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)= (r,s) (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.
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
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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
( )=E [log ( , ; )] | sub1 sub2 log ( , ; ) 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 , ) t N + v + w = = t
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.! !
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