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Outline. Motivation Representing/Modeling Causal Systems Estimation and Updating Model Search Linear Latent Variable Models Case Study: fMRI. Outline. Search I: Causal Bayes Nets Bridge Principles: Causal Structure  Testable Statistical Constraints Equivalence Classes

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Outline

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  1. Outline Motivation Representing/Modeling Causal Systems Estimation and Updating Model Search Linear Latent Variable Models Case Study: fMRI

  2. Outline • Search I: Causal Bayes Nets • Bridge Principles: Causal Structure  Testable Statistical Constraints • Equivalence Classes • Pattern Search • PAG Search • Variants • Simulation Studies on the Tetrad workbench

  3. Bridge Principles: AcyclicCausal Graph over V  Constraints on P(V) • Weak Causal Markov AssumptionV1,V2causally disconnectedV1 _||_ V2 V1 _||_ V2 v1,v2 P(V1=v1 |V2 = v2) =P(V1=v1)

  4. Bridge Principles: AcyclicCausal Graph over V  Constraints on P(V) • Determinism • (Structural Equations) • Weak Causal Markov AssumptionV1,V2causally disconnectedV1 _||_ V2 • Causal Markov Axiom • If G is a causal graph, and P a probability distribution over the variables in G, then in <G,P> satisfy the Markov Axiom iff: • every variable V is independent of its non-effects, • conditional on its immediate causes.

  5. Bridge Principles: AcyclicCausal Graph over V  Constraints on P(V) • Causal Markov Axiom • Acyclicity • d-separation criterion • Causal Graph • Independence Oracle • Z _||_ Y1 | X Z _||_ Y2 | X • Z _||_ Y1 | X,Y2 Z _||_ Y2 | X,Y1 • Y1 _||_ Y2 | X Y1 _||_ Y2 | X,Z • Z • X • Y1 • Y2

  6. Faithfulness Constraints on a probability distribution P generated by a causal structure G hold for all parameterizations of G. Tax Rate b3 Revenues := b1Rate + b2Economy + eRev Economy := b3Rate + eEcon b1 Economy b2 Tax Revenues Faithfulness: b1≠ -b3b2 b2≠ -b3b1

  7. Equivalence Classes • Equivalence: • Independence Equivalence: M1╞ (X _||_ Y | Z)  M2╞ (X _||_ Y | Z) • Distribution Equivalence: q1q2M1(q1) = M2(q2), and vice versa) • Independence (d-separation equivalence) • DAGs : Patterns • PAGs : Partial Ancestral Graphs • Intervention Equivalence Classes • Measurement Model Equivalence Classes • Linear Non-Gaussian Model Equivalence Classes • Etc.

  8. d-separation/Independence Equivalence D-separation Equivalence Theorem (Verma and Pearl, 1988) Two acyclic graphs over the same set of variables are d-separation equivalent iff they have: • the same adjacencies • the same unshielded colliders

  9. Colliders Y: Non-Collider Z Z X X Z X X Z Y: Collider Y Y Y Y Shielded Unshielded X X Z Z Y Y

  10. D-separation X is d-separated from Y by Z inGiff • Every undirected path between X and Y inG is inactive relative to Z An undirected path is inactive relative to Z iff any node on the path is inactive relative to Z A node N is inactive relative to Z iff a) N is a non-collider in Z, or b) N is a collider that is not in Z, and has no descendant in Z V Undirected Paths between X , Y: X Y W Z1 • 1) X --> Z1 <-- W --> Y Z2 • 2) X <-- V --> Y

  11. D-separation X is d-separated from Y by Z inGiff • Every undirected path between X and Y inG is inactive relative to Z An undirected path is inactive relative to Z iff any node on the path is inactive relative to Z A node N is inactive relative to Z iff a) N is a non-collider in Z, or b) N is a collider that is not in Z, and has no descendant in Z V Undirected Paths between X , Y: X Y W Z1 • 1) X --> Z1 <-- W --> Y No X d-sep Y relative to Z =  ? Z2 • 2) X <-- V --> Y Yes • X d-sep Y relative to Z = {V} ? No • X d-sep Y relative to Z = {V, Z1} ? Yes • X d-sep Y relative to Z = {W, Z2} ?

