Program on Quasi-Monte Carlo and High-Dimensional Sampling Methods for Applied Mathematics Opening Workshop, Deterministic Sampling for Bayesian Computation - Roshan Vengazhiyil, Aug 31, 2017
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The document discusses deterministic sampling techniques for Bayesian computation, highlighting the limitations of traditional Markov Chain Monte Carlo (MCMC) methods and proposing alternatives such as Quasi-Monte Carlo (QMC) and minimum energy designs. It emphasizes the challenges of generating samples from probability densities that are only known up to a proportionality constant and suggests using surrogate models or deterministic sampling for improved efficiency. The research explores optimization strategies for experimental design and the computational demands of these methods.
![Transformation to the Unit Hypercube
• “We only need to consider point sets in
[0,1]𝑝𝑝
, otherwise transform using inverse
distribution function”.
• If 𝑥𝑥1, … , 𝑥𝑥𝑝𝑝are independent with distribution
functions 𝐹𝐹1, … , 𝐹𝐹𝑝𝑝, then transform a
uniform sample 𝑢𝑢1, … , 𝑢𝑢𝑝𝑝 using
𝐹𝐹1
−1
𝑢𝑢1 , … , 𝐹𝐹𝑝𝑝
−1
𝑢𝑢𝑝𝑝 .
15](https://image.slidesharecdn.com/170831rvengazhiyillecture-170901163534/85/Program-on-Quasi-Monte-Carlo-and-High-Dimensional-Sampling-Methods-for-Applied-Mathematics-Opening-Workshop-Deterministic-Sampling-for-Bayesian-Computation-Roshan-Vengazhiyil-Aug-31-2017-15-320.jpg)
![A New Algorithm
39
𝑓𝑓(𝑥𝑥)𝛾𝛾
Tempering:
𝛾𝛾 = 0 𝑛𝑛 QMC points in [0,1]𝑝𝑝
𝛾𝛾 = 1/(𝐾𝐾 − 1) 𝑛𝑛 MED points out of 2𝑛𝑛 points
…
𝛾𝛾 = 1 𝑛𝑛 MED points out of K𝑛𝑛 points](https://image.slidesharecdn.com/170831rvengazhiyillecture-170901163534/85/Program-on-Quasi-Monte-Carlo-and-High-Dimensional-Sampling-Methods-for-Applied-Mathematics-Opening-Workshop-Deterministic-Sampling-for-Bayesian-Computation-Roshan-Vengazhiyil-Aug-31-2017-39-320.jpg)
Program on Quasi-Monte Carlo and High-Dimensional Sampling Methods for Applied Mathematics Opening Workshop, Deterministic Sampling for Bayesian Computation - Roshan Vengazhiyil, Aug 31, 2017
- 1. Deterministic Sampling for Bayesian Computation V. Roshan Joseph 1 Joseph, V. R., Dasgupta, T., Tuo, R., and Wu, C. F. J. (2015) “Sequential exploration of complex surfaces using minimum energy designs,” Technometrics, 57, 64-74. Joseph, V. R., Wang, D., Li, G, Tuo, R. and Lv, S. (2017). “Deterministic sampling from expensive posteriors”, Manuscript in preparation. Supported by NSF DMS 1712642
- 2. Bayesian Methods • Bayesian model • Posterior where 𝐶𝐶 = ∫ 𝑝𝑝 𝒚𝒚 𝜽𝜽 𝑝𝑝(𝜽𝜽) d𝜽𝜽 is the normalizing constant. 2 𝑝𝑝 𝜽𝜽 𝒚𝒚 = 1 𝐶𝐶 𝑝𝑝 𝒚𝒚 𝜽𝜽 𝑝𝑝(𝜽𝜽)
- 3. Bayesian Computation • Many intractable high-dimensional integrals – Posterior distribution – Posterior summaries – Marginal posterior distributions – Posterior predictive distributions
- 4. Markov Chain Monte Carlo Methods • Metropolis et al. 1953, Hastings 1970, Geman and Geman 1984, Gelfand and Smith 1990, …
- 5. An Example 5
- 9. Disadvantages of MCMC • 𝑓𝑓(𝒙𝒙) may be expensive and time consuming to evaluate. • 𝑔𝑔(𝒙𝒙) may be expensive and time consuming to evaluate. 9 Simulation: Integration: 𝒙𝒙𝑖𝑖~𝑓𝑓 𝒙𝒙 , 𝑖𝑖 = 1, … , 𝑛𝑛
- 10. Simulation problem Hung, Joseph, Melkote (2009) Expensive
- 11. Integration problem Uncertainty sources Input Output Propagation of uncertainty 4 11 • Support points: Simon Mak’s talk on Tuesday
