Overview of Sampling Methods in Markov Chain Monte Carlo

Overview of Sampling Methods in Markov Chain Monte Carlo
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This content covers various sampling methods in Markov Chain Monte Carlo including Rejection Sampling, Importance Sampling, and MCMC Sampling. It delves into representing distributions, drawbacks of Importance Sampling, and the motivation behind Markov Chain Monte Carlo Sampling. The illustrations provided offer insights into adaptive sampling techniques and methods like Metropolis-Hastings and Gibbs sampling.

  • Sampling Methods
  • Markov Chain Monte Carlo
  • Rejection Sampling
  • Importance Sampling
  • MCMC Sampling
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Uploaded on Mar 03, 2025 | 0 Views

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Presentation Transcript

  1. Markov Chain Monte Carlo 10701 Recitation Pengtao Xie 3/3/2025 1

  2. Outline Sampling Methods Rejection Sampling Importance Sampling MCMC Sampling 3/3/2025 2

  3. Outline Sampling Methods Rejection Sampling Importance Sampling MCMC Sampling 3/3/2025 3

  4. How to represent a distribution Closed form representation Gaussian distribution, Dirichlet distribution, Multinomial distribution Sample based representation Draw samples from the distribution and use samples to compute expectation, variance, etc 3/3/2025 4

  5. Outline Sampling Methods Rejection Sampling Importance Sampling MCMC Sampling 3/3/2025 5

  6. May reject a lot of samples 3/3/2025 6

  7. Outline Sampling Methods Rejection Sampling Importance Sampling MCMC Sampling 3/3/2025 7

  8. Importance sampling Drawback: hard to find a Q which matches well with P, may give little importance to most samples 3/3/2025 8

  9. Outline Sampling Methods Rejection Sampling Importance Sampling MCMC Sampling 3/3/2025 9

  10. Markov Chain Monte Carlo Sampling Motivation In rejection sampling and importance sampling, Q is fixed. May reject or give little importance to most samples Idea Use an adaptive Q Methods Metropolis-Hastings Gibbs sampling 3/3/2025 10

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  24. Summary Is distribution Q adaptive? NO YES Is accept rate 1? Is accept rate 1? NO YES NO YES Rejection Sampling Importance Sampling Gibbs Sampling MH 3/3/2025 24

  25. References Slides courtesy Professor Eric Xing, 10708 Graphical Models http://www.cs.cmu.edu/~epxing/Class/10708/lectures/lecture16-MC.pdf http://www.cs.cmu.edu/~epxing/Class/10708/lectures/lecture17-MCMC.pdf MCMC theory http://www.cs.cmu.edu/~epxing/Class/10708/lectures/lecture17-MCMC.pdf Slides 15-21 3/3/2025 25

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