![What is a generative model?
●
Ng, A. Y. and Jordan, M. I. (2002). On discriminative vs. generative classifiers: A comparison of logistic regression and naive bayes. In Advances in neural information processing systems, pages 841–848.
●
Rasmus Bååth, Video Introduction to Bayesian Data Analysis, Part 1: What is Bayes?: https://www.youtube.com/watch?time_continue=366&v=3OJEae7Qb_o
Hypothesis of underlying
mechanisms
AKA: “Learning the class”
No shorcuts!
D =
[2, 8, ..., 9]
θ, μ, ξ, ...
Parameters
Model
Bayesian Inference](https://image.slidesharecdn.com/unigerlunchonlineversion-191031123714/85/Striving-to-Demystify-Bayesian-Computational-Modelling-8-320.jpg)
![Frequentist Inference
Y = [7, ..., 2]
X = [2, ..., 9]
Y = a * X + b
Y ~ N(a * X + b, σ2
) L = p(D | a, b, σ2
)
argmax(Σln(p(D | a, b, σ2
))
a b σ2
MLE
“True” Population
?](https://image.slidesharecdn.com/unigerlunchonlineversion-191031123714/85/Striving-to-Demystify-Bayesian-Computational-Modelling-21-320.jpg)
![“True” Population
D = [7, 3, 2]
Sample: N=3
Sampling Distribution
e.g. F distribution
Test
statistic
Inter-group var./
Intra-group var.
∞
H0
mean
Central Limit
Theorem
“Long range” probability
● Sampling distribution applet: http://onlinestatbook.com/stat_sim/sampling_dist/index.html
Frequentist Inference](https://image.slidesharecdn.com/unigerlunchonlineversion-191031123714/85/Striving-to-Demystify-Bayesian-Computational-Modelling-22-320.jpg)
Striving to Demystify Bayesian Computational Modelling
- 1. Striving to Demystify Bayesian Computational Modelling UNIGE R Lunch Speaker: Marco Wirthlin @marcowirthlin
- 2. Abstract and Context of the Talk Abstract Bayesian approaches to computational modelling have experienced a slow, but steady gain in recognition and usage in academia and industry alike, accompanying the growing availability of evermore powerful computing platforms at shrinking costs. Why would one use such techniques? How are those models conceived and implemented? Which is the recommended workflow? Why make life hard when there are P-values? In his talk, Marco Wirthlin will first attempt an introduction to statistical notions supporting Bayesian computation and explain the difference to the Frequentist framework. In the second half, an example of a recommended workflow is outlined on a simple toy model, with simulated data. Live coding will be used as much as possible to illustrate concepts on an implementational level in the R language. Ample literature and media references for self-learning will be provided during the talk. Context and Licence This talk was performed in the context of the “R Lunch” on the 29 of October 2019 at the University of Geneva and was organized by Elise Tancoigne (@tancoigne) & Xavier Adam (@xvrdm). Many thanks for inviting me! :D Code (if any) is licenced under the BSD (3 clause), while the text licence is CC BY-NC 4.0. Any derived work has been cited. Please contact me if you see non-attributed work (marco.wirthlin@gmail.com). ONLINE VERSION
- 3. The difference between Bayesian and Frequentist statistics is how probability theory is applied to achieve their respective goals.
- 5. Computational Models Mathematical Description Solution Algorithm Implementation Relationship between Variables Generative Structure Y = a * X + b Simulation Goal Conceptual Hygiene
- 7. Computational Models Mathematical Description Solution Algorithm Implementation Relationship between Variables Generative Structure Y = a * X + b Simulation Goal Conceptual Hygiene
- 8. What is a generative model? ● Ng, A. Y. and Jordan, M. I. (2002). On discriminative vs. generative classifiers: A comparison of logistic regression and naive bayes. In Advances in neural information processing systems, pages 841–848. ● Rasmus Bååth, Video Introduction to Bayesian Data Analysis, Part 1: What is Bayes?: https://www.youtube.com/watch?time_continue=366&v=3OJEae7Qb_o Hypothesis of underlying mechanisms AKA: “Learning the class” No shorcuts! D = [2, 8, ..., 9] θ, μ, ξ, ... Parameters Model Bayesian Inference
- 9. Golf Example (Gelman 2019) Andrew Gelman: https://mc-stan.org/users/documentation/case-studies/golf.html
- 12. Andrew Gelman: https://mc-stan.org/users/documentation/case-studies/golf.html Lets think about how golf works!
