Bayesian Analysis. Bayesian Analysis: Bayesian methods concern one s belief about θ. [Current Belief (Posterior)] (Prior Belief) x (Data) Outline

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1 Bayesian Analysis DuBois Bowman, Ph.D. Gordana Derado, M. S. Shuo Chen, M. S. Department of Biostatistics and Bioinformatics Center for Biomedical Imaging Statistics Emory University Outline I. Introduction to Bayesian Methods Bayesian concepts ior distribution Data (Likelihood) Posterior distribution Example: Normal-normal model Bayesian inference iors Posterior summaries Estimation/Posterior sampling Bayesian learning II. Implementation in fmri Studies Two-stage modeling iors Variational Bayes Spatial Bayesian Hierarchical Model III. Data Example and Analyses Inhibitory control in cocaine addicts Two-stage modeling approaches SPM: Classical/Classical SPM: Classical/Bayesian SPM: Bayesian/Bayesian Classical/Spatial Bayesian hierarchical model IV. Summary HBM 8 Advanced fmri Course Perspectives Bayesian Analysis: I. Introduction to Bayesian Methods Parameter of interest: The population difference in BOLD activity between two tasks. Collect data from a sample Regarded as realizations of random variables. Bayesian Determine the distribution of the unknown random parameter, given the data. Classical (Frequentist) Use the data to estimate the unknown fixed parameter. HBM 8 Advanced fmri Course 4 Notation Let represent the parameter of interest Group (or population) mean BOLD response for a given experimental task or contrast Assume Bayesian interpretation of here, unless otherwise noted Let Y represent the data Second level modeling: Y E.g. contrasts of interest [ Y, K, Y ]. = n Also applicable to first level modeling: Y [ Y K, Y ]., i = i it Bayesian methods concern one s belief about. [Current Belief (Posterior)] (ior Belief) x (Data) 5 6

2 Bayes Theorem Bayes Theorem = ( y) Posterior : Marginal : ( ) ( y ) ( y) ( y) ior : ( ) ( y) Data : ( y ) 7 Case I: Assume has m possible states: Case II: Assume is continuous: m ( y) = ( ) ( y ) i= = ( y ) ( ) ( y ) d 8 i i Bayes Theorem In either case, = ( y) ( ) ( y ) ( y) Denominator is constant with respect to. Bayes Theorem ( y) ( ) ( y ) Posterior : ior : Data : ( y) π( y) ( ) π( ) ( y ) p ( y ) 9 Posterior ior x Data Example: Normal- Normal Model Let: Y,,Y Data: ior: [ ] Y = K n Y i ~ Normal(, σ ), i =, K, n ( ) ~ Normal, φ Posterior: Y ~ Normal, φ - φ ( ) φ σ n = + Y φ σ n = φ φ + and ( )

3 Example: Normal- Normal Model Posterior: Y ~ Normal (, φ ) - [ ( ) ] φ + n = + φ φ σ Y φ σ n φ = and Consider as a function of prior uncertainty: σ Y Normal Y, as φ n Data will dominate when φ is large compared to σ n 3 ior obabilities Noninformative ior: Conveys little knowledge about parameters Also called diffuse or vague prior Informative ior : ecise knowledge about parameters Conjugate ior: Posterior distribution has the same parametric form as the prior Convenient mathematical form and often justifiable scientifically 4 Influence of ior Beliefs Influence of ior Beliefs Normal-Normal Model: ( ) Y ~ Normal, φ Normal-Normal Model: ( ) Y ~ Normal, φ ~ Normal φ = V ( Y v ) (, φ ), (n=5) ior Posterior Data ~ Normal φ = V (, φ ), ( ) Y v (n=5) ior Posterior Data Empiricallybased prior Density More weakly informative prior Density Influence of ior Beliefs Normal-Normal Model: ~ Normal φ = V (, φ ), ( ) Y v Diffuse or Vague prior Density ( ) Y ~ Normal, φ (n=5) ior Posterior Data Caution about ior Beliefs!! Normal-Normal Model: ( ) ~ Normal, φ U.S. Political Debate Density Posterior (D) ior (D) Posterior (R) ior (R) ( ) Y ~ Normal, φ Biased Jurors. Data

