data tell us about a model?(Computing posteriors)
Model comparison: what does the data tell us about all models?(Computing evidences)
Both of these are challenging things to compute.
I Markov-Chain Monte-Carlo (MCMC) can solve the first ofthese (kind of)
I Nested sampling (NS) promises to solve both simultaneously.
![Page 27: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/27.jpg)
Parameter estimation & model comparisonThe challenge
Parameter estimation: what does the data tell us about a model?(Computing posteriors)
Model comparison: what does the data tell us about all models?(Computing evidences)
Both of these are challenging things to compute.
I Markov-Chain Monte-Carlo (MCMC) can solve the first ofthese (kind of)
I Nested sampling (NS) promises to solve both simultaneously.
![Page 28: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/28.jpg)
Parameter estimation & model comparisonThe challenge
Parameter estimation: what does the data tell us about a model?(Computing posteriors)
Model comparison: what does the data tell us about all models?(Computing evidences)
Both of these are challenging things to compute.
I Markov-Chain Monte-Carlo (MCMC) can solve the first ofthese (kind of)
I Nested sampling (NS) promises to solve both simultaneously.
![Page 29: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/29.jpg)
Parameter estimation & model comparisonWhy is it difficult?
1. In high dimensions, posteriorP occupies a vanishinglysmall region of the prior π.
2. Worse, you don’t knowwhere this region is.
I Describing an N-dimensional posterior fully is impossible.
I Project/marginalise into 2- or 3-dimensions at best
I Sampling the posterior is an excellent compression scheme.
![Page 30: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/30.jpg)
Parameter estimation & model comparisonWhy is it difficult?
1. In high dimensions, posteriorP occupies a vanishinglysmall region of the prior π.
2. Worse, you don’t knowwhere this region is.
I Describing an N-dimensional posterior fully is impossible.
I Project/marginalise into 2- or 3-dimensions at best
I Sampling the posterior is an excellent compression scheme.
![Page 31: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/31.jpg)
Parameter estimation & model comparisonWhy is it difficult?
1. In high dimensions, posteriorP occupies a vanishinglysmall region of the prior π.
2. Worse, you don’t knowwhere this region is.
I Describing an N-dimensional posterior fully is impossible.
I Project/marginalise into 2- or 3-dimensions at best
I Sampling the posterior is an excellent compression scheme.
![Page 32: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/32.jpg)
Parameter estimation & model comparisonWhy is it difficult?
1. In high dimensions, posteriorP occupies a vanishinglysmall region of the prior π.
2. Worse, you don’t knowwhere this region is.
I Describing an N-dimensional posterior fully is impossible.
I Project/marginalise into 2- or 3-dimensions at best
I Sampling the posterior is an excellent compression scheme.
![Page 33: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/33.jpg)
Parameter estimation & model comparisonWhy is it difficult?
1. In high dimensions, posteriorP occupies a vanishinglysmall region of the prior π.
2. Worse, you don’t knowwhere this region is.
I Describing an N-dimensional posterior fully is impossible.
I Project/marginalise into 2- or 3-dimensions at best
I Sampling the posterior is an excellent compression scheme.
![Page 34: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/34.jpg)
Parameter estimation & model comparisonWhy is it difficult?
1. In high dimensions, posteriorP occupies a vanishinglysmall region of the prior π.
2. Worse, you don’t knowwhere this region is.
I Describing an N-dimensional posterior fully is impossible.
I Project/marginalise into 2- or 3-dimensions at best
I Sampling the posterior is an excellent compression scheme.
![Page 35: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/35.jpg)
Markov-Chain Monte-Carlo (MCMC)Metropolis-Hastings, Gibbs, Hamiltonian. . .
I Turn the N-dimensional problem into a one-dimensional one.I Explore the space via a biased random walk.
1. Pick random direction2. Choose step length3. If uphill, make step. . .4. . . . otherwise sometimes make step.
![Page 36: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/36.jpg)
Markov-Chain Monte-Carlo (MCMC)Metropolis-Hastings, Gibbs, Hamiltonian. . .
I Turn the N-dimensional problem into a one-dimensional one.
I Explore the space via a biased random walk.
1. Pick random direction2. Choose step length3. If uphill, make step. . .4. . . . otherwise sometimes make step.
![Page 37: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/37.jpg)
Markov-Chain Monte-Carlo (MCMC)Metropolis-Hastings, Gibbs, Hamiltonian. . .
I Turn the N-dimensional problem into a one-dimensional one.I Explore the space via a biased random walk.
1. Pick random direction2. Choose step length3. If uphill, make step. . .4. . . . otherwise sometimes make step.
![Page 38: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/38.jpg)
Markov-Chain Monte-Carlo (MCMC)Metropolis-Hastings, Gibbs, Hamiltonian. . .
I Turn the N-dimensional problem into a one-dimensional one.I Explore the space via a biased random walk.
1. Pick random direction
2. Choose step length3. If uphill, make step. . .4. . . . otherwise sometimes make step.
![Page 39: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/39.jpg)
Markov-Chain Monte-Carlo (MCMC)Metropolis-Hastings, Gibbs, Hamiltonian. . .
I Turn the N-dimensional problem into a one-dimensional one.I Explore the space via a biased random walk.
1. Pick random direction2. Choose step length
3. If uphill, make step. . .4. . . . otherwise sometimes make step.
![Page 40: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/40.jpg)
Markov-Chain Monte-Carlo (MCMC)Metropolis-Hastings, Gibbs, Hamiltonian. . .
I Turn the N-dimensional problem into a one-dimensional one.I Explore the space via a biased random walk.
1. Pick random direction2. Choose step length3. If uphill, make step. . .
4. . . . otherwise sometimes make step.
