1. Start MATLAB

2. Konrad Kording and Ian Stevenson
Bayes tutorial
Konrad Kording
and Ian Stevenson

3. Why care? Experimental results are noisy Our own sensors are noisy
See for example
Faisal AA, Selen LP, Wolpert DM (2008) Noise in the nervous system. NATURE REVIEWS NEUROSCIENCE 9: 292

4. Part I: Probabilities in decoding
Movement
onset
repetition
repetition
Obviously we may want to not classify direction of movement but estimate the direction of movement. These techniques are very similar to the ones we use here and equally use the same Bayesian formalism.
time
time

5. Part II: Probabilities in perception
Brain estimates P(stimulus|percept)
Combine information across cues

6. The tutorial Learn the concepts
Try it yourself (anything for you to type is in red courier)
Use it in the future 
Remark: references in lecture notes available at
Part I: decoding movement intent
Part II: perception: decoding the world

7. Part I: decoding movement intent
See lots of cool videos and interesting papers at:
For experimental labs see (in no particular order):
For decoding techniques see (equally in random order):
Apologies for all the labs I forgot to include
From Andy Schwartz’s group

8. Part I: Probabilities in decoding
Movement
onset

9. Simulated data 1 Neuron two directions (100,000 trials)
load datasetStart
(remember Capitalization is important for MATLAB)
neuronL
neuronR
Firing rate during left trial i
Firing rate during right trial i … …
The simulated neuron has Gaussian tuning properties – this is a very simple minded model for a neuron. It turns out though, that the assumption of Gaussianity often works rather well for decoding even when neurons do not have Gaussian statistics (see the work of Shenoy).

10. Histograms plotHistograms Left 2000 hist(neuronL) number 1000 Right
Right
2000
hist(neuronR)
number
1000 20 40 60 80
100
120
140
firing rate [sp/s]

11. Probability

12. Probabilities plotProbability Left 0.06 0.04 probability 0.02 0.06
hist(neuronL)/length(neuronL)
probability
0.02
0.06
Right
0.04
hist(neuronR)/length(neuronR)
probability
0.02 20 40 60 80
100
firing rate [sp/s]

13. Gaussian approximations
plotProbabilitiesFromGauss
Left
0.06
0.04
probability
0.02
0.06
Right
0.04
pXgMS=1/(sqrt(2*pi)*sigma)*…
exp(-(x-mu).^2/(2*sigma^2))
probability
0.02 20 40 60 80
100
firing rate [sp/s]

14. Bayes rule for decoding
If we use a situation in which p(L) is not equal to p(R) then the system will be (optimally) biased in its judgements.
In this tutorial p(L)=p(R)=.5

15. Decoding decodeOneNeuron .04 Left pSgL p(S|L) .02 .04 p(S|R) Right
.04
p(S|R)
Right
pSgR
.02
As we go to the left (low firing rates) the probability of right decreases but much less so than the probability of left. Therefore, the ratio in the lower equation goes towards zero. The opposite is true for high firing rates. 1
Decoding
pSgL./(pSgL+pSgR)
P(L|S) 20 30 40 50 60 70 80 90
100
firing rate [Hz]

16. Typical curves in the CNS1 1
p(L|S)
p(L|S)
100
Spikes [Hz]
100
Spikes [Hz]
Not very informative
Very informative
The width over which the curve goes from small values to high values will depend on the ratio of the distance between the Gaussians and their width.
This is the typical case
in the CNS

17. Combining info across neurons
Assumption (naïve Bayes)
Movement intent
Naïve Bayes is a very good technique in the limit of many observed variables (neurons) and small numbers of observations for some exciting reasons that do not fit into the notes section here. See:
Ng AY (2004) Feature selection, L 1 vs. L 2 regularization, and rotational invariance. ACM International Conference Proceeding Series
Ng AY, Jordan MI (2002) On Discriminative vs. Generative Classifiers: A comparison of logistic regression and Naive Bayes. In: NIPS, vol 14
Neuron 1
Neuron 2 …

18. Exercise: combine two neurons
Same Bayes rule as above

19. CombineTwoNeurons
combineTwoNeurons
pLgN1
pLgN2

20. Classifying based on 1 neuron
nClassifiedRightWhileLeftN1 = 30
nClassifiedLeftWhileRightN1 = 32
nClassifiedRightWhileLeftN2 = 32
nClassifiedLeftWhileRightN2 = 32
About 30% wrong in all cases
In fact it is possible to mathematically calculate the expected numbers of errors in this case.