- 16. Computational Models can have different Goals Decision making Quantifying how well we understand a phenomenon Fisherian (frequentist) statistics, hypothesis tests, p-values... Bayesian statistics, generative models, simulations... Richard McElreath: Statistical Rethinking book and lectures (https://www.youtube.com/watch?v=4WVelCswXo4)
- 17. Rasmus Bååth, Video Introduction to Bayesian Data Analysis, Part 1: What is Bayes?: https://www.youtube.com/watch?time_continue=366&v=3OJEae7Qb_o
- 18. Frequentist statistics: parameter “fixed” Bayesian statistics: parameter “variable” 乁 ( )⁰⁰ ĹĹ ⁰⁰ ㄏ Frequentist statistics: maximum likelihood Bayesian statistics: likelihood ┐('д')┌ ¯_( _ )_/¯⊙ ʖ⊙ Rasmus Bååth, Video Introduction to Bayesian Data Analysis, Part 1: What is Bayes?: https://www.youtube.com/watch?time_continue=366&v=3OJEae7Qb_o
- 19. Bayesian Inference (BI) BI Likelihoods Frequentist Statistics Graphical Models Probabilistic Programming Background Implementation
- 20. Likelihoods Normal Distribution L = p(D | θ) x ~ N(μ, σ2 ) “The probability that D belongs to a distribution with mean μ and SD σ” L = p(D | μ, σ2 ) “X is normally distributed” PDF: Fix parameters, vary data L: Fix data, vary parameters Dashboard link: https://seneketh.shinyapps.io/Likelihood_Intuition ● https://www.youtube.com/watch?v=ScduwntrMzc
- 21. Frequentist Inference Y = [7, ..., 2] X = [2, ..., 9] Y = a * X + b Y ~ N(a * X + b, σ2 ) L = p(D | a, b, σ2 ) argmax(Σln(p(D | a, b, σ2 )) a b σ2 MLE “True” Population ?
- 22. “True” Population D = [7, 3, 2] Sample: N=3 Sampling Distribution e.g. F distribution Test statistic Inter-group var./ Intra-group var. ∞ H0 mean Central Limit Theorem “Long range” probability ● Sampling distribution applet: http://onlinestatbook.com/stat_sim/sampling_dist/index.html Frequentist Inference
- 23. ● Sampling distribution applet: http://onlinestatbook.com/stat_sim/sampling_dist/index.html “When a frequentist says that the probability for "heads" in a coin toss is 0.5 (50%) she means that in infinitively many such coin tosses, 50% of the coins will show "head"”. Frequentist Inference If the H0 is very unlikely “in the long run”, we accept H1 .
- 24. Bayesian Inference 1) Build a Generative model, imitate the data structure. (model) 2) Assign principled probabilities to your parameters. (priors) 3) Update those probabilities by incorporating data. (posteriors) Probabilities: Epistemic/ontological uncertainty. How well do we understand our phenomenon? https://betanalpha.github.io/assets/case_studies/principled_bayesian_workflow.html
- 25. Y = a * X + b Y ~ N(a * X + b, σ2 ) Assign probabilities to the parameters p(D, a, b, σ2 ) = p(D | a, b, σ2 )*p(a)*p(b)*p(σ2 ) a ~ N(1, 0.1) b ~ N(4, 0.5) σ2 ~ G(1, 0.1) p(a) p(b) p(σ2 ) p(D | a, b, σ2 ) p(D | θ)
- 26. From model to code Y = a * X + b a ~ N(1, 0.1) b ~ N(4, 0.5) σ2 ~ G(1, 0.1) Y ~ N(a * X + b, σ2 ) ● More examples: https://mc-stan.org/users/documentation/case-studies
- 33. Build Generative Model Assign principled probabilities to parameters Simulate Data Check if assumptions are OK with Data Expose model to Data Bayesian Workflow ● https://betanalpha.github.io/assets/case_studies/principled_bayesian_workflow.html ● Toward a principled Bayesian workflow in cognitive science: https://arxiv.org/abs/1904.12765 Simulate data with fitted parameters
- 34. Short demo.