4 After analysis, all information about a parameter is contained in the posterior Useful summaries Posterior mean/median/mode Posterior exceedance probabilities Highest (posterior) density regions (HDR) Credible intervals Bayesian confidence intervals Bayes factors (*not discussed here) Posterior distribution obability [ Y] Posterior mean [ Y] Posterior obability Maps (PPM) Posterior (exceedance) probability: [ > γ Y ], E.g. γ = [ > Y]= obability. Mean()=.8 obability Posterior obability Maps Threshold PPM s: obability γ Two thresholds: probability α: (e.g. 9%) activation threshold γ: percentage of global mean signal (physiologically relevant size of effect) 3 [ > γ Y] > α [>γ Y] > α Posterior Sampling Joint posterior follows a known distribution, leading to direct sampling Markov Chain Monte Carlo (MCMC) Gibbs sampler (known parametric forms for conditional posterior distributions, which collectively yield samples from the joint posterior distribution) Metropolis-Hastings algorithm Variational Bayes Approximation Laplace Approximations 4 4

5 Bayesian Learning Natural framework for learning and updating Study : ( y ) π( ) p( ) π y Bayesian Analysis: Study : ( y ) π ( y ) p( ) π y II. Implementation in fmri Studies 5 HBM 8 Advanced fmri Course Two-stage modeling Estimation in SPM Use conventional two-stage modeling, emulating a random effects analysis Stage : Subject-specific analyses producing estimates of task-related activity (or contrasts) Stage : Group-level estimates of task-related activity (or contrasts), using stage estimates (contrasts) of regression coefficients as data Stage Classical Bayesian Stage Classical Bayesian, 4 3 Choice of Bayesian versus classical (frequentist) modeling at each of the two stages 7 4.Classical/Spatial Bayesian hierarchical model (BHM) [*Not in SPM] 8 Posterior obability Maps For fmri applications, It is useful to compute PPM s Each voxel contains an exceedance probability, e.g. that the difference between two tasks is greater than zero. Can display thresholded PPM s, e.g. with p>.8. Spatial Bayesian Hierarchical Model Use anatomical (or functional) parcellations (e.g. Brodmann) # regions<# subjects per group Models spatial correlations in activity within and between regions Inferences: Voxel-level Regional Task-related functional connectivity 9 3 5

6 Data Example Inhibitory Control in Cocaine Addicts N=7 subjects: patients (cocaine addicts) and 5 healthy controls (matched on several variables) Bayesian Analysis: III. Data Example and Analyses Two sessions: (7 scans per subject in each session) Cocaine addicts: e and post-treatment (behavioral therapy) Controls: Baseline and follow-up fmri Tasks: Stop signal task (Inhibitory control) Subjects were presented with visual cues and directed to respond to a GO stimulus (an uppercase alphabetical letter) by pressing a button as quickly as possible A STOP signal a brief auditory tone lasting.5 seconds was presented randomly in 6% of the trials, and subjects were instructed to refrain from pushing the button if the GO stimulus was followed by a STOP signal A successful performance required prepotent behaviors to be inhibited. HBM 8 Advanced fmri Course Objective: (Post-treatment e-treatment activity in addicts) > (Follow-up baseline activity in controls) 3 SPM: Stage Model Specification SPM: Stage Design Matrix Design description: Basis function: HRF (with time and dispersion derivatives) Inter-scan Interval:. Grand mean scaling: session specific Global normalization: scaling Note: Full design matrix included three additional conditions Same for classical and Bayesian Same for classical and Bayesian SPM: Stage Classical Estimation SPM: Stage Bayesian Estimation