21. Exercise: combine two neurons
combineTwoNeurons is already prepared so that really only this equation needs to be used.
pN1gL = probability of Neuron 1 spikes given left movement
pN1gR = probability of Neuron 1 spikes given right movement
pN2gL = probability of Neuron 1 spikes given left movement
pN2gR = probability of Neuron 1 spikes given right movement
edit combineTwoNeurons

22. Exercise: combine information from two neurons
pBoth=pN1gL.*pN2gL*.5./(pN1gL.*pN2gL*.5+pN1gR.*pN2gR*.5); 50
100
150
200
0.5 1
trial number
p(L|N1)
decoding from Neuron 1 only
p(L|N2)
decoding from Neuron 2 only
p(L|N1,N2)
decoding from both Neurons
It does help and instead of 30 mistakes it only does about 20.

23. Model comparison: Cross Validate
Train
Test
For a fascinating discussion of such effects see: Puzzlingly High Correlations in fMRI Studies of Emotion, Personality, and Social Cognition by
Edward Vul, Christine Harris, Piotr Winkielman, & Harold Pashler
In this tutorial we ignore the problem

24. Combine info across n neurons
Data: Recordings from Lee Miller’s group
Thanks to Lee Miller and Anil Cherian. The dataset was rebinned from the original results and we aplogize for the way we mangled the data. Please do not draw any scientific conclusions from this – the conversion that we did to get the matrix we are using was a pretty fast hack.
realData

25. Real Data

26. Decoding from real Neurons
realNeuronDecoding
pLeft(i)=prod(pNgLeft(i,:))
Just implementing the above equation
>99% correct

27. Relevance of decoding Gene Chip data Social Neuroscience Marketing
Estimating cellular networks
etc
Here is a partial list of techniques each of which may work great on some set of problems
Support Vector machines
Boosting
L1 regularized logistic regression
Remark: there are better techniques than
naïve Bayes for many cases

28. Part II: Probabilities in perception
Perception is noisy – both for audition and for vision
Cues can be combined.
Brain estimates P(stimulus|percept)
Combine information across cues

29. Example: McGurk Effect
Found this video on the internet but was unable to find out whose video this was. If you did the video – please contact me so I can acknowledge you.
First described by:
Harry McGurk and John MacDonald in "Hearing lips and seeing voices", Nature 264, (1976).

30. Perception (multiple modalities)
Movement intent
World state
Sensor 1
Neuron 1
Sensor 2
Neuron 2 … …
This Bayesian framework has emerged in the late 80s early 90s.
Here are some of the early influential papers:
Ghahramani Z (1995) Computational and psychophysics of sensorimotor integration. In: Brain and Cognitive Science. Massachusetts Institute of Technology, Cambridge
Yuille A, Bulthoff HH (1996) Bayesian decision theory and psychophysics. In: Knill D, Richards W (eds) Perception as Bayesian Inference. Cambridge University Press., Cambridge, U.K.
Landy MS, Maloney LT, Johnston EB, Young M (1995) Measurement and modeling of depth cue combination: in defense of weak fusion. Vision Res 35:

31. Visuo-auditory cue combination
Various labs have variants of this kind of experiment. Projectors or monitors are often used. Some scientists use headphones.
Some references
Hairston WD, Wallace MT, Vaughan JW, Stein BE, Norris JL, Schirillo JA (2003) Visual localization ability influences cross-modal bias. J Cogn Neurosci 15: 20-29
Wallace MT, Roberson GE, Hairston WD, Stein BE, Vaughan JW, Schirillo JA (2004) Unifying multisensory signals across time and space. Exp Brain Res 158:
Alais D, Burr D (2004) The ventriloquist effect results from near-optimal bimodal integration. Curr Biol 14:
Choe CS, Welch RB, Gilford RM, Juola JF (1975) The "ventriloquist effect": visual dominance or response bias. Perception and Psychophysics 18: 55-60
Typical setup: from Hairston & Schirillo 2004

32. What is different discrete estimation for direction decoding
continuous 1D estimation now
.04
Left
p(S|L)
.02

33. Likelihood plotProbabilitySpaceVision probability -10 -8 -6 -4
visual percept -2
It’s the difference between the actual source and the perceived position that matters for the probability 2 4 6 8 10
-10 -5 5 10
position of stimulus