![Page 41: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/41.jpg)
Markov-Chain Monte-Carlo (MCMC)Metropolis-Hastings, Gibbs, Hamiltonian. . .
I Turn the N-dimensional problem into a one-dimensional one.I Explore the space via a biased random walk.
1. Pick random direction2. Choose step length3. If uphill, make step. . .4. . . . otherwise sometimes make step.
![Page 42: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/42.jpg)
MCMC in action
0.4 0.2 0.0 0.2 0.4µ
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MCMC in action
0.4 0.2 0.0 0.2 0.4µ
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When MCMC failsBurn in
0.4 0.2 0.0 0.2 0.4µ
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![Page 45: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/45.jpg)
When MCMC failsBurn in
0.4 0.2 0.0 0.2 0.4µ
0.6
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1.0
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1.4
A
4 2 0 2 4position
0.0
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1.0
1.2
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sign
al
![Page 46: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/46.jpg)
When MCMC failsTuning the proposal distribution
0.4 0.2 0.0 0.2 0.4µ
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4 2 0 2 4position
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sign
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![Page 47: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/47.jpg)
When MCMC failsTuning the proposal distribution
0.4 0.2 0.0 0.2 0.4µ
0.6
0.8
1.0
1.2
1.4
A
4 2 0 2 4position
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
sign
al
![Page 48: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/48.jpg)
When MCMC failsMultimodality
4 2 0 2 4µ
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A
4 2 0 2 4position
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sign
al
![Page 49: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/49.jpg)
When MCMC failsMultimodality
4 2 0 2 4µ
0.0
0.5
1.0
1.5
2.0
A
4 2 0 2 4position
0.0
0.5
1.0
1.5
2.0
sign
al
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When MCMC failsPhase transitions
L
X
![Page 51: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/51.jpg)
When MCMC failsThe real reason. . .
I MCMC does not give you evidences!
Z = P(D|M)
=
∫P(D|Θ,M)P(Θ|M)dΘ
=
∫L(Θ)π(Θ)dΘ
= 〈L〉π
I MCMC fundamentally explores the posterior, and cannotaverage over the prior.
![Page 52: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/52.jpg)
When MCMC failsThe real reason. . .
I MCMC does not give you evidences!
Z = P(D|M)
=
∫P(D|Θ,M)P(Θ|M)dΘ
=
∫L(Θ)π(Θ)dΘ
= 〈L〉π
I MCMC fundamentally explores the posterior, and cannotaverage over the prior.
![Page 53: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/53.jpg)
When MCMC failsThe real reason. . .
I MCMC does not give you evidences!
Z = P(D|M)
=
∫P(D|Θ,M)P(Θ|M)dΘ
=
∫L(Θ)π(Θ)dΘ
= 〈L〉π
I MCMC fundamentally explores the posterior, and cannotaverage over the prior.
![Page 54: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/54.jpg)
When MCMC failsThe real reason. . .
I MCMC does not give you evidences!
Z = P(D|M)
=
∫P(D|Θ,M)P(Θ|M)dΘ
=
∫L(Θ)π(Θ)dΘ
= 〈L〉π
I MCMC fundamentally explores the posterior, and cannotaverage over the prior.
![Page 55: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/55.jpg)
When MCMC failsThe real reason. . .
I MCMC does not give you evidences!
Z = P(D|M)
=
∫P(D|Θ,M)P(Θ|M)dΘ
=
∫L(Θ)π(Θ)dΘ
= 〈L〉π
I MCMC fundamentally explores the posterior, and cannotaverage over the prior.
![Page 56: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/56.jpg)
When MCMC failsThe real reason. . .
I MCMC does not give you evidences!
Z = P(D|M)
=
∫P(D|Θ,M)P(Θ|M)dΘ
=
∫L(Θ)π(Θ)dΘ
= 〈L〉π
I MCMC fundamentally explores the posterior, and cannotaverage over the prior.
![Page 57: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/57.jpg)
When MCMC failsThe real reason. . .
I MCMC does not give you evidences!
Z = P(D|M)
=
∫P(D|Θ,M)P(Θ|M)dΘ
=
∫L(Θ)π(Θ)dΘ
= 〈L〉π
I MCMC fundamentally explores the posterior, and cannotaverage over the prior.
![Page 58: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/58.jpg)
Nested SamplingJohn Skilling’s alternative to MCMC!
New procedure:Maintain a set S of n samples, which are sequentially updated:
S0: Generate n samples from the prior π.
Sn+1: Delete the lowest likelihood sample in Sn, and replaceit with a new sample with higher likelihood
Requires one to be able to sample from the prior, subject to a hardlikelihood constraint.
![Page 59: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/59.jpg)
Nested SamplingJohn Skilling’s alternative to MCMC!
New procedure:
Maintain a set S of n samples, which are sequentially updated:
S0: Generate n samples from the prior π.
Sn+1: Delete the lowest likelihood sample in Sn, and replaceit with a new sample with higher likelihood
Requires one to be able to sample from the prior, subject to a hardlikelihood constraint.
![Page 60: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/60.jpg)
Nested SamplingJohn Skilling’s alternative to MCMC!
New procedure:Maintain a set S of n samples, which are sequentially updated:
S0: Generate n samples from the prior π.
Sn+1: Delete the lowest likelihood sample in Sn, and replaceit with a new sample with higher likelihood
Requires one to be able to sample from the prior, subject to a hardlikelihood constraint.
![Page 61: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/61.jpg)
Nested SamplingJohn Skilling’s alternative to MCMC!
New procedure:Maintain a set S of n samples, which are sequentially updated:
S0: Generate n samples from the prior π.