  12. X3 and X1 d-sep by X2? X3 and X1 d-sep by X2? Yes: X3_||_ X1 | X2 No: X3_||_ X1 | X2 D-separation

  13. X3 _||_ X1 | X2 Statistical Control≠ Experimental Control X3 _||_ X1 | X2(set)

  14. Independence Equivalence Classes:Patterns & PAGs • Patterns(Verma and Pearl, 1990): graphical representation of d-separation equivalence among models with no latent common causes (i.e., causally sufficient models) • PAGs: (Richardson 1994) graphical representation of a d-separation equivalence class that includes models with latent common causes and sample selection bias that are Markov equivalent over a set of measured variables X

  15. Patterns

  16. Patterns: What the Edges Mean

  17. Patterns

  18. Tetrad Demo Load Session: patterns1.tet Change Graph3 minimally to reduce number of equivalent DAGs maximally Compute the DAGs that are equivalent to your original 3 variable DAG

  19. Statistical Inference Background Knowledge e.g., X2prior in time to X3 Constraint Based Search

  20. Score Based Search Model Score Background Knowledge e.g., X2prior in time to X3

  21. Overview of Search Methods Constraint Based Searches • TETRAD (PC, FCI) • Very fast – capable of handling 1,000 variables • Pointwise, but not uniformly consistent Scoring Searches • Scores: BIC, AIC, etc. • Search: Hill Climb, Genetic Alg., Simulated Annealing • Difficult to extend to latent variable models • Meek and Chickering Greedy Equivalence Class (GES) • Very slow – max N ~ 30-40 • Pointwise, but not uniformly consistent

  22. Tetrad Demo Open new session Template: Search from Simulated Data Create Graph, parameterize, instantiate, generate data N=50 Choose PC search, execute Attach new search node, choose GES, execute Play (sample size, parameters, alpha value, etc.)

  23. Tetrad Demo • Open new session • Load Charity.txt • Create Knowledge: • Tangibility is exogenous • AmountDonate is Last • Tangibility direct cause of Imaginability • Perform Search • Estimate output

  24. PAGs: Partial Ancestral Graphs

  25. PAGs: Partial Ancestral Graphs

  26. PAGs: Partial Ancestral Graphs What PAG edges mean.

  27. Constraint-based Search 1) Adjacency 2) Orientation

  28. Constraint-based Search: Adjacency X and Y are adjacent if they are dependent conditional on all subsets that don’t include them X and Y are not adjacent if they are independent conditional on any subset that doesn’t include them

  29. Search: Orientation Patterns Y Unshielded Z X Y X _||_ Z | Y X _||_ Z | Y Non-Collider Collider Z X Y Z X Y Z X Y Z X Y Z X Y

  30. Search: Orientation PAGs Y Unshielded Z X Y X _||_ Z | Y X _||_ Z | Y Non-Collider Collider Z X Y Z X Y

  31. Search: Orientation Away from Collider

  32. Search: Orientation After Orientation Phase X1 || X2 X1 || X4 | X3 X2 || X4 | X3

  33. Interesting Cases L1 L2 L1 L X2 X1 M1 M2 Z Y1 Y2 Y X Z1 X Z2 Y M3

  34. Tetrad Demo Open new session Create graph for M1, M2, M3 on previous slide Search with PC and FCI on each graph, compare results

  35. Tetrad Demo Open new session Load data: regression_data X is “putative cause”, Y is putative effect, Z1,Z2 prior to both (potential confounders) Use regression to estimate effect of X on Y Apply FCI search to data

  36. Variants CPC, CFCI Lingam

  37. LiNGAM • Most of the algorithms included in Tetrad (other than KPC) assume causal graphs are to be inferred from conditional independence tests. • Usually tests that assume linearity and Gaussianity. • LiNGAM uses a different approach. • Assumes linearity and non-Gaussianity. • Runs Independent Components Analysis (ICA) to estimate the coefficient matrix. • Rearranges the coefficient matrix to get a causal order. • Prunes weak coefficients by setting them to zero.