34. Combining two cues combineVisionAudition 0.04 0.03 probability 0.02
perceived V
perceived A
0.03
probability
0.02
In reality there may be a third factor, the prior. Priors are important for sensorimotor integration, e.g.:
Adams WJ, Graf EW, Ernst MO (2004) Experience can change the 'light-from-above' prior. Nat Neurosci 7:
Brainard DH, Freeman WT (1997) Bayesian color constancy. J Opt Soc Am A Opt Image Sci Vis 14:
Ernst MO, Bulthoff HH (2004) Merging the senses into a robust percept. Trends Cogn Sci 8:
Flanagan JR, Bittner JP, Johansson RS (2008) Experience Can Change Distinct Size-Weight Priors Engaged in Lifting Objects and Judging their Weights. Current Biology
Kersten D, Mamassian P, Yuille A (2004) Object perception as Bayesian inference. Annu Rev Psychol 55:
Kording KP, Wolpert DM (2004) Bayesian integration in sensorimotor learning. Nature 427:
Miyazaki M, Nozaki D, Nakajima Y (2005) Testing Bayesian models of human coincidence timing. J Neurophysiol 94:
Miyazaki M, Yamamoto S, Uchida S, Kitazawa S (2006) Bayesian calibration of simultaneity in tactile temporal order judgment. Nat Neurosci 9:
Stocker AA, Simoncelli EP (2006) Noise characteristics and prior expectations in human visual speed perception. Nat Neurosci 9:
Tassinari H, Hudson TE, Landy MS (2006) Combining priors and noisy visual cues in a rapid pointing task. J Neurosci 26:
Posterior~pV.*pA
0.01 -8 -6 -4 -2 2 4 6 8
Position

35. What if vision is better?
combineVisionAudition2
0.09
0.08
0.07
0.06
probability
0.05
0.04
0.03
0.02
0.01 -8 -6 -4 -2 2 4 6 8
Position

36. Dependence on position of visual feedback
dependenceOnCuePosition 6 4 2
best estimate [cm] -2 -4 -6 -8 -6 -4 -2 2 4 6 8
feedback position [cm]

37. Typical experimental strategy
Measure how good one piece of information is, e.g.
Measure how good another piece of information is, e.g.
Measure weight, e.g.
Compare to prediction