Sn+1: Delete the lowest likelihood sample in Sn, and replaceit with a new sample with higher likelihood
Requires one to be able to sample from the prior, subject to a hardlikelihood constraint.
![Page 62: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/62.jpg)
Nested SamplingJohn Skilling’s alternative to MCMC!
New procedure:Maintain a set S of n samples, which are sequentially updated:
S0: Generate n samples from the prior π.
Sn+1: Delete the lowest likelihood sample in Sn, and replaceit with a new sample with higher likelihood
Requires one to be able to sample from the prior, subject to a hardlikelihood constraint.
![Page 63: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/63.jpg)
Nested SamplingJohn Skilling’s alternative to MCMC!
New procedure:Maintain a set S of n samples, which are sequentially updated:
S0: Generate n samples from the prior π.
Sn+1: Delete the lowest likelihood sample in Sn, and replaceit with a new sample with higher likelihood
Requires one to be able to sample from the prior, subject to a hardlikelihood constraint.
![Page 64: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/64.jpg)
Nested SamplingGraphical aid
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Nested SamplingGraphical aid
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Nested SamplingGraphical aid
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Nested SamplingGraphical aid
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Nested SamplingGraphical aid
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Nested SamplingGraphical aid
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Nested SamplingGraphical aid
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Nested SamplingGraphical aid
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Nested SamplingGraphical aid
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Nested SamplingGraphical aid
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Nested SamplingGraphical aid
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Nested SamplingGraphical aid
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Nested SamplingGraphical aid
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Nested SamplingGraphical aid
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Nested SamplingGraphical aid
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Nested SamplingGraphical aid
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Nested SamplingGraphical aid
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Nested SamplingGraphical aid
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Nested SamplingGraphical aid
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Nested SamplingGraphical aid
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Nested SamplingGraphical aid
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Nested SamplingGraphical aid
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Nested SamplingGraphical aid
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Nested SamplingGraphical aid
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Nested SamplingGraphical aid
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Nested SamplingGraphical aid
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Nested SamplingGraphical aid
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Nested SamplingGraphical aid
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Nested SamplingGraphical aid
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Nested SamplingGraphical aid
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Nested SamplingGraphical aid
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Nested SamplingGraphical aid
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Nested SamplingGraphical aid
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Nested SamplingGraphical aid
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Nested SamplingGraphical aid
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Nested SamplingGraphical aid
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Nested SamplingGraphical aid
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Nested SamplingGraphical aid
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Nested SamplingWhy bother?
I At each iteration, the likelihood contour will shrink in volumeby a factor of ≈ 1/n.
I Nested sampling zooms in to the peak of the posteriorexponentially.
I Nested sampling can be used to get evidences!
![Page 103: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/103.jpg)
Nested SamplingWhy bother?
I At each iteration, the likelihood contour will shrink in volumeby a factor of ≈ 1/n.
I Nested sampling zooms in to the peak of the posteriorexponentially.
I Nested sampling can be used to get evidences!
![Page 104: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/104.jpg)
Nested SamplingWhy bother?
I At each iteration, the likelihood contour will shrink in volumeby a factor of ≈ 1/n.
I Nested sampling zooms in to the peak of the posteriorexponentially.
I Nested sampling can be used to get evidences!
![Page 105: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/105.jpg)
Nested SamplingWhy bother?
I At each iteration, the likelihood contour will shrink in volumeby a factor of ≈ 1/n.
I Nested sampling zooms in to the peak of the posteriorexponentially.
I Nested sampling can be used to get evidences!
![Page 106: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/106.jpg)
Calculating evidences
I Transform to 1 dimensional integral π(θ)dθ = dX
Z =
∫L(θ)π(θ)dθ
=
∫L(X )dX
I X is the prior volume
X (L) =
∫
L(θ)>Lπ(θ)dθ
I i.e. the fraction of the prior which the iso-likelihood contour Lencloses.
![Page 107: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/107.jpg)
Calculating evidences
I Transform to 1 dimensional integral π(θ)dθ = dX
Z =
∫L(θ)π(θ)dθ
=
∫L(X )dX
I X is the prior volume
X (L) =
∫
L(θ)>Lπ(θ)dθ
I i.e. the fraction of the prior which the iso-likelihood contour Lencloses.
![Page 108: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/108.jpg)
Calculating evidences
I Transform to 1 dimensional integral
π(θ)dθ = dX
Z =
∫L(θ)π(θ)dθ
=
∫L(X )dX
I X is the prior volume
X (L) =
∫
L(θ)>Lπ(θ)dθ
I i.e. the fraction of the prior which the iso-likelihood contour Lencloses.
![Page 109: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/109.jpg)
Calculating evidences
I Transform to 1 dimensional integral π(θ)dθ = dX
Z =
∫L(θ)π(θ)dθ
=
∫L(X )dX
I X is the prior volume
X (L) =
∫
L(θ)>Lπ(θ)dθ
I i.e. the fraction of the prior which the iso-likelihood contour Lencloses.
![Page 110: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/110.jpg)
Calculating evidences
I Transform to 1 dimensional integral π(θ)dθ = dX
Z =
∫L(θ)π(θ)dθ =
∫L(X )dX
I X is the prior volume
X (L) =
∫
L(θ)>Lπ(θ)dθ
I i.e. the fraction of the prior which the iso-likelihood contour Lencloses.
![Page 111: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/111.jpg)
Calculating evidences
I Transform to 1 dimensional integral π(θ)dθ = dX
Z =
∫L(θ)π(θ)dθ =
∫L(X )dX
I X is the prior volume
X (L) =
∫
L(θ)>Lπ(θ)dθ
I i.e. the fraction of the prior which the iso-likelihood contour Lencloses.
