Estimating the Transfer Function from Neuronal Activity to BOLD

1. Estimating the Transfer Function from Neuronal Activity to BOLD Maria Joao Rosa SPM Homecoming 2008 Wellcome Trust Centre for Neuroimaging

2. Statistic formulations • P(A): probability of event A occurring • P(A|B): probability of A occurring given B occurred • P(B|A): probability of B occurring given A occurred • P(A,B): probability of A and B occurring simultaneously (joint probability of A and B) Joint probability of A and B P(A,B) = P(A|B)*P(B) = P(B|A)*P(A) • P(B|A) = P(A|B)*P(B)/P(A) Which is Bayes Rule Bayes’ Rule is very often referred to Bayes’ Theorem, but it is not really a theorem, and should more properly be referred to as Bayes’ Rule (Hacking, 2001).

3. Reverend Thomas Bayes(1702 – 1761) • Reverend Thomas Bayes was a minister interested in probability and stated a form of his famous rule in the context of solving a somewhat complex problem involving billiard balls • It was first stated by Bayes in his ‘Essay towards solving a problem in the doctrine of chances’, published in the Philosophical Transactions of the Royal Society of London in 1764.

4. Conditional probability P(A|B): conditional probability of A given B Q: When are we considering conditional probabilities? A: Almost always! Examples: • Lottery chances • Dice tossing

5. Conditional probability Examples (cont’): • P(Brown eyes|Male): (P(A|B) with A := Brown eyes, B := Male) • What is the probability that a person has brown eyes, ignoring everyone who is not a male? • Ratio: (being a male with brown eyes)/(being a male) • Probability ratio: probability that a person is both male and has brown eyes to the probability that a person is male P(Male) = P(B) = 0.52 P(Brown eyes) = P(A) = 0.78 P(Male with brown eyes) = P(A,B) = 0.38 P(A|B) = P(B|A)*P(A)/P(B) = P(A,B)/P(B) = 0.38/0.52 = 0.73.. Flipping it around (Bayes idea): You could also calculate now what’s the prob. of being a male if you have brown eyes P(B|A) = P(A|B)*P(B)/P(A) = 0.73*0.52/0.78 = 0.4871…

6. Statistic terminology • P(A) is called the marginal orprior probability of A (since it is the probability of A prior to having any information about B) Similarly: • P(B): the marginal orprior probability of B • P(A|B) is called the likelihood function for A given B. • P(B|A): the posterior probabilityof B given A(since itdepends on having information about A) Bayes Rule P(B|A) = P(A|B)*P(B)/P(A) priorprobabilities of B, A (“priors”) “posterior” probability of B given A “likelihood” function for B (for fixed A)

7. Bayes’ Theorem for a given parameter  p (data) = p (data)p () / p (data) 1/P (data) is basically a normalizing constant Posteriorlikelihood x prior The prior is the probability of the parameter and represents what was thought before seeing the data. The likelihood is the probability of the data given the parameter and represents the data now available. The posteriorrepresents what is thought given both prior information and the data just seen. It relates to the conditional density of a parameter (posterior probability) with its unconditional density (prior, since depends on information present before the experiment).

8. Data and hypotheses… • We have a hypotheses H0 (null), H1 • We have data (Y) • We want to check if the model that we have (H1) fits our data (accept H1 / rejectH0) or not(H0) Inferential statistics: what is the probability that we can reject H0 and accept H1 at some level of significance (, P) These are a-priori decisions even when we don’t know what the data will be and how it will behave. Bayes: We get some evidence for the model (“likelihood”) and then can even compare “likelihoods” of different models

9. Where does Bayes Rule come at hand? • In diagnostic cases where we’re are trying to calculate P(Disease | Symptom) we often know P(Symptom | Disease), the probability that you have the symptom given the disease, because this data has been collected from previous confirmed cases. • In scientific cases where we want to know P(Hypothesis | Result), the probability that a hypothesis is true given some relevant result, we may know P(Result | Hypothesis), the probability that we would obtain that result given that the hypothesis is true- this is often statistically calculable, as when we have a p-value.

10. Applicability to (f)mri • Let’s take fMRI as a relevant example Y = X *  +  • We have: • Measured data : Y • Model : X • Model estimates: ,  (/variance)

11. What do we get with inferential statistics? • T-statistics on the betas ( = (1,2,…)) (taking error into account) for a specific voxel  we would ONLY get that there is a chance (e.g. < 5%) that there is NO effect of (e.g. 1> 2),given the data • But what about the likelihood of the model??? • What are the chances/likelihood that 1> 2 at some voxel or region • Could we get some quantitative measure on that?

