Outline 1)Motivation 2)Representing/Modeling Causal Systems 3)Estimation and Updating 4)Model Search 5)Linear Latent Variable Models 6)Case Study: fMRI 1

Outline Search I: Causal Bayes Nets 1)Bridge Principles: Causal Structure  Testable Statistical Constraints 2)Equivalence Classes 3)Pattern Search 4)PAG Search 5)Variants 6)Simulation Studies on the Tetrad workbench 2

3 Bridge Principles: Acyclic Causal Graph over V  Constraints on P(V) Weak Causal Markov Assumption V 1,V 2 causally disconnected  V 1 _||_ V 2 V 1 _||_ V 2   v 1,v 2 P(V 1 =v 1 | V 2 = v 2 ) = P(V 1 =v 1 )

4 Bridge Principles: Acyclic Causal Graph over V  Constraints on P(V) Weak Causal Markov Assumption V 1,V 2 causally disconnected  V 1 _||_ V 2 Causal Markov Axiom If G is a causal graph, and P a probability distribution over the variables in G, then in satisfy the Markov Axiom iff: every variable V is independent of its non-effects, conditional on its immediate causes. Determinism (Structural Equations)

5 Causal Markov Axiom Acyclicity d-separation criterion Independence Oracle Causal Graph Z X Y1Y1 Z _||_ Y 1 | XZ _||_ Y 2 | X Z _||_ Y 1 | X,Y 2 Z _||_ Y 2 | X,Y 1 Y 1 _||_ Y 2 | XY 1 _||_ Y 2 | X,Z Y2Y2 Bridge Principles: Acyclic Causal Graph over V  Constraints on P(V)

6 Faithfulness Constraints on a probability distribution P generated by a causal structure G hold for all parameterizations of G. Revenues :=  1 Rate +  2 Economy +  Rev Economy :=  3 Rate +  Econ Faithfulness:  1 ≠ -  3  2  2 ≠ -  3  1 Tax Rate Economy Tax Revenues 11 33 22

7 Equivalence Classes Independence (d-separation equivalence) DAGs : Patterns PAGs : Partial Ancestral Graphs Intervention Equivalence Classes Measurement Model Equivalence Classes Linear Non-Gaussian Model Equivalence Classes Etc. Equivalence: Independence Equivalence: M 1 ╞ (X _||_ Y | Z)  M 2 ╞ (X _||_ Y | Z) Distribution Equivalence:  1  2 M 1 (  1 ) = M 2 (  2 ), and vice versa)

8 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 d-separation/Independence Equivalence

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

10 D-separation X is d-separated from Y by Z in G iff Every undirected path between X and Y in G 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 X Y Z1Z1 Z2Z2 V W Undirected Paths between X, Y: 1) X --> Z 1 Y 2) X Y

11 D-separation X is d-separated from Y by Z in G iff Every undirected path between X and Y in G 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 X Y Z1Z1 Z2Z2 V W Undirected Paths between X, Y: 1) X --> Z 1 Y 2) X Y X d-sep Y relative to Z = {V} ? X d-sep Y relative to Z = {V, Z 1 } ? X d-sep Y relative to Z = {W, Z 2 } ? No Yes No X d-sep Y relative to Z =  ? Yes

12 D-separation X 3 and X 1 d-sep by X 2 ? Yes: X 3 _||_ X 1 | X 2 X 3 and X 1 d-sep by X 2 ? No: X 3 _||_ X 1 | X 2

13 Statistical Control ≠ Experimental Control X 3 _||_ X 1 | X 2 X 3 _||_ X 1 | X 2 (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 1)Load Session: patterns1.tet 2)Change Graph3 minimally to reduce number of equivalent DAGs maximally 3)Compute the DAGs that are equivalent to your original 3 variable DAG

19 Constraint Based Search Background Knowledge e.g., X 2 prior in time to X 3 Statistical Inference

20 Score Based Search Background Knowledge e.g., X 2 prior in time to X 3 Model Score

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 ~ Pointwise, but not uniformly consistent

22 Tetrad Demo 1)Open new session 2)Template: Search from Simulated Data 3)Create Graph, parameterize, instantiate, generate data N=50 4)Choose PC search, execute 5)Attach new search node, choose GES, execute 6)Play (sample size, parameters, alpha value, etc.)

