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

Stephen Roberts

Pattern Analysis & Machine Learning Research GroupUniversity of Oxford

http://www.robots.ox.ac.uk/~parg

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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

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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

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Variational Bayes - 1

Log posterior bounded below by Free Energy

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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!

x1

x2

x2

p(x1,x2)

q(x1)q(x2)

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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 nature of

degrades performance – need non-factored proposals – extra computation (e.g. some VB models with mixture model nodes)

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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

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Simple example - ICA

• ICA (Bell & Sejnowski, Attias, Amari….)• Bayesian ICA (Roberts 1998, Attias 1999, Miskin &

MacKay 2000, Choudrey & Roberts 2000)

)](||)([ spspKL i

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vbICA – graphical model

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vbICA – simple example

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How many sources?

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vbICA - VFE

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vbICA - RJMCMC

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vbICA – source suppression

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Mixtures of ICAs

(Choudrey & Roberts, 2001)

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VFE and RR (ARD) work

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Example (& a cautionary tale)

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… a cautionary tale

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Ridge regression…

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Variational free energy

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Recovered images

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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

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Priors

• If we have prior knowledge regarding the sources or the mixing , we can use it.

• Spatial information• Positive mixing• Positive sources• Structured observations

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Positivity

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An example

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ICA with different priors

Which is ‘correct’ though?

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Which is ‘correct’?

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Epilepsy data

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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?

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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.

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Structure priors

• 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

Potential from brain source – dipole potential (Knuth)

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Phantom head experiments

Without prior With prior

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Brain-Computer Interfaces

‘direct’ control in real-time using ‘thought’

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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.

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Cursor control – real time BCI

max median min

dT = 50msbaseline

Bayes – rejection

Bayes

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The curse of feedback

bits

t (secs)

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Information Engines

DATA ENGINE (MODEL) INFORMATION

potential entropy machine useful

1110000101010101010101001001010101001010111010100101010100101010010010100101010010101010001010010100001001001000

P(action|data)

= 0.95

“If all you have is a hammer, everything looks like a nail.”

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111000010101010101010100100101010100101011101010010101010010101001001010010101001010101000101001010000100100100010010101

Inside or outside the box?

The inferences we make, and the actions decided upon have an impact on the data

Learning with changing objectives

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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)

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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

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Foul stuff…

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What it buys us

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Thanks to John Gann

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PART II: BIRDS

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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

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Strategies, navigation & uncertainty

• Dynamics of flight gives indication of navigation strategy

• Easy to measure from GPS co-ordinates

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States – Hidden Markov Models

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Variational Bayes

P(H,V) intractable. Tractable proposal Q(H) forms strict lower bound to P(H,V)

Hidden Markov Model States S with observation model parameters θ

Learning involves minimization of F wrt S, θ

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Posteriors over states

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Influence from landscape

• Navigation states dependent upon many factors– ‘Markov pigeon property’– Visual landscape– Magnetic field (Hall effect sensor)– Other birds – etc…

• Allow for coupling between factors

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Coupled models

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Variable lag between chains

GM now loopy. Sample, node cluster or use EP (thanks Iead Rezek)

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Gaussian process pigeons…

Gives a null hypothesis as GP paths independent of landscape

Mike Osborne & Richard Mann

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Birds are known to be highly right-eye dominant

Primary attention to edges in landscape

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Bird-brained conclusions

• Simple explanatory ‘navigation states’

• Terrain information may be important

• Co-released birds may co-operate

• Perceptive field may be possible to infer from tracks

• Edges may be primary information source for navigation

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Thanks…

• Steve Reece (for being the skeptic)

• Mike Osborne & Charles Fox (for glaring at me if I start straying from the path of Bayes)

• Riz Choudrey, Will Addison, Will Penny, Richard Everson, Evengelos Roussos (ICA)

• John Gann, Duncan Lowne, Peter Sykacek (brains)

• Tim Guilford, Rob Freeman, Richard Mann, Mike Osborne (bird brains)

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Recent sequential VB

• Very successful in musical score following

– or ‘How to replace your bass player with a laptop’

– Charles Fox

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Single fragment Bayes net

• uncertain number and matchings of notes

• Inference is hard because of loops

Start-time (invese) tempo

beats

notes

data

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Start with good priors

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Converged

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Start with very poor priors

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Failed

Arrg! Floating point exception…

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