Typical Sets: What They Are and How to (Hopefully) Find Them

Although typical sets are important in understanding how/why sampling algorithms (do not) work, they are rarely taught when most astronomers are introduced to sampling methods such as Markov Chain Monte Carlo (MCMC). I introduce the idea of typical sets using some basic examples and show why they make sampling difficult in higher dimensions. I then outline how their behavior shapes various MCMC algorithms such as (Adaptive) Metropolis-Hastings, ensemble sampling, and Hamiltonian Monte Carlo. See https://github.com/joshspeagle/typical_sets for additional resources.

Transcript

1. Typical Sets: What They Are and How to (Hopefully) Find Them Josh Speagle jspeagle@cfa.harvard.edu Based on this talk by Michael Betancourt at StanCon.
2. Intended Audience • Some experience with the basics of Bayesian statistics.
3. Intended Audience • Some experience with the basics of Bayesian statistics. • Some experience using MCMC for research.
4. Intended Audience • Some experience with the basics of Bayesian statistics. • Some experience using MCMC for research. • Have heard of ensemble sampling methods such as emcee.
5. Bayesian Inference
6. Bayesian Inference Pr , M = Pr , M Pr |M Pr M Bayes’ Theorem
7. Bayesian Inference Pr , M = Pr , M Pr |M Pr M Bayes’ Theorem Parameters
8. Bayesian Inference Pr , M = Pr , M Pr |M Pr M Bayes’ Theorem Data Parameters
9. Bayesian Inference Pr , M = Pr , M Pr |M Pr M Bayes’ Theorem Data Parameters Model
10. Bayesian Inference Pr , M = Pr , M Pr |M Pr M Bayes’ Theorem
11. Bayesian Inference Pr , M = Pr , M Pr |M Pr M Bayes’ Theorem Prior
12. Bayesian Inference Pr , M = Pr , M Pr |M Pr M Bayes’ Theorem Prior Likelihood
13. Bayesian Inference Pr , M = Pr , M Pr |M Pr M Bayes’ Theorem Prior Likelihood Posterior
14. Bayesian Inference Pr , M = Pr , M Pr |M Pr M Bayes’ Theorem Prior Likelihood Posterior Evidence
15. Bayesian Inference = ℒ Bayes’ Theorem ≡ Ω ℒ Posterior Likelihood Prior Evidence
16. Bayesian Inference = ℒ Bayes’ Theorem Posterior Likelihood Prior Evidence ≡ Ω ℒ
17. Where is the posterior? ≡ Ω
18. Where is the posterior? ≡ {: =}
19. Where is the posterior? ≡ 0 ∞
20. Where is the posterior? ≡ 0 ∞ =
21. Where is the posterior? ≡ 0 ∞ “Amplitude” “Volume” =
22. = Where is the posterior? ≡ 0 ∞ “Typical Set”
23. Typical Sets: Gaussian Example
24. Typical Sets: Gaussian Example ∝ 0 ∞ − 2 2
25. Typical Sets: Gaussian Example ∝ 0 ∞ − 2 2 ∝ 0 ∞ − 2 2 −1
26. Typical Distance Typical Sets: Gaussian Example
27. = Where is the posterior? ≡ 0 ∞ “Typical Set”
28. = Where is the posterior? ≡ 0 ∞ “Typical Set”
29. = Where is the posterior? ≡ 0 ∞ “Typical Set”
30. = Where is the posterior? ≡ 0 ∞ “Typical Set” MCMC wants to draw samples from this “shell”
31. Tension in the Metropolis Update ′ = min 1, ′ ′ ′
32. Tension in the Metropolis Update ′ = min 1, ′ ′ ′ Proposal
33. Tension in the Metropolis Update ′ = min 1, ′ ′ ′ “Volume”
34. Tension in the Metropolis Update ′ = min 1, ′ ′ ′ “Volume” “Amplitude”
35. Metropolis-Hastings
36. Metropolis-Hastings ′ = Normal ′ = , =
37. Metropolis-Hastings ′ = Normal ′ = , = Typical Distance
38. Metropolis-Hastings ′ = Normal ′ = , = Typical Distance
39. Metropolis-Hastings ′ = Normal ′ = , =
40. Metropolis-Hastings ′ = Normal ′ = , =
41. Ideal Metropolis-Hastings ′ = Normal ′ = , = Typical Separation
42. Ideal Metropolis-Hastings ′ = Normal ′ = , = Typical Separation M-H
43. Ideal Metropolis-Hastings ′ = Normal ′ = , = s Typical Separation Adaptive M-H
44. Ensemble Sampling
45. Ensemble Sampling
46. Ensemble Sampling
47. Ensemble Sampling
48. Ensemble Sampling
49. Ensemble Sampling
50. emcee ′ = min 1, ′ −1 ~ = 1 from 1 , 0 otherwise “Stretch” factor
51. Ideal Typical Separation emcee M-H
52. Ideal Typical Separation emcee M-H emcee
53. Ideal Typical Separation emcee M-H emcee
54. Ideal Typical Separation emcee M-H emcee After weighting by acceptance probability
55. emcee ′ = min 1, ′ −1 ~ = 1 from 1 , 0 otherwise “Stretch” factor
56. emcee ′ = min 1, ′ −1 ~ = 1 from 1 , 0 otherwise “Stretch” factor 
57. Summary • Volume scales as . • The posterior density depends on both volume and amplitude. • Most of the posterior is concentrated in a “shell” around the best solution called the typical set. • MCMC draws samples from the typical set.
58. But what about corner plots?
59. But what about corner plots? 2-dimensional projection of D-dimensional shell
60. But what about corner plots? 2-dimensional projection of D-dimensional shell
61. But what about corner plots? 2-dimensional projection of D-dimensional shell
62. Hamiltonian Monte Carlo
63. Hamiltonian Monte Carlo
64. Hamiltonian Monte Carlo
65. Hamiltonian Monte Carlo Treat the particle at position q as a point mass with mass matrix M and momentum p. Pr , ∝ , = − −1 2 Hamiltonian
66. Hamiltonian Monte Carlo Pr , ∝ , = − −1 2 Treat the particle at position q as a point mass with mass matrix M and momentum p. = = −1 = − = ln Hamiltonian Hamilton’s Equations
67. Hamiltonian Monte Carlo ′, −′ , = min 1, Pr ′, −′ Pr , ∼ Normal = , =
68. Typical Distance Hamiltonian Monte Carlo ∼ Normal = , =
69. Typical Distance Hamiltonian Monte Carlo ∼ Normal = , =
70. Ideal Typical Separation M-H emcee Hamiltonian Monte Carlo ∼ Normal = , =
71. Ideal Typical Separation M-H emcee Hamiltonian Monte Carlo ∼ Normal = , = HMC