![Page 112: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/112.jpg)
Calculating evidences
I Transform to 1 dimensional integral π(θ)dθ = dX
Z =
∫L(θ)π(θ)dθ =
∫L(X )dX
I X is the prior volume
X (L) =
∫
L(θ)>Lπ(θ)dθ
I i.e. the fraction of the prior which the iso-likelihood contour Lencloses.
![Page 113: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/113.jpg)
Calculating evidences
I Transform to 1 dimensional integral π(θ)dθ = dX
Z =
∫L(θ)π(θ)dθ =
∫L(X )dX
I X is the prior volume
X (L) =
∫
L(θ)>Lπ(θ)dθ
I i.e. the fraction of the prior which the iso-likelihood contour Lencloses.
![Page 114: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/114.jpg)
Nested SamplingCalculating evidences
![Page 115: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/115.jpg)
Nested SamplingCalculating evidences
L
X
![Page 116: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/116.jpg)
Nested SamplingCalculating evidences
L
X
L1L2
L3
L4
L5
![Page 117: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/117.jpg)
Nested SamplingCalculating evidences
L
XX2 X1X3X4X5
L1L2
L3
L4
L5
![Page 118: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/118.jpg)
Nested SamplingCalculating evidences
L
X
∫L(X )dX ≈∑
i
Li(Xi−1 − Xi)
X2 X1X3X4X5
L1L2
L3
L4
L5
![Page 119: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/119.jpg)
Nested SamplingCalculating evidences
L
XX2 X1X3X4X5
L1L2
L3
L4
L5
∫L(X )dX ≈∑
i
Liwi
![Page 120: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/120.jpg)
Estimating evidencesEvidence error
L
X
![Page 121: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/121.jpg)
Estimating evidencesEvidence error
L
X
![Page 122: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/122.jpg)
Estimating evidencesEvidence error
L
X
![Page 123: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/123.jpg)
Estimating evidencesEvidence error
L
logX
![Page 124: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/124.jpg)
Estimating evidencesEvidence error
L
logX
Z =∫LdX
![Page 125: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/125.jpg)
Estimating evidencesEvidence error
L
logX
Z =∫XL d(logX )
![Page 126: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/126.jpg)
Estimating evidencesEvidence error
logX
L
XL
![Page 127: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/127.jpg)
Estimating evidencesEvidence error
logX
L
XL
H
![Page 128: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/128.jpg)
Estimating evidencesEvidence error
logX
L
XL
H
H =∫log
dPdX
dX
![Page 129: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/129.jpg)
Estimating evidencesEvidence error
I approximate compression:
∆ logX ∼ −1
n± 1
n
logXi ∼ −i
n±√i
n
I # of steps to get to H:
iH ∼ nH
I estimate of volume at H:
logXH ≈ −H ±√
H
n
I estimate of evidence error:
logZ ≈∑
wiLi ±√
H
n
![Page 130: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/130.jpg)
Estimating evidencesEvidence error
I approximate compression:
∆ logX ∼ −1
n
± 1
n
logXi ∼ −i
n±√i
n
I # of steps to get to H:
iH ∼ nH
I estimate of volume at H:
logXH ≈ −H ±√
H
n
I estimate of evidence error:
logZ ≈∑
wiLi ±√
H
n
![Page 131: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/131.jpg)
Estimating evidencesEvidence error
I approximate compression:
∆ logX ∼ −1
n± 1
n
logXi ∼ −i
n±√i
n
I # of steps to get to H:
iH ∼ nH
I estimate of volume at H:
logXH ≈ −H ±√
H
n
I estimate of evidence error:
logZ ≈∑
wiLi ±√
H
n
![Page 132: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/132.jpg)
Estimating evidencesEvidence error
I approximate compression:
∆ logX ∼ −1
n± 1
n
logXi ∼ −i
n
±√i
n
I # of steps to get to H:
iH ∼ nH
I estimate of volume at H:
logXH ≈ −H ±√
H
n
I estimate of evidence error:
logZ ≈∑
wiLi ±√
H
n
![Page 133: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/133.jpg)
Estimating evidencesEvidence error
I approximate compression:
∆ logX ∼ −1
n± 1
n
logXi ∼ −i
n±√i
n
I # of steps to get to H:
iH ∼ nH
I estimate of volume at H:
logXH ≈ −H ±√
H
n
I estimate of evidence error:
logZ ≈∑
wiLi ±√
H
n
![Page 134: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/134.jpg)
Estimating evidencesEvidence error
I approximate compression:
∆ logX ∼ −1
n± 1
n
logXi ∼ −i
n±√i
n
I # of steps to get to H:
iH ∼ nH
I estimate of volume at H:
logXH ≈ −H ±√
H
n
I estimate of evidence error:
logZ ≈∑
wiLi ±√
H
n
![Page 135: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/135.jpg)
Estimating evidencesEvidence error
I approximate compression:
∆ logX ∼ −1
n± 1
n
logXi ∼ −i
n±√i
n
I # of steps to get to H:
iH ∼ nH
I estimate of volume at H:
logXH ≈ −H ±√
H
n
I estimate of evidence error:
logZ ≈∑
wiLi ±√
H
n
![Page 136: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/136.jpg)
Estimating evidencesEvidence error
I approximate compression:
∆ logX ∼ −1
n± 1
n
logXi ∼ −i
n±√i
n
I # of steps to get to H:
iH ∼ nH
I estimate of volume at H:
logXH ≈ −H ±√
H
n
I estimate of evidence error:
logZ ≈∑
wiLi ±√
H
n
![Page 137: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/137.jpg)
Nested samplingParameter estimation
I NS can also be used to sample the posterior
I The set of dead points are posterior samples with anappropriate weighting factor
![Page 138: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/138.jpg)
Nested samplingParameter estimation
I NS can also be used to sample the posterior
I The set of dead points are posterior samples with anappropriate weighting factor
![Page 139: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/139.jpg)
Nested samplingParameter estimation
I NS can also be used to sample the posterior
I The set of dead points are posterior samples with anappropriate weighting factor
![Page 140: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/140.jpg)
When NS succeeds
0.4 0.2 0.0 0.2 0.4µ
0.6
0.8
1.0
1.2
1.4
A
4 2 0 2 4position
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
sign
al
![Page 141: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/141.jpg)
When NS suceeds
0.4 0.2 0.0 0.2 0.4µ
0.6
0.8
1.0
1.2
1.4
A
4 2 0 2 4position
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
sign
al
![Page 142: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/142.jpg)
When NS succeeds
4 2 0 2 4µ
0.0
0.5
1.0
1.5
2.0
A
4 2 0 2 4position
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
sign
al
![Page 143: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/143.jpg)
When NS suceeds
4 2 0 2 4µ
0.0
0.5
1.0
1.5
2.0
A
4 2 0 2 4position
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
sign
al
![Page 144: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/144.jpg)
Sampling from a hard likelihood constraint
“It is not the purpose of this introductory paper todevelop the technology of navigation within such avolume. We merely note that exploring a hard-edgedlikelihood-constrained domain should prove to be neithermore nor less demanding than exploring alikelihood-weighted space.”
