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On Computational and Probabilistic Inference Rajat Mani Thomas - PowerPoint PPT Presentation

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On Computational and Probabilistic Inference Rajat Mani Thomas Objectives: Revisiting Bayesian inference . A look at Likelihood and the prior Probabilistic programming: Automating Bayesian (like) Inference Probabilistic Toolkit:

  1. On Computational and Probabilistic Inference Rajat Mani Thomas

  2. Objectives: Revisiting Bayesian inference . ● A look at Likelihood and the prior ● Probabilistic programming: Automating Bayesian (like) Inference ● Probabilistic Toolkit: (i) Markov Chain Monte Carlo, ● (ii) Variational inference Deep probabilistic models ● Probabilistic programming@ICTP - April - 2019

  3. What is the main problem we are trying to solve ? Inference Probabilistic programming@ICTP - April - 2019

  4. What is the main problem we are trying to solve ? Inference Probabilistic programming@ICTP - April - 2019

  5. What is the main problem we are trying to solve ? Inference Probabilistic programming@ICTP - April - 2019

  6. What is the main problem we are trying to solve ? Inference Probabilistic programming@ICTP - April - 2019

  7. What is the main problem we are trying to solve ? Inference Posterior Likelihood Prior Probabilistic programming@ICTP - April - 2019

  8. What is the main problem we are trying to solve ? Inference Posterior Likelihood Prior Probabilistic programming@ICTP - April - 2019

  9. The Likelihood function Generative model of the Data: Think Simulations ● The plausibility of a given parameter in generating a particular outcome. ● Scoring function. ● What has now appeared is that the mathematical concept of probability is ... inadequate to express our mental confidence or [lack of confidence] in making ... inferences, and that the mathematical quantity which usually appears to be appropriate for measuring our order of preference among different possible populations does not in fact obey the laws of probability. To distinguish it from probability, I have used the term "likelihood" to designate this quantity.... — R. A. Fisher, Statistical Methods for Research Workers [2] Probabilistic programming@ICTP - April - 2019

  10. Prior Prior as a searchlight Probabilistic programming@ICTP - April - 2019

  11. Prior Prior as a searchlight Probabilistic programming@ICTP - April - 2019

  12. The two ( hypothesized ) ways to Intelligence Deep Learning Programs with R.V. and Probabilistic calculations Probabilistic programming@ICTP - April - 2019

  13. The two ( hypothesized ) ways to Intelligence Deep Learning Programs with R.V. and Probabilistic calculations Probabilistic programming@ICTP - April - 2019

  14. The two ( hypothesized ) ways to Intelligence Deep Learning Probabilistic programming@ICTP - April - 2019

  15. The two ( hypothesized ) ways to Intelligence Programs with R.V. and Probabilistic calculations Probabilistic programming@ICTP - April - 2019

  16. The two ( hypothesized ) ways to Intelligence Programs with R.V. and Probabilistic calculations Probabilistic programming@ICTP - April - 2019

  17. The case for probabilistic programming languages Democratize model building and Inference Probabilistic programming@ICTP - April - 2019

  18. What is a probabilistic program? Any imperative or functional program with two additional constructs: 1. Ability to draw values at random from distributions 2. Ability to condition variables with observations Probabilistic programming@ICTP - April - 2019

  19. PP - A CS way of doing Statistical Inference Computer Science Parameters —- Program — Outputs Generation Probabilistic Programming Parameters —- Program — Observations Inference Statistics Unknowns —- Generative model — Observations Probabilistic programming@ICTP - April - 2019

  20. An example of a PP Probabilistic programming@ICTP - April - 2019

  21. An example of a PP Probabilistic programming@ICTP - April - 2019

  22. An example of a PP Probabilistic programming@ICTP - April - 2019

  23. An example of a PP Probabilistic programming@ICTP - April - 2019

  24. Program traces Probabilistic programming@ICTP - April - 2019

  25. Program traces F.wood (NIPS, 2015) Probabilistic programming@ICTP - April - 2019

  26. Program traces F.wood (NIPS, 2015) Probabilistic programming@ICTP - April - 2019

  27. Getting to the posterior Probabilistic programming@ICTP - April - 2019

  28. Getting to the posterior Probabilistic programming@ICTP - April - 2019

  29. Getting to the posterior Probabilistic programming@ICTP - April - 2019

  30. Getting to the posterior : Importance Sampling 1. Run the program N times generate traces: 2. Approximate the posterior: Probabilistic programming@ICTP - April - 2019

  31. Algorithms that make it feasible Variational MCMC Inference Probabilistic programming@ICTP - April - 2019

  32. Markov Chain Monte Carlo Assumes very little… Can you run your program and generate samples? - SIMULATION Can you calculate (even an un-normalized) density of observation? - SCORE Probabilistic programming@ICTP - April - 2019

  33. Markov Chain Monte Carlo - 101 Probabilistic programming@ICTP - April - 2019

  34. Markov Chain Monte Carlo - 101 Probabilistic programming@ICTP - April - 2019

  35. Markov Chain Monte Carlo - 101 Probabilistic programming@ICTP - April - 2019

  36. Markov Chain Monte Carlo - 101 Probabilistic programming@ICTP - April - 2019

  37. Markov Chain Monte Carlo - 101 Probabilistic programming@ICTP - April - 2019

  38. Markov Chain Monte Carlo - 101 Probabilistic programming@ICTP - April - 2019

  39. MCMC - 101 : Metropolis-Hastings Probabilistic programming@ICTP - April - 2019

  40. Variational Inference 1. Choose a “nice” family of distributions 2. Cast inference as an optimization problem Probabilistic programming@ICTP - April - 2019

  41. Variational Inference 1. Choose a “nice” family of distributions 2. Cast inference as an optimization problem Probabilistic programming@ICTP - April - 2019

  42. Variational Inference Probabilistic programming@ICTP - April - 2019

  43. Variational Inference Probabilistic programming@ICTP - April - 2019

  44. Advantages Amortized inference: Easily expressive language for model Write your Generative model, PP takes care of the inference What would it look like? Simulation Power spectra Probabilistic programming@ICTP - April - 2019

  45. Examples of what PP can do Construct a world in which 20% of balls go into the basket Probabilistic programming@ICTP - April - 2019

  46. Deep Learning + Probabilistic Programming An example of the Variational Autoencoder Probabilistic programming@ICTP - April - 2019

  47. Deep Learning + Probabilistic Programming An example of the Variational Autoencoder Probabilistic programming@ICTP - April - 2019

  48. Canonical Autoencoder Probabilistic programming@ICTP - April - 2019

  49. Canonical Autoencoder Probabilistic programming@ICTP - April - 2019

  50. Canonical Autoencoder Probabilistic programming@ICTP - April - 2019

  51. Variational Inference Probabilistic programming@ICTP - April - 2019

  52. Variational Autoencoder Probabilistic programming@ICTP - April - 2019

  53. Variational Autoencoder Probabilistic programming@ICTP - April - 2019

  54. Variational Autoencoder Deep NN Probabilistic programming@ICTP - April - 2019

  55. Deep Mixed Probabilistic Models A new paradigm for model building Probabilistic programming@ICTP - April - 2019

  56. Deep Mixed Probabilistic Models A new paradigm for model building Probabilistic programming@ICTP - April - 2019

  57. Deep Mixed Probabilistic Models A new paradigm for model building , Probabilistic programming@ICTP - April - 2019

  58. Conclusions 1. Probabilistic programming languages are making it easy to run inference on anything that can be written as a computer code 2. Advances in MCMC techniques like Hamiltonian Monte Carlo and Variational Inference are the workhorses of inference algorithms 3. Mixed programming paradigms will be the way forward Probabilistic programming@ICTP - April - 2019

  59. References: Probabilistic Programming Language: —————- Frank Wood (NIPS Tutorial on Probabilistic Programming), Probabilistic Programming Applications: —————- Josh Tenenbaum (MIT), Noah Goodman (Stanford) MCMC/HMC: ———————— Michael Betancourt, https://arxiv.org/pdf/1701.02434.pdf Variational Inference: —————- Max Welling, Dirk Kingma, Danilo Rezende, David Blei Frameworks for (Deep) Probabilistic Programming: Stan, TFP (Google, Tensorflow), Pyro (UBER, pytorch), Pyprob (Atilim, Frank Wood’s lab), Probabilistic C, ... Probabilistic programming@ICTP - April - 2019

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