38. Near universal finding
Vision & Audition
Priors & Likelihood for movement
Vision & Proprioception
Texture & Disparity
Cognitive science
Here is a general list of Bayesian approaches to human behavior. It is highly incomplete.
Alais D, Burr D (2004) The ventriloquist effect results from near-optimal bimodal integration. Curr Biol 14:
Albrecht DW, Zukerman I, Nicholson. AE (1998) Bayesian Models for Keyhole Plan Recognition in an Adventure Game. User Modeling and User-Adapted Interaction 8: 5-47
Barber MJ, Clark JW, Anderson CH (2003) Neural representation of probabilistic information. Neural Comput 15:
Battaglia PW, Schrater PR (2007) Humans trade off viewing time and movement duration to improve visuomotor accuracy in a fast reaching task. J Neurosci 27:
Beierholm U, Shams L, Kording K, Ma WJ (2008) Comparing Bayesian models for multisensory cue combination without mandatory integration. In: Neural Information Processing Systems, Vancouver
Brainard DH, Freeman WT (1997) Bayesian color constancy. J Opt Soc Am A Opt Image Sci Vis 14:
Bresciani JP, Dammeier F, Ernst MO (2006) Vision and touch are automatically integrated for the perception of sequences of events. J Vis 6:
Chater N, Tenenbaum JB, Yuille A (2006) Probabilistic models of cognition: conceptual foundations. Trends Cogn Sci 10:
Courville AC, Daw ND, Touretzky DS (2006) Bayesian theories of conditioning in a changing world. Trends Cogn Sci 10:
Ernst M, O., (2006a) A Bayesian view on multimodal cue integration. In: Knoblich G, Thornton, I., M., Grosjean, M., Shiffrar, M., (ed) Human body perception from the inside out. Oxford University Press., New York, pp
Ernst MO (2006b) A Bayesian view on multimodal cue integration. A Bayesian view on multimodal cue integration. In: Knoblich G, M. , Grosjean I, Thornton M (eds). Oxford University Press, New York, pp
Ernst MO, Banks MS (2002) Humans integrate visual and haptic information in a statistically optimal fashion. Nature 415:
Jacobs RA (1999) Optimal integration of texture and motion cues to depth. Vision Res 39:
Kersten D, Yuille A (2003) Bayesian models of object perception. Curr Opin Neurobiol 13:
Knill D, Richards W (1996) Perception as Bayesian Inference. Cambridge University Press
Knill DC (2003) Mixture models and the probabilistic structure of depth cues. Vision Res 43:
Knill DC (2007) Robust cue integration: a Bayesian model and evidence from cue-conflict studies with stereoscopic and figure cues to slant. J Vis 7:
Kording K (2007) Decision theory: what "should" the nervous system do? Science 318:
Körding KP, Ku SP, Wolpert D (2004) Bayesian Integration in force estimation. Journal of Neurophysiology 92:
Kording KP, Ku SP, Wolpert DM (2004) Bayesian integration in force estimation. J Neurophysiol 92:
Kording KP, Tenenbaum JB (2006) Causal inference in sensorimotor integration. In: Scholkopf B, Platt J, Hoffman T (eds) Advances in Neural Information Processing Systems vol 19, pp
Kording KP, Tenenbaum JB, Shadmehr R (2006 ) Multiple timescales and uncertainty in motor adaptation. In: Advances in Neural Information Processing Systems, vol 19
Kording KP, Tenenbaum JB, Shadmehr R (2007) The dynamics of memory as a consequence of optimal adaptation to a changing body. Nat Neurosci 10:
Kording KP, Wolpert DM (2004a) Bayesian integration in sensorimotor learning. Nature 427:
Kording KP, Wolpert DM (2004b) Probabilistic Inference in Human Sensorimotor Processing. Advances in Neural Information Processing Systems
Kording KP, Wolpert DM (2006a) Bayesian decision theory in sensorimotor control. Trends Cogn Sci 10:
Kording KP, Wolpert DM (2006b) Bayesian decision theory in sensorimotor control. Trends Cogn Sci
Laurens J, Droulez J (2007) Bayesian processing of vestibular information. Biol Cybern 96:
MacNeilage PR, Ganesan N, Angelaki DE (2008) Computational approaches to spatial orientation: from transfer functions to dynamic Bayesian inference. J Neurophysiol 100:
Miyazaki M, Nozaki D, Nakajima Y (2005) Testing Bayesian models of human coincidence timing. J Neurophysiol 94:
Miyazaki M, Yamamoto S, Uchida S, Kitazawa S (2006) Bayesian calibration of simultaneity in tactile temporal order judgment. Nat Neurosci 9:
Najemnik J, Geisler WS (2005) Optimal eye movement strategies in visual search. Nature 434:
Roach NW, Heron J, McGraw PV (2006) Resolving multisensory conflict: a strategy for balancing the costs and benefits of audio-visual integration. Proc Biol Sci 273:
Rowland B, Stanford T, Stein BE (2007) A Bayesian model unifies multisensory spatial localization with the physiological properties of the superior colliculus. Exp Brain Res DOI /s
Sato Y, Toyoizumi T, Aihara K (2007) Bayesian inference explains perception of unity and ventriloquism aftereffect: identification of common sources of audiovisual stimuli. Neural Comput 19:
Sobel D, Tenenbaum JB, Gopnik A (2004) Children's causal inferences from indirect evidence: Backwards blocking and Bayesian reasoning in preschoolers. . Cognitive Science 28:
Steyvers I, Tenenbaum JB, Wagenmakers EJ, Blum B (2003) Inferring causal networks from observations and interventions. Cognitive Science 27:
Stocker A, Simoncelli E (2005a) Sensory Adaptation within a Bayesian Framework for Perception. In: Weiss Y, Scholkopf B, J. P (eds) Advances in Neural Information Processing Systems. MIT Press, Vancouver BC Canada
Stocker A, Simoncelli EP (2005b) Constraining a Bayesian model of human visual speed perception. In: Adv. Neural Information Processing Systems, v17,
Stocker AA, Simoncelli EP (2006) Noise characteristics and prior expectations in human visual speed perception. Nat Neurosci 9:
Tassinari H, Hudson TE, Landy MS (2006) Combining priors and noisy visual cues in a rapid pointing task. J Neurosci 26:
Tenenbaum JB, Griffiths TL, Kemp C (2006) Theory-based Bayesian models of inductive learning and reasoning. Trends Cogn Sci 10:
Tenenbaum JB, Niyogi S (2003) Learning causal laws. In: Twenty-Fifth Annual Conference of the Cognitive Science Society
Todorov E (2004) Optimality principles in sensorimotor control. Nat Neurosci 7:
Todorov E (2005) Stochastic optimal control and estimation methods adapted to the noise characteristics of the sensorimotor system. Neural Comput 17:
Todorov E, Ghahramani Z (2004) Analysis of the synergies underlying complex hand manipulation. Conf Proc IEEE Eng Med Biol Soc 6:
van Ee R, Adams WJ, Mamassian P (2003) Bayesian modeling of cue interaction: Bistability in stereoscopic slant perception. Journal of the Optical Society of America, A 20:
Wei K, Kording K (2009) Relevance of error: what drives motor adaptation? J Neurophysiol 101:
Wei K, Kording KP (2008) Relevance of error: what drives motor adaptation? J Neurophysiol
Wolpert DM (2007) Probabilistic models in human sensorimotor control. Hum Mov Sci
Yuille A, Bulthoff HH (1996) Bayesian decision theory and psychophysics. In: Knill D, Richards W (eds) Perception as Bayesian Inference. Cambridge University Press., Cambridge, U.K.
Yuille A, Kersten D (2006) Vision as Bayesian inference: analysis by synthesis? Trends Cogn Sci 10:
Yuille AL, Bulthoff HH (1996 ) Bayesian decision theory and psychophysics In: Perception as Bayesian inference Cambridge University Press, Cambridge pp
Zupan LH, Merfeld DM, Darlot C (2002) Using sensory weighting to model the influence of canal, otolith and visual cues on spatial orientation and eye movements. Biological cybernetics 86:
countless papers in many domains