— John Skilling
![Page 145: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/145.jpg)
Sampling from a hard likelihood constraint
“It is not the purpose of this introductory paper todevelop the technology of navigation within such avolume. We merely note that exploring a hard-edgedlikelihood-constrained domain should prove to be neithermore nor less demanding than exploring alikelihood-weighted space.”
— John Skilling
![Page 146: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/146.jpg)
Sampling within an iso-likelihood contourPrevious attempts
Rejection Sampling MultiNest; F. Feroz & M. Hobson (2009).
I Suffers in high dimensions
Hamiltonian sampling F. Feroz & J. Skilling (2013).
I Requires gradients and tuning
Diffusion Nested Sampling B. Brewer et al. (2009).
I Very promisingI Too many tuning parameters
![Page 147: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/147.jpg)
Sampling within an iso-likelihood contourPrevious attempts
Rejection Sampling MultiNest; F. Feroz & M. Hobson (2009).
I Suffers in high dimensions
Hamiltonian sampling F. Feroz & J. Skilling (2013).
I Requires gradients and tuning
Diffusion Nested Sampling B. Brewer et al. (2009).
I Very promisingI Too many tuning parameters
![Page 148: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/148.jpg)
Sampling within an iso-likelihood contourPrevious attempts
Rejection Sampling MultiNest; F. Feroz & M. Hobson (2009).
I Suffers in high dimensions
Hamiltonian sampling F. Feroz & J. Skilling (2013).
I Requires gradients and tuning
Diffusion Nested Sampling B. Brewer et al. (2009).
I Very promisingI Too many tuning parameters
![Page 149: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/149.jpg)
Sampling within an iso-likelihood contourPrevious attempts
Rejection Sampling MultiNest; F. Feroz & M. Hobson (2009).
I Suffers in high dimensions
Hamiltonian sampling F. Feroz & J. Skilling (2013).
I Requires gradients and tuning
Diffusion Nested Sampling B. Brewer et al. (2009).
I Very promisingI Too many tuning parameters
![Page 150: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/150.jpg)
Sampling within an iso-likelihood contourPrevious attempts
Rejection Sampling MultiNest; F. Feroz & M. Hobson (2009).
I Suffers in high dimensions
Hamiltonian sampling F. Feroz & J. Skilling (2013).
I Requires gradients and tuning
Diffusion Nested Sampling B. Brewer et al. (2009).
I Very promisingI Too many tuning parameters
![Page 151: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/151.jpg)
Sampling within an iso-likelihood contourPrevious attempts
Rejection Sampling MultiNest; F. Feroz & M. Hobson (2009).
I Suffers in high dimensions
Hamiltonian sampling F. Feroz & J. Skilling (2013).
I Requires gradients and tuning
Diffusion Nested Sampling B. Brewer et al. (2009).
I Very promisingI Too many tuning parameters
![Page 152: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/152.jpg)
Sampling within an iso-likelihood contourPrevious attempts
Rejection Sampling MultiNest; F. Feroz & M. Hobson (2009).
I Suffers in high dimensions
Hamiltonian sampling F. Feroz & J. Skilling (2013).
I Requires gradients and tuning
Diffusion Nested Sampling B. Brewer et al. (2009).
I Very promising
I Too many tuning parameters
![Page 153: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/153.jpg)
Sampling within an iso-likelihood contourPrevious attempts
Rejection Sampling MultiNest; F. Feroz & M. Hobson (2009).
I Suffers in high dimensions
Hamiltonian sampling F. Feroz & J. Skilling (2013).
I Requires gradients and tuning
Diffusion Nested Sampling B. Brewer et al. (2009).