7 SPM: Stage Results SPM: Stage Model Specification Similar for classical and Bayesian Same for classical and Bayesian Design: Conditions: 5 d.f. [7 images parameters] SPM: Stage Classical Estimation SPM: Stage Bayesian Estimation 39 4 SPM: Stage Results Spatial Bayesian Hierarchical Model [clusts,post_sim,post_sim,pmean,pmedian,cont_pmean, cov_pmed,glocs,cont_sim,bf,pp,misc]=stmod(7,,.); Outputs: Simulations from joint posterior distribution of all model parameters Posterior summaries of voxel, regional, and covariance parameters: E.g., posterior means, medians, probs., Bayes factors, etc. Information on prior distributions Similar for classical and Bayesian Inputs: ) Number of iterations or posterior samples drawn, ) Number of burn-in iterations (eventually discarded), 3) ior covariance weighting (percent reduction in covariances/) 4 4 7

8 Results: SPM Results: SPM. Classical/Classical Voxel-level t-statistic maps p=.5 (uncorrected). Classical/Bayesian Voxel-level PPM s Activation threshold (γ)= α= Results: SPM Results: Spatial BHM 3. Bayesian/Bayesian Voxel-level PPM Activation threshold (γ)= α= Classical/Spatial BHM Voxel-level PPM Likelihood of voxel-level activations α= Results: Spatial BHM Results: Spatial BHM 45/46 4/3 4. Classical/Spatial BHM Regional-level PPM Likelihood of regionallevel activations α=.5 9 BA : BA 45/46: BA 8: BA 9: BA 4/3: 8 p=.5 (Blue) p=.5 (Black) p=.5 (Green) p=.55 (Pink) p=.53 (Red) Functional Connectivity Inter-regional taskrelated functional connections in cocaine addicts following treatment FC>.3 BA 4/3: Anterior cingulate

9 Results: Spatial BHM Functional Connectivity Posterior probs. of FC changes (or differences) Δ s =Group difference in connectivity during session s (Δ pre > Δ post ) >.9 Bayesian Analysis: IV. Summary BA 4/3: Anterior cingulate 49 HBM 8 Advanced fmri Course Summary: Strengths SPM : C/B Fast computations SPM Bayesian SPM 3: B/B SPM Local spatial corrs. No smoothing in pre-process. 4. C/SBHM PPM s to quantify evidence (also Bayes Factors) Flexible inferences from joint posterior distribution May circumvent (mitigate) multiple testing issues Fast computations Regional inferences FC inferences Intra/inter-regional spatial corrs. No or minimal smoothing SPM : C/C Conventional testing methodology Fast computations SPM Classical 5 Summary: Weaknesses SPM : C/B SPM 3: B/B 4. C/SBHM Must specify prior distributions (somewhat subjectively) Unconventional thresholding methods Bayesian Long computations Unconventional thresholding methods Personal implementation Small sample constraint on no. of regions Classical SPM : C/C 5 References. Bowman, F. D., Caffo, B. A, Bassett, S., and Kilts, C. (8). Bayesian Hierarchical Framework for Spatial Modeling of fmri Data. NeuroImage 39, Guillaume Flandin and William D. Penny (7): Bayesian fmri data analysis with sparse spatial basis function priors, NeuroImage 34(3), Friston K.J., Glaser D.E., Henson R.N.A., Kiebel S., Phillips C., Ashburner J. (). Classical and in Neuroimaging: applications. NeuroImage 6(): Friston K, Penny W, Phillips C, Kiebel S, Hinton G, Ashburner J. (). Classical and Bayesian inference in neuroimaging: theory. NeuroImage 6(): Penny,W., Kiebel,S., and Friston,K. (3). Variational Bayesian inference for fmri time series. NeuroImage 9, Penny,W., Trujillo-Barreto,N.J., and Friston,K. (5) Bayesian fmri time series analysis with spatial priors. NeuroImage 4(), Website: