An Introduction to (Dynamic) Nested Sampling

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An Introduction to (Dynamic) Nested Sampling Josh Speagle [email protected]

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Introduction

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Background Pr , M = Pr , M Pr |M Pr M Bayes’ Theorem

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Background Pr , M = Pr , M Pr |M Pr M Bayes’ Theorem Prior

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Background Pr , M = Pr , M Pr |M Pr M Bayes’ Theorem Prior Likelihood

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Background Pr , M = Pr , M Pr |M Pr M Bayes’ Theorem Prior Likelihood Posterior

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Background Pr , M = Pr , M Pr |M Pr M Bayes’ Theorem Prior Likelihood Posterior Evidence

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Background = ℒ Bayes’ Theorem ≡ Ω ℒ Posterior Likelihood Prior Evidence

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Background = ℒ Bayes’ Theorem ≡ Ω ℒ Posterior Likelihood Prior Evidence

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Posterior Estimation via Sampling , −1 , … , 2 , 1 ∈ Ω Samples

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Posterior Estimation via Sampling , −1 , … , 2 , 1 ∈ Ω , −1 , … , 2 , 1 ⇒ = =1 − i Samples Weights

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Posterior Estimation via Sampling , −1 , … , 2 , 1 ∈ Ω , −1 , … , 2 , 1 ⇒ = =1 − i Samples Weights 1, 1, … , 1, 1 MCMC

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Posterior Estimation via Sampling , −1 , … , 2 , 1 ∈ Ω , −1 , … , 2 , 1 ⇒ = =1 − i Samples Weights 1, 1, … , 1, 1 MCMC , −1 −1 , … , 2 2 , 1 1 Importance Sampling

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Posterior Estimation via Sampling , −1 , … , 2 , 1 ∈ Ω , −1 , … , 2 , 1 ⇒ = =1 − i Samples Weights Nested Sampling?

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Motivation: Integrating the Posterior Ω

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Motivation: Integrating the Posterior {: =} =

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Motivation: Integrating the Posterior 0 ∞ =

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Motivation: Integrating the Posterior 0 ∞ “Amplitude” “Volume” =

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Motivation: Integrating the Posterior 0 ∞ = “Typical Set”

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Motivation: Integrating the Posterior 0 ∞ ∝ −1 “Typical Set”

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Motivation: Integrating the Posterior Ω

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Motivation: Integrating the Posterior ≡ Ω ℒ

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Motivation: Integrating the Posterior ≡ Ω ℒ ≡ :ℒ > “Prior Volume” Feroz et al. (2013)

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Motivation: Integrating the Posterior = 0 ∞ ≡ :ℒ > “Prior Volume” Feroz et al. (2013)

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Motivation: Integrating the Posterior = 0 1 ℒ ≡ :ℒ > “Prior Volume” Feroz et al. (2013)

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Motivation: Integrating the Posterior ≈ =1 ℒ , … , … ≡ :ℒ > “Prior Volume” Feroz et al. (2013) ℒ , … , … can be rectangles, trapezoids, etc. Amplitude Differential Volume Δ

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Motivation: Integrating the Posterior ≈ =1 ≡ :ℒ > “Prior Volume” Feroz et al. (2013) = ℒ , … , …

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Motivation: Integrating the Posterior = We get posteriors “for free” ≈ =1 Importance Weight = ℒ , … , …

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Motivation: Integrating the Posterior ≈ =1 = ℒ , … , … Directly proportional to typical set. = We get posteriors “for free” Importance Weight ~ , … , …

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Motivation: Sampling the Posterior Sampling directly from the likelihood ℒ is hard. Pictures from this 2010 talk by Skilling.

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Motivation: Sampling the Posterior Sampling uniformly within bound ℒ > is easier. Pictures from this 2010 talk by Skilling.

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Motivation: Sampling the Posterior Sampling uniformly within bound ℒ > is easier. Pictures from this 2010 talk by Skilling. −1

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Motivation: Sampling the Posterior Sampling uniformly within bound ℒ > is easier. Pictures from this 2010 talk by Skilling. −1

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Motivation: Sampling the Posterior Sampling uniformly within bound ℒ > is easier. Pictures from this 2010 talk by Skilling. −1

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Motivation: Sampling the Posterior Sampling uniformly within bound ℒ > is easier. Pictures from this 2010 talk by Skilling. +1

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Motivation: Sampling the Posterior Sampling uniformly within bound ℒ > is easier. Pictures from this 2010 talk by Skilling. +1 MCMC: Solving a Hard Problem once. vs Nested Sampling: Solving an Easier Problem many times.

