# Estimating the Transfer Function from Neuronal Activity to bold

 Date 20.08.2017 Size 16.29 Kb. #28211

## 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)

## 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

## 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
• ?

## (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

• 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