- 35. Thank you for your attention… and endurance!
- 36. Additional Slides Sources, links and more!
- 37. Who to follow on Twitter? ● Chris Fonnesbeck @fonnesbeck (pyMC3) ● Thomas Wiecki @twiecki (pyMC3) Blog: https://twiecki.io/ (nice intros) ● Bayes Dose @BayesDose (general info and papers) ● Richard McElreath @rlmcelreath (ecology, Bayesian statistics expert) All his lectures: https://www.youtube.com/channel/UCNJK6_DZvcMqNSzQdEkzvzA ● Michael Betancourt @betanalpha (Stan) Blog: https://betanalpha.github.io/writing/ Specifically: https://betanalpha.github.io/assets/case_studies/principled_bayesian_workflow.html ● Rasmus Bååth @rabaath Great video series: http://www.sumsar.net/blog/2017/02/introduction-to-bayesian-data-analysis-part-one/ ● Frank Harrell @f2harrell (statistics sage) Great Blog: http://www.fharrell.com/ ● Andrew Gelman @StatModeling (statistics sage) https://statmodeling.stat.columbia.edu/ ● Judea Pearl @yudapearl Book of Why: http://bayes.cs.ucla.edu/WHY/ (more about causality, BN and DAG) ● AND MANY MORE!
- 38. All sources in one place! About Generative vs. Discriminative models: Ng, A. Y. and Jordan, M. I. (2002). On discriminative vs. generative classifiers: A comparison of logistic regression and naive bayes. In Advances in neural information processing systems, pages 841–848. Rasmus Bååth: Video Introduction to Bayesian Data Analysis, Part 1: What is Bayes?: https://www.youtube.com/watch?time_continue=366&v=3OJEae7Qb_o When to use ML vs. Statistical Modelling: Frank Harrell's Blog: http://www.fharrell.com/post/stat-ml/ http://www.fharrell.com/post/stat-ml2/ Frequentist approach: How do sampling distributions work (applet): http://onlinestatbook.com/stat_sim/sampling_dist/index.html Bayesian inference and computation: John Kruschke: Doing Bayesian Data Analysis: A Tutorial with R, JAGS, and Stan Chapter 5 Rasmus Bååth: http://www.sumsar.net/blog/2017/02/introduction-to-bayesian-data-analysis-part-two/ Richard McElreath: Statistical Rethinking book and lectures (https://www.youtube.com/watch?v=4WVelCswXo4 ) Many model examples in Stan: https://mc-stan.org/users/documentation/case-studies About Bayesian Neural Networks: https://alexgkendall.com/computer_vision/bayesian_deep_learning_for_safe_ai/ https://twiecki.io/blog/2018/08/13/hierarchical_bayesian_neural_network/ Volatility Examples: Hidden Markov Models: https://github.com/luisdamiano/rfinance17 Volatility Garch Model and Bayesian Workflow: https://luisdamiano.github.io/personal/volatility_stan2018.pdf Dictionary: Stats ↔ ML https://ubc-mds.github.io/resources_pages/terminology/ The Bayesian Workflow: https://betanalpha.github.io/assets/case_studies/principled_bayesian_workflow.html Algorithm explanation applet for MCMC exploration of the parameter space: http://elevanth.org/blog/2017/11/28/build-a-better-markov-chain/ Probabilistic Programming Conference Talks: https://www.youtube.com/watch?v=crvNIGyqGSU