23 Tetrad Demo 1)Open new session 2)Load Charity.txt 3)Create Knowledge: a.Tangibility is exogenous b.AmountDonate is Last c.Tangibility direct cause of Imaginability 4)Perform Search 5)Estimate output

24 PAGs: Partial Ancestral Graphs

25 PAGs: Partial Ancestral Graphs

26 PAGs: Partial Ancestral Graphs What PAG edges mean.

27 1) Adjacency 2) Orientation Constraint-based Search

Constraint-based Search: Adjacency 1.X and Y are adjacent if they are dependent conditional on all subsets that don’t include them 2.X and Y are not adjacent if they are independent conditional on any subset that doesn’t include them

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

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

Search: Orientation Away from Collider

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

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

35 Tetrad Demo 1)Open new session 2)Create graph for M1, M2, M3 on previous slide 3)Search with PC and FCI on each graph, compare results

36 Tetrad Demo 1)Open new session 2)Load data: regression_data 3)X is “putative cause”, Y is putative effect, Z1,Z2 prior to both (potential confounders) 4)Use regression to estimate effect of X on Y 5)Apply FCI search to data

37 Variants 1)CPC, CFCI 2)Lingam

LiNGAM 1.Most of the algorithms included in Tetrad (other than KPC) assume causal graphs are to be inferred from conditional independence tests. 2.Usually tests that assume linearity and Gaussianity. 3.LiNGAM uses a different approach. 4.Assumes linearity and non-Gaussianity. 5.Runs Independent Components Analysis (ICA) to estimate the coefficient matrix. 6.Rearranges the coefficient matrix to get a causal order. 7.Prunes weak coefficients by setting them to zero.

ICA  Although complicated, the basic idea is very simple.  a11 X a1n Xn = e1 ...  an1 X ann Xn = en  Assume e1,...,en are i.i.d.  Try to maximize the non-Gaussianity of w1 X wn Xn = ?  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.

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.

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.

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!)

Hands On 1)Attach a Generalized SEM IM. 2)Attach a data set, simulate 1000 points. 3)Attach a Search box and run LiNGAM. 4)Attach another search box to Data and run PC. 5)Compare PC to LiNGAM.

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.

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.

Hands On 1.Simulate data from a “complicated” DAG using a SEM IM. 1.Choose the Search from Simulated Data item from the Templates menu. 2.Make a random 20 node 20 edge DAG. 3.Parameterize as a linear SEM, accepting defaults. 4.Run CPC. 5.Attach another search box to data. 6.Run PC. 7.Layout the PC graph using Fruchterman-Reingold. 8.Copy the layout to the CPC graph. 9.Open PC and CPC simultaneously and note the differences.

Special Variants of Algorithms 1.CFCI 1.Same idea as for CPC but for FCI instead. 2.KPC 1.The PC algorithm typically uses independence tests that assume linearity. 2.The KPC algorithm makes two changes: 1.It uses a non-parametric independence test. 2.It adds some steps to orient edges that are unoriented in the PC pattern.

Special Variants of Algorithms 1.PcLiNGAM 1.If some variables are Gaussian (more than one), others non- Gaussian, this algorithm applies. 2.Runs PC, then orients the unoriented edges (if possible) using non- Gaussianity. 2.LiNG 1.Extends LiNGAM to orient cycles using non-Gaussianity

Special Variants of Algorithms 1.JCPC 1.Uses a Markov blanket style test to add/remove individual edges, using CPC style orientation. 2.Allows individual adjacencies in the graph to be revised from the initial estimate using the PC adjacency search.

50 Simulation Studies with Tetrad