- 12. Two Possible Solutions 1. Approximate 𝑓𝑓(𝑥𝑥) using an easy-to- evaluate surrogate model ̂𝑓𝑓(𝑥𝑥) and generate MCMC sample using ̂𝑓𝑓(𝑥𝑥). – High dimensional function approximation is hard! 2. Use a deterministic sample that is well- spaced instead of a random sample. – QMC 12
- 13. Deterministic Sample • Quasi-Monte Carlo (QMC): 50-point Sobol sequence 13
- 14. Deterministic Sample • Quasi-Monte Carlo (QMC): 50-point Sobol sequence 14
- 15. Transformation to the Unit Hypercube • “We only need to consider point sets in [0,1]𝑝𝑝 , otherwise transform using inverse distribution function”. • If 𝑥𝑥1, … , 𝑥𝑥𝑝𝑝are independent with distribution functions 𝐹𝐹1, … , 𝐹𝐹𝑝𝑝, then transform a uniform sample 𝑢𝑢1, … , 𝑢𝑢𝑝𝑝 using 𝐹𝐹1 −1 𝑢𝑢1 , … , 𝐹𝐹𝑝𝑝 −1 𝑢𝑢𝑝𝑝 . 15
- 16. Limitations in Bayesian problems • 𝑥𝑥1, … , 𝑥𝑥𝑝𝑝 are rarely independent. • Joint density is known only up to a proportionality constant: – Distribution function is unknown. – Inverse distribution function is unknown. – So this rarely works! 16
- 17. Another recommended strategy • Use an importance sampling density whose inverse distribution function can be easily obtained. • However, finding an importance sampling density in a Bayesian problem is very hard. – So this rarely works! 17
- 18. Research Problem • How to generate a deterministic sample directly from a probability density that is known only up to a proportionality constant? 18
- 19. Minimum Energy Designs • Experimental region: • Experimental design: – View the n points as charged particles inside a box. – They will occupy positions that will minimize the total potential energy. 19
- 20. MED-continued • Let q(xi) be the charge at xi. • Then, minimize 20
- 21. Charge function-Intuition • Charge should be inversely proportional to density value. 21
- 23. Limiting distribution 23 Theorem: There exists a probability measure 𝑃𝑃 such that 𝑃𝑃𝑛𝑛 converges to 𝑃𝑃. Moreover, 𝑃𝑃 has a density 𝑓𝑓 over 𝑋𝑋 with 𝑓𝑓 𝒙𝒙 ∝ 1 𝑞𝑞2𝑝𝑝 𝒙𝒙 .
- 24. Charge Function • So if we choose the charge function to be then we can obtain the target distribution. 24
- 27. Sphere Packing Problems • Minimum Riesz energy points – Borodachov, Hardin, Saff (2008a,b) 27
- 28. Uniform Distribution • MED for 𝑛𝑛 = 25, 𝑝𝑝 = 2 • The effective sample size for each dimension is only 𝑛𝑛1/𝑝𝑝 . 28
- 30. Choice of s 30 MaxPro Low discrepancy and good space-filling!
- 31. MaxPro Design 31 Joseph, V. R., Gul, E., and Ba, S. (2015). “Maximum Projection Designs for Computer Experiments,” Biometrika, 102, 371-380.
- 33. A Greedy Algorithm • It can get stuck in a local optimum, but good designs are produced with a good starting point: • Requires a global optimization at each step-> Computationally very expensive! 33
- 34. Complex probability distributions where C is the (unknown ) normalizing constant. C is not needed! 𝑓𝑓 𝑥𝑥 = 1 𝐶𝐶 ℎ(𝑥𝑥) 34
- 36. 36
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- 38. Computational time & Number of evaluations • Global optimization using Generalized simulated annealing (GSA) 38
- 39. A New Algorithm 39 𝑓𝑓(𝑥𝑥)𝛾𝛾 Tempering: 𝛾𝛾 = 0 𝑛𝑛 QMC points in [0,1]𝑝𝑝 𝛾𝛾 = 1/(𝐾𝐾 − 1) 𝑛𝑛 MED points out of 2𝑛𝑛 points … 𝛾𝛾 = 1 𝑛𝑛 MED points out of K𝑛𝑛 points
- 40. 40
- 41. 41
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- 44. 44
- 46. Computational time & Number of evaluations 46 21 hours 0.5 hours
- 47. Conclusions 47 0 Cost of evaluations MCMC QMC+ MCMC QMC+ Function approximation +MCMC