- 39. Dictionary: Stats ML↔ Check: https://ubc-mds.github.io/resources_pages/terminology/ for more terminology Statistics Estimation/Fitting Hypothesis Data Point Regression Classification Covariates Parameters Response Factor Likelihood Machine learning / AI ~ Learning ~ Classification rule ~ Example/ Instance ~ Supervised Learning ~ Supervised Learning ~ Features ~ Features ~ Label ~ Factor (categorical variables) ~ Cost Function (sometimes)
- 40. Slides about Bayesian Computation
- 41. Bayesian Inference ● John Kruschke: Doing Bayesian Data Analysis: A Tutorial with R, JAGS, and Stan Chapter 5 ● Rasmus Bååth: http://www.sumsar.net/blog/2017/02/introduction-to-bayesian-data-analysis-part-two/ ● Richard McElreath: Statistical Rethinking book and lectures (https://www.youtube.com/watch?v=4WVelCswXo4)
- 42. Bayesian Inference ● John Kruschke: Doing Bayesian Data Analysis: A Tutorial with R, JAGS, and Stan Chapter 5 ● Rasmus Bååth: http://www.sumsar.net/blog/2017/02/introduction-to-bayesian-data-analysis-part-two/ ● Richard McElreath: Statistical Rethinking book and lectures (https://www.youtube.com/watch?v=4WVelCswXo4)
- 43. Bayesian Inference ● John Kruschke: Doing Bayesian Data Analysis: A Tutorial with R, JAGS, and Stan Chapter 5 ● Rasmus Bååth: http://www.sumsar.net/blog/2017/02/introduction-to-bayesian-data-analysis-part-two/ ● Richard McElreath: Statistical Rethinking book and lectures (https://www.youtube.com/watch?v=4WVelCswXo4) Discrete Values: Just sum it up! :) Cont. Values: Integration over complete parameter space... :(
- 44. Bayesian Inference: ● John Kruschke: Doing Bayesian Data Analysis: A Tutorial with R, JAGS, and Stan Chapter 5 ● Rasmus Bååth: http://www.sumsar.net/blog/2017/02/introduction-to-bayesian-data-analysis-part-two/ ● Richard McElreath: Statistical Rethinking book and lectures (https://www.youtube.com/watch?v=4WVelCswXo4) Averaging over the complete parameter space via integration is impractical! Solution: We sample from the conjugate probability distribution with smart MCMC algorithms! (Subject of another talk)
- 45. Bayesian Inference ● http://elevanth.org/blog/2017/11/28/build-a-better-markov-chain/ ● John Kruschke: Doing Bayesian Data Analysis: A Tutorial with R, JAGS, and Stan Chapter 5 ● Rasmus Bååth: http://www.sumsar.net/blog/2017/02/introduction-to-bayesian-data-analysis-part-two/ ● Richard McElreath: Statistical Rethinking book and lectures (https://www.youtube.com/watch?v=4WVelCswXo4) Lets compute this and sample from it!
- 46. Principles of Bayesian Modeling What we know about a cognitive phenomenon before the measurement. Knowledge gained by the measurement. Knowledge gained after measurement. To-date scientific knowledge. Model/ Hypothesis Data Updated believes
- 48. Requirements: ● If: The prior distribution is specified by a function that is easily evaluated. This simply means that if you specify a value for θ, then the value of p(θ) is easily determined, ● And If: The value of the likelihood function, p(D|θ), can be computed for any specified values of D and θ. ● Then: The method produces an approximation of the posterior distribution, p(θ|D), in the form of a large number of θ values sampled from that distribution (same as we would sample people from a determined population). MCMC Requirements
- 51. True param: 0.7 Source: Kruschke, J. (2014). HMCStep-Sizes