12. What do we get with Bayes statistics? Here, the idea (Bayes) is to use our post-hoc knowledge (our data) to estimate the model, ( also allowing us to compare hypotheses (models) and see which fits our data best) priorprobabilities of Y, X (“priors”) P(X|Y) = P(Y|X)*P(X)/P(Y) “posterior”distribution for X given Y “likelihood” of Y given X i.e. P(|Y) = P(Y|)*P()/P(Y) Now to Steve about the practical sides in SPM…

13. Bayes for Beginners: Applications

14. Bayes in SPM SPM uses priors for estimation in… • spatial normalization • segmentation • EEG source localisation and Bayesian inference in… • Posterior Probability Maps (PPM) • Dynamic Causal Modelling (DCM)

15. Null hypothesis significance testing • Standard approach in science is the null hypothesis significance test (NHST) • Low p value suggests “there is not nothing” • Assumption is H0 = noise; randomness H0 = molecules are randomly arranged in space Looking unlikely… Kreuger (2001) American Psychologist

16. Something vs nothing If the underlying effect δ~= 0, no matter how small, the test statistic grows in size – is this physiological? …If there is any effect.. Our interpretations ultimately depend on p(H0) “Risky” vs “safe” research… Better to be explicit – incorporate subjectivity when specifying hypotheses. Belief change = p(H0) – p(H0 | D)

17. The case for the defence • Law of large numbers means that the test statistic will identify a consistent trend (δ ~= 0) with a sufficient sample size • In SPM, we look at images of statistics, not effect sizes • A highly significant statistic may reflect a small non-physiological difference, with large N • BUT… as long as we are aware of this, classical inference works well for common sample sizes

18. lpost=ld + lpMpost= ld Md + lp Mp lpost lpost-1 ld-1 lp-1 Mp Mpost Md Posterior Probability Distribution precision  = 1/2

19. (1) Bayesian model comparison BUT!!! What is p(H0) for randomness?! Reframe the question – compare alternative hypotheses/models: H1 H2 H3 H4 vs vs vs vs etc… If only one model, then p(y) is a normalising constant… Bayes: For model Hi : Model evidence for Hi

20. Practical example (1)

21. SPC H=2 V1 Photic V5 0.86 0.75 1.42 0.89 Attention 0.55 -0.02 0.56 Motion H=3 Photic SPC Model Evidence: 0.85 0.70 0.85 1.36 Attention V1 0.03 Bayes factor: -0.02 0.57 V5 Motion 0.23 Attention Dynamic causal modelling (DCM) H=1 Photic SPC 0.85 0.70 0.84 V1 -0.02 0.57 V5 Motion 0.58 Attention

22. (2) Priors about the null hypothesis General Linear Model: with What are the priors? • In “classical” SPM, no (flat) priors • In “full” Bayes, priors might be from theoretical arguments or from independent data • In “empirical” Bayes, priors derive from the same data, assuming a hierarchical model for generation of the data Parameters of one level can be made priors on distribution of parameters at lower level Parameters and hyperparameters at each level can be estimated using EM algorithm

23. Shrinkage prior General Linear Model: Shrinkage prior: In the absence of evidence to the contrary, parameters will shrink to zero 0

24. Bayesian Inference PPMs Posterior Likelihood Prior SPMs Changes with search volume Bayesian test Classical T-test

25. Practical example (2)

26. SPM5 Interface

27. (2) Posterior Probability Maps Activation threshold  Mean (Cbeta_*.img) PPM (spmP_*.img) Posterior probability distribution p(b |Y) Probability  Std dev (SDbeta_*.img)

28. (3) Use informative priors (cutting edge!) • Spatial constraints on EEG sources, e.g. using fMRI blobs • Spatial constraints on fMRI activity (e.g. grey matter) ?

29. (4) Tasters – The Bayesian Brain

30. (4a) Taster: Modelling behaviour… Ernst & Banks (2002) Nature

31. (4a) Taster: Modelling behaviour… Ernst & Banks (2002) Nature

32. (4b) Taster: Modelling the brain… Friston (2005) Phil Trans R Soc B

33. Acknowledgements and further reading • Previous MFD talks • Jean & Guillame’s SPM course slides • Krueger (2001) Null hypothesis significance testing Am Psychol 56: 16-26 • Penny et al. (2004) Comparing dynamic causal models. Neuroimage 22: 1157-1172 • Friston & Penny (2003) Posterior probability maps and SPMs Neuroimage 19: 1240-1249 • Friston (2005) A theory of cortical responses Phil Trans R Soc B • www.ualberta.ca/~chrisw/BayesForBeginners.pdf • www.fil.ion.ucl.ac.uk/spm/doc/books/hbf2/pdfs/Ch17.pdf

34. Bayes’ ending Bunhill Fields Burial Ground off City Road, EC1