I Very promisingI Too many tuning parameters
![Page 154: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/154.jpg)
PolyChord“Hit and run” slice sampling
L0
x0
![Page 155: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/155.jpg)
PolyChord“Hit and run” slice sampling
L0
x0
1) From an initial starting point x0,choose a random direction
![Page 156: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/156.jpg)
PolyChord“Hit and run” slice sampling
L0
x0
1) From an initial starting point x0,choose a random direction
![Page 157: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/157.jpg)
PolyChord“Hit and run” slice sampling
L0
x0
2) Place random initial boundof width w
![Page 158: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/158.jpg)
PolyChord“Hit and run” slice sampling
L0
x0
2) Place random initial boundof width w
![Page 159: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/159.jpg)
PolyChord“Hit and run” slice sampling
L0
x0
3) Extend bounds by“stepping out” procedure
![Page 160: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/160.jpg)
PolyChord“Hit and run” slice sampling
L0
x0
3) Extend bounds by“stepping out” procedure
![Page 161: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/161.jpg)
PolyChord“Hit and run” slice sampling
L0
x0
3) Extend bounds by“stepping out” procedure
![Page 162: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/162.jpg)
PolyChord“Hit and run” slice sampling
L0
x0
3) Extend bounds by“stepping out” procedure
![Page 163: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/163.jpg)
PolyChord“Hit and run” slice sampling
L0
x0
3) Extend bounds by“stepping out” procedure
![Page 164: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/164.jpg)
PolyChord“Hit and run” slice sampling
L0
x0
4) Sample uniformly along this chord
![Page 165: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/165.jpg)
PolyChord“Hit and run” slice sampling
L0
x0
4) Sample uniformly along this chord
x1
![Page 166: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/166.jpg)
PolyChord“Hit and run” slice sampling
L0
x0
x1
If x1 is drawn outside of L0,then contract bounds,
![Page 167: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/167.jpg)
PolyChord“Hit and run” slice sampling
L0
x0
If x1 is drawn outside of L0,then contract bounds,
![Page 168: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/168.jpg)
PolyChord“Hit and run” slice sampling
L0
x0
Draw again until x1 is within L0.
![Page 169: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/169.jpg)
PolyChord“Hit and run” slice sampling
L0
x0
Draw again until x1 is within L0.
x1
![Page 170: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/170.jpg)
PolyChord“Hit and run” slice sampling
L0
x0
x1
5) Repeat this procedure,starting with x1
![Page 171: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/171.jpg)
PolyChord“Hit and run” slice sampling
L0
x0
x1
![Page 172: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/172.jpg)
PolyChord“Hit and run” slice sampling
L0
x0
x1
![Page 173: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/173.jpg)
PolyChord“Hit and run” slice sampling
L0
x0
x1
x2
![Page 174: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/174.jpg)
PolyChord“Hit and run” slice sampling
L0
x0
x2
![Page 175: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/175.jpg)
PolyChord“Hit and run” slice sampling
L0
x0
x2
![Page 176: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/176.jpg)
PolyChord“Hit and run” slice sampling
L0
x0
x2x3
![Page 177: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/177.jpg)
PolyChord“Hit and run” slice sampling
L0
x0
x3
![Page 178: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/178.jpg)
PolyChord“Hit and run” slice sampling
L0
x0
x3
![Page 179: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/179.jpg)
PolyChord“Hit and run” slice sampling
L0
x0
x3x4
![Page 180: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/180.jpg)
PolyChord“Hit and run” slice sampling
L0
x0
x3
![Page 181: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/181.jpg)
PolyChord“Hit and run” slice sampling
L0
x0
x3
x4
![Page 182: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/182.jpg)
PolyChord“Hit and run” slice sampling
L0
x0x4
![Page 183: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/183.jpg)
PolyChord“Hit and run” slice sampling
L0
x0x4
![Page 184: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/184.jpg)
PolyChord“Hit and run” slice sampling
L0
x0x4
xN
![Page 185: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/185.jpg)
PolyChord“Hit and run” slice sampling
L0
x0
xN
![Page 186: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/186.jpg)
PolyChordKey points
I This procedure satisfies detailed balance.
I Works even if L0 contour is disjoint.
I Need N reasonably large ∼ O(ndims) so that xN isde-correlated from x1.
![Page 187: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/187.jpg)
PolyChordKey points
I This procedure satisfies detailed balance.
I Works even if L0 contour is disjoint.
I Need N reasonably large ∼ O(ndims) so that xN isde-correlated from x1.
![Page 188: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/188.jpg)
PolyChordKey points
I This procedure satisfies detailed balance.
I Works even if L0 contour is disjoint.
I Need N reasonably large ∼ O(ndims) so that xN isde-correlated from x1.
![Page 189: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/189.jpg)
PolyChordKey points
I This procedure satisfies detailed balance.
I Works even if L0 contour is disjoint.
I Need N reasonably large ∼ O(ndims) so that xN isde-correlated from x1.
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PolyChordKey points
I This procedure satisfies detailed balance.
I Works even if L0 contour is disjoint.
I Need N reasonably large ∼ O(ndims) so that xN isde-correlated from x1.
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Issues with Slice Sampling
1. Does not deal well with correlated distributions.
2. Need to “tune” w parameter.
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Issues with Slice Sampling
1. Does not deal well with correlated distributions.
2. Need to “tune” w parameter.
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Issues with Slice Sampling
1. Does not deal well with correlated distributions.
2. Need to “tune” w parameter.
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PolyChord’s solutionsCorrelated distributions
Contour fromcorrelateddistribution
Affine transformation y = Lx
y
x Contour withdimensions∼ O(1)
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PolyChord’s solutionsCorrelated distributions
I We make an affine transformation to remove degeneracies,and “whiten” the space.
I Samples remain uniformly sampled
I We use the covariance matrix of the live points and allinter-chain points
I Cholesky decomposition is the required skew transformation
I w = 1 in this transformed space
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PolyChord’s solutionsCorrelated distributions
I We make an affine transformation to remove degeneracies,and “whiten” the space.
I Samples remain uniformly sampled
I We use the covariance matrix of the live points and allinter-chain points
I Cholesky decomposition is the required skew transformation
I w = 1 in this transformed space
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PolyChord’s solutionsCorrelated distributions
I We make an affine transformation to remove degeneracies,and “whiten” the space.