39. Classical interpretation
Auditory input
Visual input
Auditory estimate
Visual estimate
The problem in this framework is that the weights need to change constantly as sensory uncertainty is affected by various factors
Position
Probabilistic framework led to fundamental reinterpretation
(What follows is some new development)

40. But, back to the assumptions
Position
Vision
Audition … …

41. Some more psychophysics

42. Causality / Relevance P(causal) P(not causal) Position Position or
Vision
Audition
Vision
Audition
See
Kording KP, Beierholm U, Ma WJ, Quartz S, Tenenbaum JB, Shams L (2007) Causal inference in multisensory perception. PLoS ONE 2: e943
Also see:
Knill DC (2003) Mixture models and the probabilistic structure of depth cues. Vision Res 43:
Knill DC (2007) Robust cue integration: a Bayesian model and evidence from cue-conflict studies with stereoscopic and figure cues to slant. J Vis 7:
Beierholm U, Shams L, Kording K, Ma WJ (2008) Comparing Bayesian models for multisensory cue combination without mandatory integration. In: Neural Information Processing Systems, Vancouver
Shams L, Ma WJ, Beierholm U (2005) Sound-induced flash illusion as an optimal percept. Neuroreport 16:
irrelevant

43. Causal inference: estimation for audition
If causal
otherwise
Remark – we ignore priors here. They are important but they would just make the derivation here significantly (p<.005) more complicated.
causal
irrelevant

44. Nonlinear motor adaptation
From Wei and Kording, J Neurophys, 2009

45. Exercises: causal inference
edit dependenceOnCuePosition2
causal
irrelevant
The code is to be changed before the normalization. Otherwise the alpha constant must be different.
Explore how the data I just showed (sublinear integration)
can be replicated in a causal inference model
alpha=.00005;
pV=(1-alpha)*exp(-(x-muV).^2/(2*sigmaV^2))+alpha;

46. Causal inference 4 2 best estimate [cm] -2 -4 -8 -6 -4 -2 2 4 6 8-2 -4 -8 -6 -4 -2 2 4 6 8
feedback position [cm]

47. Why talk about causal inference
Bayesian approach is modular
To solve a new problem we can almost entirely built on previous work
Take advantage of other’s experiments

48. Thank you check www.koerding.com/tutorial for more information
send to: or
with any questions