Bayes, birds and brains: applications of inference and probabilistic modelling

1. Bayes, birds and brains: applications of inference and probabilistic modelling Stephen Roberts Pattern Analysis & Machine Learning Research Group University of Oxford http://www.robots.ox.ac.uk/~parg

2. Introduction • Bayesian inference has profound impact in principled handling of uncertainty in practical computation • What this talks aims to do: • Give an overview of Bayesian inference applied to several real-world problem domains • What it does not aim to do: • Give endless equations – these are important and elegant, but are in the open literature

4. What’s wrong with sampling? • Nothing – apart from speed and the occasional frequentist way the samples are used… • Not much we can do for speed, lots of clever methods out there which help • Bayesian sampling (using Gaussian processes) • Bayes-Hermite Quadrature [O’Hagan, 1992] • Bayesian Monte Carlo [Rasmussen and Ghahramani, 2003] • Variational Bayes

5. Variational Bayes - 1 Log posterior bounded below by Free Energy

6. p(x1,x2) x2 x1 q(x1)q(x2) x2 Variational Bayes - 2 • A slow and (often) painful derivation leads to an iterative node update for DAGs • This converges to a local optimum – like EM and many other energy minimization approaches – get the priors right!

7. Variational Bayes – 3 • Some relief via Variational Message Passing • Same update equations as VB but at fraction of pain • Conjugate exponential family only • Pearl-style message passing on graphical model using sufficient statistics only • For many applications the factored natureof degrades performance – need non-factored proposals – extra computation (e.g. some VB models with mixture model nodes)

8. Priors & model selection • Sensitivity to priors • posterior distributions conjugate with priors • empirically can be a problem – know the domain • Model selection • evaluate set of models for VFE. Rank or integrate • use VFE in ‘quasi-RJMCMC’ approach • use ridge regression (ARD, weight decay) priors

9. Simple example - ICA • ICA (Bell & Sejnowski, Attias, Amari….) • Bayesian ICA (Roberts 1998, Attias 1999, Miskin & MacKay 2000, Choudrey & Roberts 2000)

10. vbICA – graphical model

11. vbICA – simple example

12. How many sources?

13. vbICA - VFE

14. vbICA - RJMCMC

15. vbICA – source suppression

16. Mixtures of ICAs (Choudrey & Roberts, 2001)

17. VFE and RR (ARD) work

18. Example (& a cautionary tale)

19. … a cautionary tale

20. Ridge regression…

21. Variational free energy

22. Recovered images

23. A cautionary conclusion… • In high noise regimes use ARD to focus on a small subset of models • These are then investigated in more detail using variational free energy

24. Priors • If we have prior knowledge regarding the sources or the mixing , we can use it. • Spatial information • Positive mixing • Positive sources • Structured observations

25. Positivity

26. An example

27. ICA with different priors Which is ‘correct’ though?

28. Which is ‘correct’?

29. Epilepsy data

32. Structure priors • To be an ICA matrix, must lie on manifold of decorrelating matrices. These form ‘great circles’ in the matrix space. • Can parameterize using co-ordinates on the manifold. • Where do priors lie?

33. Gaussian priors Gaussian priors on the mixing process just form great circles – they have little impact if we already compute on the decorrelating manifold as they are aligned with the manifold.

34. Structure priors Potential from brain source – dipole potential (Knuth) • Sensor coupling has spatial structure, close by sensors have similar coupling weights • Gaussian process prior: still gives great circle in matrix space but very informative as not aligned along decorrelating manifold

35. Phantom head experiments Without prior With prior

36. Brain-Computer Interfaces ‘direct’ control in real-time using ‘thought’

37. Motor cortex • When we plan a movement, changes take place in the motor cortex, whether or not the movement takes place. • When we change cognitive task, changes take place in the cortex.

38. Cursor control – real time BCI Bayes – rejection Bayes baseline dT = 50ms max median min

39. The curse of feedback bits t (secs)

40. 11100001010101010101010010010101010010101110101001010101001010100100101001010100101010100010100101000010010010001110000101010101010101001001010101001010111010100101010100101010010010100101010010101010001010010100001001001000 Information Engines “If all you have is a hammer, everything looks like a nail.” P(action|data) = 0.95 DATA ENGINE (MODEL) INFORMATION potential entropy machine useful

41. 111000010101010101010100100101010100101011101010010101010010101001001010010101001010101000101001010000100100100010010101111000010101010101010100100101010100101011101010010101010010101001001010010101001010101000101001010000100100100010010101 Inside or outside the box? The inferences we make, and the actions decided upon have an impact on the data Learning with changing objectives

42. Sequential Bayesian inference • Particle filter (SIR) • Humble (variational) Kalman filter: Bayesian inference assuming generalized (non-) linear Gaussian system • Adaptive system using sequential variational Bayes • BCI application • (Musical score following)

43. Generalised non-linear dynamic classifier Copes with missing inputs & labels, input noise and bit errors on labels as well as time-delayed information Penny, Sykacek, Lowne

44. Foul stuff…

45. What it buys us

47. Thanks to John Gann

48. PART II: BIRDS

49. Hidden Markov birds… • Global Positioning System (GPS) • 15g units • Strapped to back of bird • Gives position every second Roberts, Guilford, Biro, Lau 2004,5 JTB