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How Nested Sampling Works

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Estimating the Prior Volume ≈ =1 ≡ :ℒ > “Prior Volume” Feroz et al. (2013) = ℒ , … , …

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Estimating the Prior Volume ≈ =1 ≡ :ℒ > “Prior Volume” Feroz et al. (2013) = ℒ , … , … 

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Estimating the Prior Volume ≈ =1 ≡ :ℒ > “Prior Volume” Feroz et al. (2013) = ℒ , … , …  ???

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Estimating the Prior Volume ≈ =1 ≡ :ℒ > “Prior Volume” Feroz et al. (2013) = ℒ , … , …  ??? ~ ~ Unif PDF CDF Probability Integral Transform

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Estimating the Prior Volume ≈ =1 ≡ :ℒ > “Prior Volume” Feroz et al. (2013) = ℒ , … , …  ??? ~ ~ Unif PDF CDF Probability Integral Transform Need to sample from the constrained prior.

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Estimating the Prior Volume Pictures from this 2010 talk by Skilling. +1 Posterior

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Estimating the Prior Volume Pictures from this 2010 talk by Skilling. +1 +1 = ~ Unif Posterior

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Pictures from this 2010 talk by Skilling. +1 +1 = =0 0 0 , … , ~ Unif i. i. d. Estimating the Prior Volume Posterior

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Pictures from this 2010 talk by Skilling. +1 +1 = =0 0 0 , … , ~ Unif 0 ≡ 1 i. i. d. Estimating the Prior Volume Posterior

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Pictures from this 2010 talk by Skilling. +1 +1 = =0 0 , … , ~ Unif i. i. d. 0 ≡ 1 Estimating the Prior Volume Posterior

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Estimating the Prior Volume ≈ =1 ≡ :ℒ > “Prior Volume” Feroz et al. (2013) = ℒ , … , …  ???

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Estimating the Prior Volume ≈ =1 ≡ :ℒ > “Prior Volume” Feroz et al. (2013) = ℒ , … , …  Pr 1 , 2 , … , where 1st & 2nd moments can be computed

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Nested Sampling Algorithm ℒ > ℒ−1 > ⋯ > ℒ2 > ℒ1 > 0 ℒ > ℒ−1 > ⋯ > ℒ2 > ℒ1 > 0 1 2 −1

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Nested Sampling Algorithm (Ideal) ℒ > ℒ−1 > ⋯ > ℒ2 > ℒ1 > 0 Samples sequentially drawn from constrained prior ℒ>ℒ . ℒ > ℒ−1 > ⋯ > ℒ2 > ℒ1 > 0 1 2 −1

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Nested Sampling Algorithm (Naïve) ℒ > ℒ−1 > ⋯ > ℒ2 > ℒ1 > 0 1 ~ 1. Samples sequentially drawn from prior . 2. New point only accepted if ℒ+1 > ℒ.

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Nested Sampling Algorithm (Naïve) ℒ > ℒ−1 > ⋯ > ℒ2 > ℒ1 > 0 1 ~ 2 1. Samples sequentially drawn from prior . 2. New point only accepted if ℒ+1 > ℒ.

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ℒ > ℒ−1 > ⋯ > ℒ2 > ℒ1 > 0 Nested Sampling Algorithm (Naïve) ℒ > ℒ−1 > ⋯ > ℒ2 > ℒ1 > 0 1 ~ 2 −1 1. Samples sequentially drawn from prior . 2. New point only accepted if ℒ+1 > ℒ.

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Building a Nested Sampling Algorithm

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Components of the Algorithm 1. Adding more particles. 2. Knowing when to stop. 3. What to do after stopping.

