Estimating the Transfer Function from Neuronal Activity to bold



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Estimating the Transfer Function from Neuronal Activity to BOLD

  • Maria Joao Rosa
  • SPM Homecoming 2008
  • Wellcome Trust Centre for Neuroimaging

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).

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.

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

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…

Statistic terminology

  • P(A) is called the marginal or prior probability of A (since it is the probability of A prior to having any information about B)
  • Similarly:
  • P(B): the marginal or prior probability of B
  • P(A|B) is called the likelihood function for A given B.
  • P(B|A): the posterior probability of B given A (since it depends on having information about A)
    • Bayes Rule
    • P(B|A) = P(A|B)*P(B)/P(A)
  • “likelihood” function for B (for fixed A)
  • “posterior” probability of B given A
  • prior probabilities of B, A (“priors”)
  • It relates to the conditional density of a parameter (posterior probability) with its unconditional density (prior, since depends on information present before the experiment).
  • The likelihood is the probability of the data given the parameter and represents the data now available.
  • Bayes’ Theorem for a given parameter 
  • p (data) = p (data) p () / p (data)
  • 1/P (data) is basically
  • a normalizing constant
  • The prior is the probability of the parameter and represents what was thought before seeing the data.
  • The posterior represents what is thought given both prior information and the data just seen.

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 / reject H0) 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

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.

Applicability to (f)mri

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

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?

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)
  • “posterior” distribution for X given Y
  • “likelihood” of Y given X
  • prior probabilities of Y, X (“priors”)
  • Now to Steve about the practical sides in SPM…
    • P(X|Y) = P(Y|X)*P(X)/P(Y)
    • i.e. P(|Y) = P(Y|)*P()/P(Y)

Bayes for Beginners: Applications

SPM uses priors for estimation in…

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

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

Something vs nothing

  • …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)
  • If the underlying effect δ ~= 0, no matter how small, the test statistic grows in size – is this physiological?

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
  • Mp
  • p-1
  • Mpost
  • post-1
  • d-1
  • Md
  • post = d + p Mpost = d Md + p Mp
  • post
  • Posterior Probability Distribution
  • precision  = 1/2

(1) Bayesian model comparison

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

Practical example (1)

Dynamic causal modelling (DCM)

  • V1
  • V5
  • SPC
  • Motion
  • Photic
  • Attention
  • 0.85
  • 0.57
  • -0.02
  • 0.84
  • 0.58
  • H=1
  • V1
  • V5
  • SPC
  • Motion
  • Photic
  • Attention
  • 0.86
  • 0.56
  • -0.02
  • 1.42
  • 0.55
  • 0.75
  • 0.89
  • H=2
  • 0.70
  • V1
  • V5
  • SPC
  • Motion
  • Photic
  • Attention
  • 0.85
  • 0.57
  • -0.02
  • 1.36
  • 0.70
  • 0.85
  • 0.23
  • H=3
  • Attention
  • 0.03
  • Model Evidence:
  • Bayes factor:

(2) Priors about the null hypothesis

  • General Linear Model:
  • What are the priors?
  • with
  • 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

Shrinkage prior

  • General Linear Model:
  • Shrinkage prior:
  • 0
  • In the absence of evidence
  • to the contrary, parameters
  • will shrink to zero
  • Bayesian Inference
  • Likelihood
  • Prior
  • Posterior
  • SPMs
  • PPMs
  • Bayesian test
  • Classical T-test

Practical example (2)

  • SPM5 Interface
  • (2) Posterior Probability Maps
  • Mean (Cbeta_*.img)
  • Std dev (SDbeta_*.img)
  • PPM (spmP_*.img)
  • Activation threshold
  • Probability
  • Posterior probability distribution p( |Y)

(3) Use informative priors (cutting edge!)

  • Spatial constraints on fMRI activity (e.g. grey matter)
  • Spatial constraints on EEG sources, e.g. using fMRI blobs
  • ?

(4) Tasters – The Bayesian Brain

(4a) Taster: Modelling behaviour…

  • Ernst & Banks (2002) Nature

(4a) Taster: Modelling behaviour…

  • Ernst & Banks (2002) Nature

(4b) Taster: Modelling the brain…

  • Friston (2005) Phil Trans R Soc B

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

Bayes’ ending

  • Bunhill Fields Burial Ground
  • off City Road, EC1


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