Bayes for Beginners Reverend Thomas Bayes ( ) Velia Cardin Marta Garrido

“In addition to WLS estimators and classical inference, SPM2 also supports Bayesian estimation and inference. In this instance the statistical parametric maps become posterior probability maps Posterior Probability Maps (PPMs), where the posterior probability is a probability of an effect given the data. There is no multiple comparison problem in Bayesian inference and the posterior probabilities do not require adjustment.”BayesianPPM

Overview Velia Bayes vs Frequentist approach An example Bayes theorem Marta Bayesian Inference Posterior Probability Distribution Bayes in SPM Summary

vs Bayesian statistics Frequentist Statistics

With a frequentist approach…. The frequentist conclusion is restricted to the data at hand, it doesn’t take into account previous, valuable information. In general, we want to relate an event (E) to a hypothesis (H) …and the probability of E given H If the p-value is sufficiently small, you reject the null hypothesis, but… it doesn’t say anything about the probability of H i. We obtain a p-value that is the probability, given a true H 0, for the outcome to be more or equally extreme as the observed outcome.

Conclusions depends on previous evidence. Bayesian approach is not data analysis per se, it brings different types of evidence to answer the questions of importance. In general, we want to relate an event (E) to a hypothesis (H) and the probability of E given H Given a prior state of knowledge or belief, it tells how to update beliefs based upon observations (current data). The probability of a H being true is determined. With a Bayesian approach… You can compare the probabilities of different H for a same E A probability distribution of the parameter or hypothesis is obtained

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We took a random sample of 40 people, 10 of them were young stars, being 3 of them addicted to drugs. From the other 30, just one. Our hypothesis is: “Young actors have more probability of becoming drug-addicts” Drug-addicted young actors control YA+ YA- D+D-

With a frequentist approach we will test: H i :’Conditions A and B have different effects’ Young actors have a different probability of becoming drug addicts than the rest of the people H 0 :’There is no difference in the effect of conditions A and B’ This is not what we want to know!!! …and we have strong believes that young actors have more probability of becoming drug addicts!!! The statistical test of choice is  2 and Yates’ correction:  2 = 3.33 p=0.07 We can’t reject the null hypothesis, and the information the p is giving us is basically that if we “do this experiment” many times, 7% of the times we will obtain this result if there is no difference between both conditions.

We want to know if 1 (0.025)29 (0.725) 10 (0.25) 30 (0.75) 3 (0.075)7 (0.175) 4 (0.1)36 (0.9) 40 (1) p(D+  YA+) p (D+  YA+) = p (D+ and YA+) / p (YA+) p (D+  YA+)  p (YA+  D+) Reformulating This is Bayes’ Theorem !!! YA+ YA- D-D+ total p (YA+  D+) = p (D+ and YA+) / p (D+) p (D+  YA+) = / 0.25 = 0.3 p (YA+  D+) = / 0.1 =  0.75 p (D+ and YA+) = p (YA+  D+) * p (D+) p (D+  YA-) = p (D+ and YA-) / p (YA-) p (D+  YA+) > p (D+  YA-) p (D+  YA-) = / 0.75 = p (D+  YA+) > p (D+  YA-) 0.3 > With a Bayesian approach… p (D+  YA-) p (YA+  D+) p (YA+  D+) * p (D+) p (YA+) p (D+  YA+) Substituting p (D+ and YA+) on p (D+  YA+) = p (D+ and YA+) / p (YA+)

It relates 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 Posterior  likelihood x prior 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.

In fMRI…. Classical –‘What is the likelihood of getting these data given no activation occurred?’ Bayesian (SPM2) –‘What is the chance of getting these parameters, given these data?

What you know about the model after the data arrive,, is what you knew before,, and what the data told you,. In order to make probability statements about  given y we begin with a model Bayesian Inference joint prob. distribution where discrete case continuous case or likelihood prior posterior

Likelihood:p(y|  ) =  (M d, d -1 ) Prior:p(  ) =  (M p, p -1 ) Posterior: p(  y) ∝ p(y|  )*  p(  ) =  (M post, post -1 ) MpMp p -1 M post post -1 d -1 MdMd post = d + p M post = d M d + p M p  post Posterior Probability Distribution precision = 1/  2

The effects of different precisions p = d p < d p > d p ≈ 0

Multivariate Distributions

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

Shrinkage Priors Small, variable effect Large, variable effect Small, consistent effect Large, consistent effect

Thresholding p(  | y) = 0.95 

Summary In Bayesian estimation we… 1.…start with the formulation of a model that we hope is adequate to describe the situation of interest. 2.…observe the data and when the information available changes it is necessary to update the degrees of belief (probability). 3.…evaluate the fit of the model. If necessary we compute predictive distributions for future observations. priors over the parameters posterior distributions new priors over the parameters Prejudices or scientific judgment? The selection of a prior is subjective and arbitrary. It is reasonable to draw conclusions in the light of some reason. Bayesian methods use probability models for quantifying uncertainty in inferences based on statistical data analysis.

(Bayes’ original essay!!!) A. Gelman, J.B. Carlin, H.S. Stern and D.B. Rubin, 2 nd ed. Bayesian Data Analysis. Chapman & Hall/CRC. Mackay D. Information Theory, Inference and Learning Algorithms. Chapter 37: Bayesian inference and sampling theory. Cambridge University Press, Berry D, Stangl D. Bayesian Methods in Health-Realated Research. In: Bayesian Biostatistics. Berry D and Stangl K (eds). Marcel Dekker Inc, Friston KJ, Penny W, Phillips C, Kiebel S, Hinton G, Ashburner J. Classical and Bayesian inference in neuroimaging: theory. Neuroimage Jun;16(2): References

Bayes for Beginners Reverend Thomas Bayes ( ) …good-bayes!!! “We don’t see what we don’t seek.” E. M. Forster