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Adding More Particles ℒ1 1 > ⋯ > ℒ2 1 > ℒ1 1 > 0 Skilling (2006)

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Adding More Particles ℒ1 1 > ⋯ > ℒ2 1 > ℒ1 1 > 0 Skilling (2006) +1 1 = =0

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Adding More Particles ℒ1 1 > ⋯ > ℒ2 1 > ℒ1 1 > 0 ℒ2 2 > ⋯ > ℒ2 2 > ℒ1 2 > 0 Skilling (2006)

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Adding More Particles ℒ1 1 > ⋯ > ℒ2 1 > ℒ1 1 > 0 ℒ2 2 > ⋯ > ℒ2 2 > ℒ1 2 > 0 Skilling (2006) +1 1 = =0 +1 2 = =0

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Adding More Particles ℒ1 1 > ℒ2 2 > ⋯ > ℒ2 1 > ℒ2 2 > ℒ1 2 > ℒ1 1 > 0 Skilling (2006)

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Adding More Particles ℒ1 1 > ℒ2 2 > ⋯ > ℒ2 1 > ℒ2 2 > ℒ1 2 > ℒ1 1 > 0 Skilling (2006)

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Adding More Particles ℒ1 1 > ℒ2 2 > ⋯ > ℒ2 1 > ℒ2 2 > ℒ1 2 > ℒ1 1 > 0 Skilling (2006)

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Adding More Particles ℒ1 1 > ℒ2 2 > ⋯ > ℒ2 1 > ℒ2 2 > ℒ1 2 > ℒ1 1 > 0 Skilling (2006)

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Adding More Particles ℒ1 1 > ℒ2 2 > ⋯ > ℒ2 1 > ℒ2 2 > ℒ1 2 > ℒ1 1 > 0 Skilling (2006)

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Adding More Particles ln +1 = =0 ln 0 , … , ~ Unif Skilling (2006)

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Adding More Particles ln +1 = =0 ln 0 , … , ~ 2 1 , 2 ~ Unif ⇒ 1 , 2 Order Statistics Skilling (2006)

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Adding More Particles ln +1 = =0 ln 0 , … , ~ Beta 2,1 Skilling (2006) 1 , 2 ~ Unif ⇒ 1 , 2 Order Statistics

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Adding More Particles ℒ1 1 > ℒ2 2 > ⋯ > ℒ2 1 > ℒ2 2 > ℒ1 2 > ℒ1 1 > 0 Skilling (2006)

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Adding More Particles ℒ1 1 > ℒ2 2 > ⋯ > ℒ2 1 > ℒ2 2 > ℒ1 2 > ℒ1 1 > 0 One run with 2 “live points” = 2 runs with 1 live point Skilling (2006) live points dead points

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Adding More Particles ℒ1 1 > ℒ2 2 > ⋯ > ℒ2 1 > ℒ2 2 > ℒ1 2 > ℒ1 1 > 0 One run with K “live points” = K runs with 1 live point Skilling (2006) live points dead points

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Adding More Particles ln +1 = =0 ln 0 , … , ~ Beta 2,1 Skilling (2006) 1 , 2 ~ Unif ⇒ 1 , 2 Order Statistics live points dead points

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Adding More Particles 0 , … , ~ Beta K, 1 Skilling (2006) 1 , … , ~ Unif ⇒ 1 , … , Order Statistics ln +1 = =0 ln

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Adding More Particles 0 , … , ~ Beta K, 1 Skilling (2006) ln +1 = =0 ln + 1 1 , … , ~ Unif ⇒ 1 , … , Order Statistics

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Stopping Criteria = + in

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Stopping Criteria + in >

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Stopping Criteria = =1 + in

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Stopping Criteria ≤ =1 + ℒ K Uniform slab Maximum Likelihood

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Stopping Criteria ≲ =1 + ℒ K Uniform slab Maximum Likelihood

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“Recycling” the Final Set of Particles

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“Recycling” the Final Set of Particles

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“Recycling” the Final Set of Particles … , +1 , … ~ 1 , … , ∼ Unif

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“Recycling” the Final Set of Particles … , +1 , … ~ 1 , … , ∼ Unif ℒ > ℒ−1 > ⋯ > ℒ2 > ℒ1 > 0

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“Recycling” the Final Set of Particles … , +1 , … ~ 1 , … , ∼ Unif ℒ+ > ⋯ > ℒ+1 > ℒ > ℒ−1 > ⋯ > ℒ2 > ℒ1 > 0

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“Recycling” the Final Set of Particles … , +1 , … ~ 1 , … , ∼ Unif ℒ+ > ⋯ > ℒ+1 > ℒ > ℒ−1 > ⋯ > ℒ2 > ℒ1 > 0 + = +1 −+1