I Samples remain uniformly sampled
I We use the covariance matrix of the live points and allinter-chain points
I Cholesky decomposition is the required skew transformation
I w = 1 in this transformed space
![Page 198: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/198.jpg)
PolyChord’s solutionsCorrelated distributions
I We make an affine transformation to remove degeneracies,and “whiten” the space.
I Samples remain uniformly sampled
I We use the covariance matrix of the live points and allinter-chain points
I Cholesky decomposition is the required skew transformation
I w = 1 in this transformed space
![Page 199: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/199.jpg)
PolyChord’s solutionsCorrelated distributions
I We make an affine transformation to remove degeneracies,and “whiten” the space.
I Samples remain uniformly sampled
I We use the covariance matrix of the live points and allinter-chain points
I Cholesky decomposition is the required skew transformation
I w = 1 in this transformed space
![Page 200: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/200.jpg)
PolyChord’s solutionsCorrelated distributions
I We make an affine transformation to remove degeneracies,and “whiten” the space.
I Samples remain uniformly sampled
I We use the covariance matrix of the live points and allinter-chain points
I Cholesky decomposition is the required skew transformation
I w = 1 in this transformed space
![Page 201: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/201.jpg)
PolyChord’s Additions
I Parallelised up to number of live points with openMPI.
I Novel method for identifying and evolving modes separately.
I Implemented in CosmoMC, as “CosmoChord”, with fast-slowparameters.
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PolyChord’s Additions
I Parallelised up to number of live points with openMPI.
I Novel method for identifying and evolving modes separately.
I Implemented in CosmoMC, as “CosmoChord”, with fast-slowparameters.
![Page 203: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/203.jpg)
PolyChord’s Additions
I Parallelised up to number of live points with openMPI.
I Novel method for identifying and evolving modes separately.
I Implemented in CosmoMC, as “CosmoChord”, with fast-slowparameters.
![Page 204: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/204.jpg)
PolyChord’s Additions
I Parallelised up to number of live points with openMPI.
I Novel method for identifying and evolving modes separately.
I Implemented in CosmoMC, as “CosmoChord”, with fast-slowparameters.
![Page 205: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/205.jpg)
PolyChord in actionPrimordial power spectrum PR(k) reconstruction
logPR(k)
log k
As
(kk∗
)ns−1
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PolyChord in actionPrimordial power spectrum PR(k) reconstruction
logPR(k)
log k
(k1,P1)
(k2,P2)
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PolyChord in actionPrimordial power spectrum PR(k) reconstruction
logPR(k)
log k
(k1,P1)
(k2,P2)
(k3,P3)
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PolyChord in actionPrimordial power spectrum PR(k) reconstruction
logPR(k)
log k
(k1,P1)
(k2,P2)
(k3,P3)
(k4,P4)
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PolyChord in actionPrimordial power spectrum PR(k) reconstruction
logPR(k)
log k
(k1,P1)
(k2,P2)
(k3,P3)
(k4,P4)
(kNknots,PNknots
)
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Planck dataPrimordial power spectrum PR(k) reconstruction
I Temperature data TT+lowP
I Foreground (14) & cosmological (4 + 2 ∗ Nknots − 2)parameters
I Marginalised plots of PR(k)
I
P(PR|k ,Nknots) =
∫δ(PR − f (k ; θ))P(θ)dθ
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Planck dataPrimordial power spectrum PR(k) reconstruction
I Temperature data TT+lowP
I Foreground (14) & cosmological (4 + 2 ∗ Nknots − 2)parameters
I Marginalised plots of PR(k)
I
P(PR|k ,Nknots) =
∫δ(PR − f (k ; θ))P(θ)dθ
![Page 212: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/212.jpg)
Planck dataPrimordial power spectrum PR(k) reconstruction
I Temperature data TT+lowP
I Foreground (14) & cosmological (4 + 2 ∗ Nknots − 2)parameters
I Marginalised plots of PR(k)
I
P(PR|k ,Nknots) =
∫δ(PR − f (k ; θ))P(θ)dθ
![Page 213: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/213.jpg)
Planck dataPrimordial power spectrum PR(k) reconstruction
I Temperature data TT+lowP
I Foreground (14) & cosmological (4 + 2 ∗ Nknots − 2)parameters
I Marginalised plots of PR(k)
I
P(PR|k ,Nknots) =
∫δ(PR − f (k ; θ))P(θ)dθ
![Page 214: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/214.jpg)
Planck dataPrimordial power spectrum PR(k) reconstruction
I Temperature data TT+lowP
I Foreground (14) & cosmological (4 + 2 ∗ Nknots − 2)parameters
I Marginalised plots of PR(k)
I
P(PR|k ,Nknots) =
∫δ(PR − f (k ; θ))P(θ)dθ
![Page 215: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/215.jpg)
0 internal knotsPrimordial power spectrum PR(k) reconstruction
10−4 10−3 10−2 10−1
k/Mpc
2.0
2.5
3.0
3.5
4.0
log
(10
10P R
)10 100 1000
`
0σ
1σ
2σ
3σ
![Page 216: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/216.jpg)
1 internal knotsPrimordial power spectrum PR(k) reconstruction
10−4 10−3 10−2 10−1
k/Mpc
2.0
2.5
3.0
3.5
4.0
log
(10
10P R
)10 100 1000
`
0σ
1σ
2σ
3σ
![Page 217: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/217.jpg)