  38. ICA • Although complicated, the basic idea is very simple. • a11 X1 + ... + a1n Xn = e1 • ... • an1 X1 + ... + annXn = en • Assume e1,...,en are i.i.d. • Try to maximize the non-Gaussianityof w1 X1 + ... + wnXn= ? • There are n ways to do it up to symmetry! (Cf. Central Limit Theorem, Hyavarinen et al., 2002) • You can use the coefficients for e1, or for e2, or for... • All other linear combinations of e1,...,en are more Gaussian.

  39. ICA • This equation is usually denoted Wx = s • But also X = BX + s where B is the coefficient matrix • So Wx = (I – B)x = e • s is the vector of independent components • x is the vector of variables • Just showed that under strong conditions we can estimate W. • So we can estimate B! (But with unknown row order) • Using assumptions of linearity and non-Gaussianity (of all but one variable) alone. • More sophisticated analyses allow errors to be non-i.i.d.

  40. LiNGAM • LiNGAM runs ICA to estimate the coefficient matrix B. • The order of the errors is not fixed by ICA, so some rearranging of the B matrix needs to be done. • Rows of the B matrix are swapped so the it is lower triangular. • a[i][j] should be non-zero (representing an edge) just in case ij • Typically, a cutoff is used to determine if a matrix element is zero. • The rearranged matrix corresponds to the idea of a causal order.

  41. LiNGAM • Once you know which nodes are adjacent in the graph and what the causal order is, you can infer a complete DAG. • Review: • Use data from a linear non-Gaussian model (all but one variable non-Gaussian) • Infer a complete DAG (more than a pattern!)

  42. Hands On • Attach a Generalized SEM IM. • Attach a data set, simulate 1000 points. • Attach a Search box and run LiNGAM. • Attach another search box to Data and run PC. • Compare PC to LiNGAM.

  43. Special Variants of Algorithms • PC Pattern • PC Pattern enforces the requirement that the output of the algorithm will be a pattern. • PCD • PCD adds corrective code to PC for the case where some variables stand in deterministic relationships. • This results in fewer edges being removed from the graph. • For example, if X _||_ Y | Z but Z determines Y, X---Y is not taken out.

  44. Special Variants of Algorithms • CPC • The PC algorithm may jump too quickly to the conclusion that a collider and noncolliders should be oriented, X->Y<-Z, X---Y---Z • The CPC algorithm uses a much more conservative test for colliders and noncolliders, double and triple checking to make sure they should be oriented, against different adjacents to X and to Z. • The result is a graph with fewer but more accurate orientations.

  45. Hands On • Simulate data from a “complicated” DAG using a SEM IM. • Choose the Search from Simulated Dataitem from the Templates menu. • Make a random 20 node 20 edge DAG. • Parameterize as a linear SEM, accepting defaults. • Run CPC. • Attach another search box to data. • Run PC. • Layout the PC graph using Fruchterman-Reingold. • Copy the layout to the CPC graph. • Open PC and CPC simultaneously and note the differences.

  46. Special Variants of Algorithms • CFCI • Same idea as for CPC but for FCI instead. • KPC • The PC algorithm typically uses independence tests that assume linearity. • The KPC algorithm makes two changes: • It uses a non-parametric independence test. • It adds some steps to orient edges that are unoriented in the PC pattern.

  47. Special Variants of Algorithms • PcLiNGAM • If some variables are Gaussian (more than one), others non-Gaussian, this algorithm applies. • Runs PC, then orients the unoriented edges (if possible) using non-Gaussianity. • LiNG • Extends LiNGAM to orient cycles using non-Gaussianity

  48. Special Variants of Algorithms • JCPC • Uses a Markov blanket style test to add/remove individual edges, using CPC style orientation. • Allows individual adjacencies in the graph to be revised from the initial estimate using the PC adjacency search.

  49. Simulation Studies with Tetrad

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