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“Recycling” the Final Set of Particles … , + , … ~ −+1 1 , … , ∼ Unif ℒ+ > ⋯ > ℒ+1 > ℒ > ℒ−1 > ⋯ > ℒ2 > ℒ1 > 0 + = +1 −+1

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“Recycling” the Final Set of Particles … , + , … ~ −+1 1 , … , ∼ Unif ℒ+ > ⋯ > ℒ+1 > ℒ > ℒ−1 > ⋯ > ℒ2 > ℒ1 > 0 + = +1 −+1 Rényi Representation

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“Recycling” the Final Set of Particles … , , … ∼ =1 =1 +1 , 1 , … , +1 ∼ Expo Rényi Representation … , + , … ~ −+1 1 , … , ∼ Unif

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“Recycling” the Final Set of Particles … , , … ∼ =1 =1 +1 , 1 , … , +1 ∼ Expo Rényi Representation … , + , … ~ −+1 1 , … , ∼ Unif 1 2 3 −1 …

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“Recycling” the Final Set of Particles … , , … ∼ =1 =1 +1 , 1 , … , +1 ∼ Expo Rényi Representation … , + , … ~ −+1 1 , … , ∼ Unif 1 2 3 +1 …

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“Recycling” the Final Set of Particles ln = =1 ln + 1

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“Recycling” the Final Set of Particles ln + = =1 ln + 1 + ln − + 1 + 1

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“Recycling” the Final Set of Particles ln + = =1 ln + 1 + =1 ln − + 1 − + 2

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“Recycling” the Final Set of Particles ln + = =1 ln + 1 + =1 ln − + 1 − + 2 Exponential Shrinkage Uniform Shrinkage

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Nested Sampling Errors

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Nested Sampling Uncertainties Pictures from this 2010 talk by Skilling.

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Nested Sampling Uncertainties Pictures from this 2010 talk by Skilling.

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Nested Sampling Uncertainties Pictures from this 2010 talk by Skilling. Statistical uncertainties • Unknown prior volumes

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Nested Sampling Uncertainties Pictures from this 2010 talk by Skilling. Statistical uncertainties Sampling uncertainties • Unknown prior volumes • Number of samples (counting) • Discrete point estimates for contours • Particle path dependencies

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Sampling Error: Poisson Uncertainties = ∼ ? ? ? Δ ln Based on Skilling (2006) and Keeton (2011) “Distance” from prior to posterior

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Sampling Error: Poisson Uncertainties ≡ Ω ln Based on Skilling (2006) and Keeton (2011) Kullback-Leibler divergence from to  “information gained”.

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Sampling Error: Poisson Uncertainties ≡ Ω ln = 1 0 1 ℒ ln ℒ − ln Based on Skilling (2006) and Keeton (2011) Kullback-Leibler divergence from to  “information gained”.

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Sampling Error: Poisson Uncertainties = ∼ Δ ln Based on Skilling (2006) and Keeton (2011)

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Sampling Error: Poisson Uncertainties = ∼ Δ ln Based on Skilling (2006) and Keeton (2011) ln

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Sampling Error: Poisson Uncertainties = ∼ Δ ln Based on Skilling (2006) and Keeton (2011) ln ∼ ln

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Sampling Error: Poisson Uncertainties = ∼ Δ ln Based on Skilling (2006) and Keeton (2011) ln ∼ ln ∼ Δ ln 2

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Sampling Error: Poisson Uncertainties = ∼ Δ ln Based on Skilling (2006) and Keeton (2011) ln ∼ ln ∼ Δ ln 2 ∼ Δ ln

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Sampling Error: Poisson Uncertainties = ∼ Δ ln Based on Skilling (2006) and Keeton (2011) ln ∼ ln ∼ Δ ln 2 ∼ =1 + Δ Δ ln = 1/

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Sampling Error: Monte Carlo Noise Formalism following Higson et al. (2017) and Chopin and Robert (2010) = Ω = 1 0 1 X ℒ

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Sampling Error: Monte Carlo Noise = Ω = 1 0 1 X ℒ Formalism following Higson et al. (2017) and Chopin and Robert (2010) X = ℒ = ℒ

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Sampling Error: Monte Carlo Noise = Ω = 1 0 1 X ℒ Formalism following Higson et al. (2017) and Chopin and Robert (2010) X = ℒ = ℒ

