Probabilistic Graphical Models and Their Inference: with Applications on Functional Network Estimation Wei Liu! University of Utah Graphical Model Applications probability graph Theory theory probabilistic Computer Vision! graphical models Image Understanding! Web Search! Speech Recognition! Natural Language Processing! Chemical Reaction! Bioinformatics 2 Unified Models z1 z2 zn 1 zn zn+1 − ⇡ zn ⇤ x1 x2 xn 1 xn xn+1 − xn µ N Hidden Markov Models! Kalman Filters! Mixture of Gaussians! Neural Networks! Boltzmann Machines! Probabilistic Principal Component Analysis (2) (2) (2) zn 1 zn zn+1 − (1) (1) (1) zn 1 zn zn+1 − xn 1 xn xn+1 − 3 Outline Overview of probabilistic graphical model.! ! Undirected graph: Markov random field.! ! Graphical model inference on single subject fMRI.! ! Inference on group of subject.! ! Multi-Task learning on Autism patients.! ! Lesion detection. 4 Graphs chain regular lattice directed graph tree general graph chain graph 5 Probability <—> Directed Graph P (A, B, C)=P (A) P (B,C A) · | = P (A) P (B A) P (C B,A) · | · | full connected P (A, B, C)=P (A) P (B A) P (C B) A C B · | · | head to tail ⊥⊥ | P (A, B, C)=P (A) P (B A) P (C A) B C A · | · | ⊥⊥ | tail to tail! P (A, B, C)=P (A) P (B) P (C A, B) A B C · · | ⊥⊥ | head to head 6 Markov Random Field =( , ) G V E ∈V V = ,... { } (, ) G N ∈E⇔ ∈N G ∈V ( )=( ) | V− | N 7 Gibbs Distribution X X G⇔ 1 1 P (X)= exp U(X) ,U(X)= V (x ). Z −T c c c ∈C cliques examples U(X)=β (r,s) ψ(xr,xs) ∈V 8 Hidden Markov Model • X is defined on MRF. ! • Y is assumed to be generated from X! • Inverse problem: Given Y, estimate X. Other forms exist: conditional random field. No Bayesian interpretation. 9 Statistical Inference What statistical question we can ask: () () regular lattice Inference methods: directed graph chain general tree graph 10 fMRI Experiments J. C. Snow, Nature 2011 11 fMRI Data • Blood oxygen level dependent (BOLD) indirectly measure neuronal activity.! • 3D volumes sampled at time interval. ! • Fast scan, but noisy image. ! • Spatio-temporal dependency. 12 Task v.s. Resting • Paradigm signal.! • No paradigm signal.! • Subjects in active cognitive • Subject in scanner, eyes closed/ activity.! open to fixed cross.! • Linear model of stimulus and • Correlation between BOLD BOLD response. signals.! paradigm BOLD BOLD task resting-state 13 Spatial Coherent Connectivity Voxel correla- tions y in 2d dimensional space yij p(y x ) ij| ij xij xik Connectivity map x in 2d dimensional MRF , k ∈{ } i j ( ) Original d di- | mensional im- age space 14 1 Hierarchical Model Existing Methods:! Proposed:! • Bottom-up or top-down.! • a hierarchical graph including group • Subject is estimate and subjects.! independently.! • Joint estimation of both levels.! • Estimation is one way. • Bayesian, data driven and parameter estimation. fMRI sub1 sub2 sub3 sub1 sub2 sub3 sub1 sub2 sub3 time courses sub1 sub2 sub3 sub1 sub2 sub3 group functional network map group sub1 sub2 sub3 group [Bellec, 2010] [Calhoun, 2001b] [Varoquaux, 2010, 2011] [Van Den Heuvel, 2008] [Beckmann, 2009] Our method: HMRF [Esposito, 2005] [Filippini, 2009] 15 A Single graph data sub1 sub2 sub3 sub1 sub2 sub3 between-level links network map group Within-subject links • Within-subject piecewise constant constraint.! • Between-subject (between- level) dependency. 16 An Abstract Graphical representation group subjects BOLD J 17 Emission Distribution • von Mises-Fisher distribution: Multivariate Gaussian on sphere.! • Gaussian mixture -> vMF mixture. 18 Inference group Monte Carlo Expectation Maximization (θ)=E [log (, ; θ)] | sub1 sub2 log (, ; θ) ≈ Gibbs sampling Schedule Sampling voxels Sampling voxels in group map in subjects 19 Consistency Test by Variance 20 Autism Classification for Multi-Sites features samples grp1 grp2 21 Learning Multiple Tasks w (w ,σ) t ∼N ξ + λ v + λ w = = t λ λ 22 Lesion Detection by Active Learning • Multi-modality, longitudinal, complex patterns. • Existing methods: high false-positive/negative, 2D, single object. • A slight user involvement significantly improves result. • Computer active, user passive (less burden). unlabeled supervised learning semi-supervised active learning 23 Multi-task Learning 24 a MRF Prior Prior: K = 2 (FG) K = 3 (BG) mixture model MRF Likelihood: 25 Active Learning Demo 26 Learning Deep structures 27 Conclusion Multivariate distribution -> graph.! ! Unified model.! ! Inference is difficult, but approximation possible.! ! Application in hierarchical data.! ! Application in other domain: chemical reaction.! ! Deep Learning, Boltzmann machines.! ! 28.