![Stochastic Variational Inference (Hoffman et al. 2013)
● Minimize exclusive divergence by maximizing ELBO:
KL(q(θ;λ)|| p(θ|d)) = log p(d|θ) - ELBO
● ELBO can be estimated by stochastic sampling of parameters
ELBO = E[log q(θ;λ) - log p(d,θ)]
● Optimization works well if samples have low variance
○ Relies on tricks like reparametrization, dependency structure handling and
Rao-Blackwellization](https://image.slidesharecdn.com/introductiontoprobabilisticprogrammingwithpyro-190424081008/85/Introduction-to-probabilistic-programming-with-pyro-27-320.jpg)
Introduction to probabilistic programming with pyro
- 1. Introduction to Probabilistic Programming with Pyro Ahmad Salim Al-Sibahi
- 2. Is this a cat or a dog?
- 3. Could be worse. Not sure how, but it could be. CC BY Kassy at SketchPort
- 4. Disposition ● What is probabilistic programming? ● Probabilistic models in Pyro ● Techniques for Bayesian Inference
- 5. Probabilistic Programming is Machine Learning...
- 6. It is all about inference!
- 7. Inference is answering questions about data! Image Observed Data (d) Cat or Dog? Category Latent Variable (𝛉)
- 8. Inference is answering questions about data! Sequence of Amino Acids Folded Structure Observed Data (d) Latent Variable (𝛉)
- 9. Inference is answering questions about data! Voucher Observed Data (d) Payment Information Latent Variable (𝛉) Sender: Asterix SARL Recipient: Hamlet-Kierkegaard ApS Issue Date: April 1st 2015 Total Amount: EUR 98,691.00
- 10. Modelling as joint distributions p(d,𝛉)
- 11. Traditional Inference ℓ(𝛉’|d) ∝ p(d, 𝛉’) Find 𝛉’ maximizing likelihood:
- 12. Traditional Inference ℓ(𝛉’|d) ∝ p(d| 𝛉’) Find 𝛉’ maximizing likelihood:
- 13. Limitations ● Only a single value is given as output 𝛉’ ● Not possible to incorporate domain knowledge ● Deep neural networks are not humanly interpretable and require lots of data
- 14. Bayesian Inference p(𝛉|d) ∝ p(d, 𝛉) Find probability distribution
- 15. Bayesian Inference p(𝛉|d) ∝ p(d|𝛉)p(𝛉) Find probability distribution
- 16. Bayesian inference allows quantification of uncertainty
- 17. Probabilistic Programming Language Frameworks i HackPPL - Facebook Pyro - Uber and Linux Foundation TensorFlow Probability - Google Infer.Net - Microsoft PyMC3 - NumFOCUS
- 19. Logistic Regression ● Given height and weight of person, then infer whether male or female ● Use logistic regression!
- 21. Techniques for Bayesian Inference
- 22. The challenge of inference p(𝛉|d) ∝ p(d|𝛉)p(𝛉)
- 23. The challenge of inference p(𝛉|d) = p(d|𝛉)p(𝛉) p(d)
- 24. The challenge of inference p(𝛉|d) = p(d|𝛉)p(𝛉) ∫𝛉 p(d|𝛉) Can be arbitrarily complex (Multi-modal, saddlepoints, discontinuous) Often intractable
- 26. Variational Inference ● Posit a guide q(θ;λ) to approximate posterior p(θ|d) ● Find optimal λ’ so that q(θ;λ’) has a good fit w.r.t. p(θ|d)
- 27. Stochastic Variational Inference (Hoffman et al. 2013) ● Minimize exclusive divergence by maximizing ELBO: KL(q(θ;λ)|| p(θ|d)) = log p(d|θ) - ELBO ● ELBO can be estimated by stochastic sampling of parameters ELBO = E[log q(θ;λ) - log p(d,θ)] ● Optimization works well if samples have low variance ○ Relies on tricks like reparametrization, dependency structure handling and Rao-Blackwellization
- 28. Amortized Inference (Ritchie et al. 2016) ● Assume that guide can be factorized into local and global parameters: q(θ;λ) = q(θG;λG) ∏i q(θi;λi, λG) ● Learn a function f that maps local parameters from data q(θ|d;λ) = q(θG;λG) ∏i q(θi;f(di), λG) ● The function f can be a deep neural network!
- 29. Reflections on Variational Inference Pros ● Work with expressive probabilistic programming models ● Quantifies uncertainty ● Scalable to large datasets Cons ● Discrete variables can introduce high variance ● Can be over-confident in prediction ● No guarantees on approximation correctness
- 30. Sampling ● Estimate posterior p(θ|d) by drawing many high-probability samples of parameters θ1, …, θn ● Samples can be proposed by custom distribution q(θ) and accepted when p(θi|d) is sufficiently high
- 31. Hamiltonian Monte Carlo (Duane et al. 1987) ● Markov chain Monte Carlo-based ● Relies on gradient information ● No U-Turn Sampler (NUTS) allows automated step-size adaptation (Hoffman and Gelman 2011) θ0 θ1 θ2 θ3 Animation https://chi-feng.github.io/mcmc-demo/
- 32. Reflections on Sampling Pros ● Most flexible technique for inference ● Captures precise correlations between parameters ● Provides precise characterization of true posterior with enough samples Cons ● Not efficiently scalable to large datasets ● Sensitive to landscape of true posterior distribution ● Latent variables can pose a problem for differentiation
- 33. Wrapping up
- 34. Probabilistic Programming ● Probabilistic Programming is important because it allows incorporating domain knowledge and quantifying uncertainty ● Pyro allows specifying both classical and deep probabilistic models ● Modern inference algorithms are very powerful, and easily accessible in Pyro
- 35. Hands-on