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Sampling Error: Monte Carlo Noise = Ω = 1 0 1 X ℒ Formalism following Higson et al. (2017) and Chopin and Robert (2010) X = ℒ = ℒ

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Sampling Error: Monte Carlo Noise Formalism following Higson et al. (2017) and Chopin and Robert (2010) ≈ =1 + = Ω = 1 0 1 X ℒ

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Exploring Sampling Uncertainties One run with K “live points” = K runs with 1 live point! ℒ > ℒ−1 > ⋯ > ℒ2 > ℒ1 > 0

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Exploring Sampling Uncertainties One run with K “live points” = K runs with 1 live point! ℒ1 1 > ⋯ > ℒ2 1 > ℒ1 1 > 0 ℒ2 2 > ⋯ > ℒ2 2 > ℒ1 2 > 0

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Exploring Sampling Uncertainties One run with K “live points” = K runs with 1 live point! ℒ ∙ = ℒ 1 , ℒ 2 , … Original run “strand” “strand”

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Exploring Sampling Uncertainties ℒ ∙ = ℒ 1 , ℒ 2 , … Original run “strand” “strand”

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Exploring Sampling Uncertainties ℒ ∙ = ℒ 1 , ℒ 2 , … Original run “strand” “strand” ℒ ∙ ′ = ℒ 1 ′ , ℒ 2 ′ , … We would like to sample K paths from the set of all possible paths P ℒ , … . However, we don’t have access to it.

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Exploring Sampling Uncertainties ℒ ∙ = ℒ 1 , ℒ 2 , … Original run “strand” “strand” We would like to sample K paths from the set of all possible paths P ℒ , … . However, we don’t have access to it. Use bootstrap estimator. ℒ ∙ ′ = ℒ 1 , ℒ 1 , ℒ 2 , …

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Exploring Sampling Uncertainties ℒ ∙ = ℒ 1 , ℒ 2 , … Original run “strand” “strand” ′ ≈ =1 ′ ′ ′ = ′ ′ We would like to sample K paths from the set of all possible paths P ℒ , … . However, we don’t have access to it. Use bootstrap estimator.

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Nested Sampling In Practice

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Nested Sampling In Practice Higson et al. (2017) arxiv:1704.03459

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Method 0: Sampling from the Prior Higson et al. (2017) arxiv:1704.03459

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Method 0: Sampling from the Prior Higson et al. (2017) arxiv:1704.03459 Sampling from the prior becomes exponentially more inefficient as time goes on.

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Method 1: Constrained Uniform Sampling Feroz et al. (2009) Proposal: Bound the iso-likelihood contours in real time and sample from the newly constrained prior.

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Method 1: Constrained Uniform Sampling Feroz et al. (2009) Issues: • How to ensure bounds always encompass iso-likelihood contours? • How to generate flexible bounds?

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Method 1: Constrained Uniform Sampling Feroz et al. (2009) Issues: • How to ensure bounds always encompass iso-likelihood contours? • How to generate flexible bounds? Bootstrapping. Easier with uniform (transformed) prior.

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Method 1: Constrained Uniform Sampling Feroz et al. (2009) Ellipsoids Balls/Cubes Buchner (2014) MultiNest

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Method 2: “Evolving” Previous Samples Proposal: Generate independent samples subject to the likelihood constraint by “evolving” copies of current live points.

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Method 2: “Evolving” Previous Samples Proposal: Generate independent samples subject to the likelihood constraint by “evolving” copies of current live points.

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Method 2: “Evolving” Previous Samples Proposal: Generate independent samples subject to the likelihood constraint by “evolving” copies of current live points. • Random walks (i.e. MCMC) • Slice sampling • Random trajectories (i.e. HMC) PolyChord

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Method 2: “Evolving” Previous Samples Issues: • How to ensure samples are independent (thinning) and properly distributed within likelihood constraint? • How to generate efficient proposals? • Random walks (i.e. MCMC) • Slice sampling • Random trajectories (i.e. HMC) PolyChord

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Example: Gaussian Shells Feroz et al. (2013)

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Example: Eggbox Feroz et al. (2013)

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Summary: (Static) Nested Sampling

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Summary: (Static) Nested Sampling 1. Estimates the evidence .

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Summary: (Static) Nested Sampling 1. Estimates the evidence . 2. Estimates the posterior .

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Summary: (Static) Nested Sampling 1. Estimates the evidence . 2. Estimates the posterior . 3. Possesses well-defined stopping criteria.

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Summary: (Static) Nested Sampling 1. Estimates the evidence . 2. Estimates the posterior . 3. Possesses well-defined stopping criteria. 4. Combining runs improves inference.

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Summary: (Static) Nested Sampling 1. Estimates the evidence . 2. Estimates the posterior . 3. Possesses well-defined stopping criteria. 4. Combining runs improves inference. 5. Sampling and statistical uncertainties can be simulated from a single run.

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Dynamic Nested Sampling

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Dynamic Nested Sampling Higson et al. (2017) arxiv:1704.03459

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Dynamic Nested Sampling Higson et al. (2017) arxiv:1704.03459

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Dynamic Nested Sampling Higson et al. (2017) arxiv:1704.03459

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Dynamic Nested Sampling Higson et al. (2017) arxiv:1704.03459

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Dynamic Nested Sampling Higson et al. (2017) arxiv:1704.03459

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Dynamic Nested Sampling ℒmax 1 > ℒ 1 1 > ⋯ > ℒ 2 1 > ℒ1 1 > ℒ min 1 ℒ2 2 > ⋯ > ℒ2 2 > ℒ1 2 > 0

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Dynamic Nested Sampling ℒ2 2 > ⋯ > ℒ1 1 … > ℒ1 1 > ℒ2 2 > ℒ1 2 > 0 2 live points 2 live points 1 + 2 live points

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Dynamic Nested Sampling ℒ2 2 > ⋯ > ℒ1 1 … > ℒ1 1 > ℒ2 2 > ℒ1 2 > 0 2 live points 2 live points 1 + 2 live points ln +1 = =1 ln

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Dynamic Nested Sampling 2 live points 2 live points 1 + 2 live points ln +1 = =1 ln +1 ≥ : +1 ∼ Beta(+1 , 1) Constant/Increasing ℒ2 2 > ⋯ > ℒ1 1 … > ℒ1 1 > ℒ2 2 > ℒ1 2 > 0

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Dynamic Nested Sampling ℒ2 2 > ⋯ > ℒ1 1 … > ℒ1 1 > ℒ2 2 > ℒ1 2 > 0 2 live points 2 live points 1 + 2 live points ln +1 = =1 ln +1 ≥ : +1 ∼ Beta(+1 , 1) +> < : +1 , … + ∼ , … , −+1 Constant/Increasing Decreasing sequence

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Static Nested Sampling Exponential shrinkage Uniform shrinkage ln + = =1 ln + 1 + =1 ln − + 1 − + 2

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Dynamic Nested Sampling ln = =1 1 ln + 1 + =1 2 ln 1 − + 1 1 − + 2 + ⋯ + =1 final ln final − + 1 final − + 2 Exponential shrinkage Uniform shrinkage

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Benefits of Dynamic Nested Sampling • Can accommodate new “strands” within a particular range of prior volumes without changing overall statistical framework. • Particles can be adaptively added until stopping criteria are reached, allowing targeted estimation.

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Sampling Uncertainties (Static) ℒ ∙ = ℒ 1 , ℒ 2 , … Original run “strand” “strand” “strand” We would like to sample K paths from the set of all possible paths P ℒ , … . However, we don’t have access to it. Use bootstrap estimator. ℒ ∙ ′ = ℒ 1 , ℒ 1 , ℒ 2 , …

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Sampling Uncertainties (Dynamic) ℒ ∙ = ℒ 1 , ℒ 2 , … Original run ℒ min 1 = −∞ (originated from the prior) “strand” ℒ min 2 = 2 (originated interior to the prior)

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Sampling Uncertainties (Dynamic) ℒ ∙ = ℒ 1 , ℒ 2 , … Original run ℒ min 1 = −∞ (originated from the prior) “strand” ℒ min 2 = 2 (originated interior to the prior) We would like to sample: paths from P ℒ , … and paths from P ℒ , … .

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Dynamic Nested Sampling • Can accommodate new “strands” within a particular range of prior volumes without changing overall statistical framework. • Particles can be adaptively added until stopping criteria are reached, allowing targeted estimation.