2 internal knotsPrimordial power spectrum PR(k) reconstruction
10−4 10−3 10−2 10−1
k/Mpc
2.0
2.5
3.0
3.5
4.0
log
(10
10P R
)10 100 1000
`
0σ
1σ
2σ
3σ
![Page 218: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/218.jpg)
3 internal knotsPrimordial power spectrum PR(k) reconstruction
10−4 10−3 10−2 10−1
k/Mpc
2.0
2.5
3.0
3.5
4.0
log
(10
10P R
)10 100 1000
`
0σ
1σ
2σ
3σ
![Page 219: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/219.jpg)
4 internal knotsPrimordial power spectrum PR(k) reconstruction
10−4 10−3 10−2 10−1
k/Mpc
2.0
2.5
3.0
3.5
4.0
log
(10
10P R
)10 100 1000
`
0σ
1σ
2σ
3σ
![Page 220: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/220.jpg)
5 internal knotsPrimordial power spectrum PR(k) reconstruction
10−4 10−3 10−2 10−1
k/Mpc
2.0
2.5
3.0
3.5
4.0
log
(10
10P R
)10 100 1000
`
0σ
1σ
2σ
3σ
![Page 221: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/221.jpg)
6 internal knotsPrimordial power spectrum PR(k) reconstruction
10−4 10−3 10−2 10−1
k/Mpc
2.0
2.5
3.0
3.5
4.0
log
(10
10P R
)10 100 1000
`
0σ
1σ
2σ
3σ
![Page 222: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/222.jpg)
7 internal knotsPrimordial power spectrum PR(k) reconstruction
10−4 10−3 10−2 10−1
k/Mpc
2.0
2.5
3.0
3.5
4.0
log
(10
10P R
)10 100 1000
`
0σ
1σ
2σ
3σ
![Page 223: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/223.jpg)
8 internal knotsPrimordial power spectrum PR(k) reconstruction
10−4 10−3 10−2 10−1
k/Mpc
2.0
2.5
3.0
3.5
4.0
log
(10
10P R
)10 100 1000
`
0σ
1σ
2σ
3σ
![Page 224: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/224.jpg)
Bayes FactorsPrimordial power spectrum PR(k) reconstruction
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
0 1 2 3 4 5 6 7 8
Bayes
factor
B 0,N
w.r.t.ΛCDM
Number of internal knots N
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Marginalised plotPrimordial power spectrum PR(k) reconstruction
10−4 10−3 10−2 10−1
k/Mpc
2.0
2.5
3.0
3.5
4.0
log
(10
10P R
)10 100 1000
`
0σ
1σ
2σ
3σ
![Page 226: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/226.jpg)
Object detectionToy problem
![Page 227: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/227.jpg)
Object detectionEvidences
I logZ ratio: −251 : −156 : −114 : −117 : −136
I odds ratio: 10−60 : 10−19 : 1 : 0.04 : 10−10
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Object detectionEvidences
I logZ ratio: −251 : −156 : −114 : −117 : −136
I odds ratio: 10−60 : 10−19 : 1 : 0.04 : 10−10
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Object detectionEvidences
I logZ ratio: −251 : −156 : −114 : −117 : −136
I odds ratio: 10−60 : 10−19 : 1 : 0.04 : 10−10
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PolyChord vs. MultiNestGaussian likelihood
102
103
104
105
106
107
108
109
1010
1 2 4 8 16 32 64 128 256
Number
ofLikelihoodevaluations,
NL
Number of dimensions, D
PolyChordMultiNest
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PolyChord vs. MultiNestGaussian likelihood
102
103
104
105
106
107
108
109
1010
1 2 4 8 16 32 64 128 256 512
Number
ofLikelihoodevaluations,
NL
Number of dimensions, D
PolyChord 2.0PolyChord 1.0
MultiNest
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ConclusionsThe future of nested sampling
I We are at the beginning of a new era of sampling algorithms
I Plenty of more work in to be done in exploring new versions ofnested sampling
I Nested sampling is just the beginning
I arXiv:1506.00171
I http://ccpforge.cse.rl.ac.uk/gf/project/polychord/
![Page 233: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/233.jpg)
ConclusionsThe future of nested sampling
I We are at the beginning of a new era of sampling algorithms
I Plenty of more work in to be done in exploring new versions ofnested sampling
I Nested sampling is just the beginning
I arXiv:1506.00171
I http://ccpforge.cse.rl.ac.uk/gf/project/polychord/
![Page 234: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/234.jpg)
ConclusionsThe future of nested sampling
I We are at the beginning of a new era of sampling algorithms
I Plenty of more work in to be done in exploring new versions ofnested sampling
I Nested sampling is just the beginning
I arXiv:1506.00171
I http://ccpforge.cse.rl.ac.uk/gf/project/polychord/
![Page 235: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/235.jpg)
ConclusionsThe future of nested sampling
I We are at the beginning of a new era of sampling algorithms
I Plenty of more work in to be done in exploring new versions ofnested sampling
I Nested sampling is just the beginning
I arXiv:1506.00171
I http://ccpforge.cse.rl.ac.uk/gf/project/polychord/
![Page 236: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/236.jpg)
ConclusionsThe future of nested sampling
I We are at the beginning of a new era of sampling algorithms
I Plenty of more work in to be done in exploring new versions ofnested sampling
I Nested sampling is just the beginning
I arXiv:1506.00171
I http://ccpforge.cse.rl.ac.uk/gf/project/polychord/
![Page 237: PolyChord: Next Generation Nested Sampling - Sampling](https://reader034.vdocuments.net/reader034/viewer/2022051506/589302e31a28abc85f8b799e/html5/thumbnails/237.jpg)
ConclusionsThe future of nested sampling
I We are at the beginning of a new era of sampling algorithms
I Plenty of more work in to be done in exploring new versions ofnested sampling
I Nested sampling is just the beginning
I arXiv:1506.00171
I http://ccpforge.cse.rl.ac.uk/gf/project/polychord/