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Allocating Samples Higson et al. (2017) arxiv:1704.03459 ∼ > ∼ , = + 1 − Posterior weight Evidence weight Weight Function

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Allocating Samples Higson et al. (2017) arxiv:1704.03459

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Allocating Samples 3-D correlated multivariate Normal Static Dynamic (posterior) Dynamic (evidence)

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Allocating Samples 3-D correlated multivariate Normal Static Dynamic (posterior) Dynamic (evidence)

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How Many Samples is Enough?

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How Many Samples is Enough? • Ill-posed question: depends on application!

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How Many Samples is Enough? • In any sampling-based approach to estimating with , how many samples are necessary?

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How Many Samples is Enough? • In any sampling-based approach to estimating with , how many samples are necessary? Assume general case: we want D-dimensional and i densities to be “close”. “True” posterior constructed over same “domain”.

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How Many Samples is Enough? • In any sampling-based approach to estimating with , how many samples are necessary? Assume general case: we want D-dimensional and i densities to be “close”. = =1 ln We want access to P , but we don’t know .

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How Many Samples is Enough? • In any sampling-based approach to estimating with , how many samples are necessary? Assume general case: we want D-dimensional and i densities to be “close”. ′ = =1 ′ ln ′ We want access to Pr , but we don’t know . Use bootstrap estimator.

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How Many Samples is Enough? • In any sampling-based approach to estimating with , how many samples are necessary? Assume general case: we want D-dimensional and i densities to be “close”. ′ = =1 ′ ln ′ We want access to Pr , but we don’t know . Use bootstrap estimator. Random variable

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How Many Samples is Enough? • In any sampling-based approach to estimating with , how many samples are necessary? Assume general case: we want D-dimensional and i densities to be “close”. Possible stopping criterion: fractional (%) variation in H. We want access to Pr , but we don’t know . Use bootstrap estimator.

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Dynamic Nested Sampling Summary 1. Can sample from multi-modal distributions. 2. Can simultaneously estimate the evidence and posterior . 3. Combining independent runs improves inference (“trivially parallelizable”). 4. Can simulate uncertainties (sampling and statistical) from a single run. 5. Enables adaptive sample allocation during runtime using arbitrary weight functions. 6. Possesses evidence/posterior-based stopping criteria.

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Examples and Applications

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Dynamic Nested Sampling with dynesty dynesty.readthedocs.io • Pure Python. • Easy to use. • Modular. • Open source. • Parallelizable. • Flexible bounding/sampling methods. • Thorough documentation!

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Example:

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Example: Linear Regression (Posterior) = {, , ln }

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Example: Linear Regression (Posterior) Corner Plot

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Example: Linear Regression (Posterior) Corner Plot Trace Plot

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Example: Multivariate Normal (Evidence) = 1 , 2 , 3

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Example: Multivariate Normal (Evidence) “Summary” Plot

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Example: Multivariate Normal (Errors) Static Dynamic “Summary” Plot

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Example: 50-D Multivariate Normal “Summary” Plot = 1 , … , 50

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Example: Eggbox = 1 , 2

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Example: Eggbox

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Example: 2-D and 10-D LogGamma = 1 , … ,

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Example: 2-D and 10-D LogGamma 10-D 2-D = 1 , … ,

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Application:

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Application: • All results are preliminary but agree with results from MCMC methods (derived using emcee). • Samples allocated with 100% posterior weight, automated stopping criterion (2% fractional error in simulated KLD). • dynesty was substantially (~3-6x) more efficient at generating good samples than emcee, before thinning.

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Application: Modeling Galaxy SEDs = ln ∗ , ln , 5 , 6 , 2 D=15 With: Joel Leja, Ben Johnson, Charlie Conroy

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Application: Modeling Galaxy SEDs With: Joel Leja, Ben Johnson, Charlie Conroy

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Application: Modeling Galaxy SEDs Fig: Joel Leja With: Joel Leja, Ben Johnson, Charlie Conroy

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Application: Supernovae Light Curves Fig: Open Supernova Catalog (LSQ12dlf), James Guillochon With: James Guillochon, Kaisey Mandel = , 4 , 3 , 3 , 2 D=12

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Application: Supernovae Light Curves With: James Guillochon, Kaisey Mandel

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Application: Molecular Cloud Distances With: Catherine Zucker, Doug Finkbeiner, Alyssa Goodman = 2 , 5 , 5 , D=13

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Application: Molecular Cloud Distances With: Catherine Zucker, Doug Finkbeiner, Alyssa Goodman