Report

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 Bridge Principles: Acyclic Causal Graph over V Constraints on P(V) Weak Causal Markov Assumption V1,V2 causally disconnected V1 _||_ V2 V1 _||_ V2 v1,v2 P(V1=v1 | V2 = v2) = P(V1=v1) 3 Bridge Principles: Acyclic Causal Graph over V Constraints on P(V) Weak Causal Markov Assumption Determinism V1,V2 causally disconnected V1 _||_ V2 (Structural Equations) 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. 4 Bridge Principles: Acyclic Causal Graph over V Constraints on P(V) Causal Markov Axiom Acyclicity d-separation criterion Causal Graph Z X Independence Oracle Y1 Y2 Z _||_ Y1 | X Z _||_ Y2 | X Z _||_ Y1 | X,Y2 Z _||_ Y2 | X,Y1 Y1 _||_ Y2 | X Y1 _||_ Y2 | X,Z 5 Faithfulness Constraints on a probability distribution P generated by a causal structure G hold for all parameterizations of G. Tax Rate b3 b1 Tax Revenues Economy Revenues := b1Rate + b2Economy + eRev Economy := b3Rate + eEcon b2 Faithfulness: b 1 ≠ - b 3b 2 b 2 ≠ - b 3b 1 6 Equivalence Classes Equivalence: • Independence Equivalence: M1 ╞ (X _||_ Y | Z) M2 ╞ (X _||_ Y | Z) • Distribution Equivalence: q1 q2 M1(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. 7 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 8 Colliders Y: Collider X Y: Non-Collider Z X Z Shielded X Unshielded Z Y Z X Y Y Y Y Z X X Z Y 9 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 Z1 1) X --> Z1 <-- W --> Y 2) X <-- V --> Y V X Undirected Paths between X , Y: W Y Z2 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 Z1 Z2 1) X --> Z1 <-- W --> Y 2) X <-- V --> Y V X Undirected Paths between X , Y: W Y X d-sep Y relative to Z = ? No X d-sep Y relative to Z = {V} ? Yes X d-sep Y relative to Z = {V, Z1 } ? No X d-sep Y relative to Z = {W, Z2 } ? Yes 11 D-separation X3 and X1 d-sep by X2? X1 X3 X2 T X1 X2 Yes: X3 _||_ X1 | X2 X3 and X1 d-sep by X2? X3 No: X3 _||_ X1 | X2 12 Statistical Control ≠ Experimental Control T X1 X3 X2 X3 _||_ X1 | X2 T X1 X2 X3 X3 _||_ X1 | X2(set) I 13 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 14 Patterns E xam ple Possible E dges X1 X2 X1 X2 X1 X3 X1 X2 15 X2 X4 Patterns: What the Edges Mean X1 X1 X1 X2 X2 X2 X 1 a n d X 2 a re n o t a d ja c e n t in a n y m e m b e r o f th e e q u iv a le n c e c la s s X 1 X 2 (X 1 is a cau se o f X 2 ) in ev ery m em b er o f th e eq u iv alen ce class. X 1 X 2 in so m e m em b ers o f th e eq u iv alen ce class, an d X 2 X 1 in o th ers. 16 Patterns X2 X1 P attern X4 X3 R ep resen ts X3 X2 X1 X2 X1 X4 17 X3 X4 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 18 Constraint Based Search D ata E quivalence C lass of C ausal G raphs X1 X1 X1 X2 X2 X2 X3 X3 X3 Statistical Inference Causal Markov Axiom (D-separation) D iscovery A lgorithm Statistical C onstraints X1 Background Knowledge e.g., X2 prior in time to X3 19 X3 | X2 Score Based Search Equivalence C lass of C ausal G raphs X1 X2 X3 D a ta X1 X1 X2 X3 X2 X3 Equivalence C lass of C ausal G raphs X1 X2 X3 Model Score X1 X2 X3 X1 X2 X3 Equivalence C lass of C ausal G raphs X1 X2 X3 Background Knowledge e.g., X2 prior in time to X3 20 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 21 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.) 22 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 23 PAGs: Partial Ancestral Graphs X2 X1 PAG X3 R e p r e s e n ts X2 X1 X2 X1 T1 X3 X3 e tc . X1 X2 X2 X1 T1 T1 X3 X3 24 T2 PAGs: Partial Ancestral Graphs Z1 Z2 PAG X Y Represents Z1 Z2 Z1 Z2 T1 X3 X3 Y Y etc. T1 Z1 Z2 Z1 Z2 T2 T1 X3 X3 Y 25 Y PAGs: Partial Ancestral Graphs What PAG edges mean. X1 X2 X 1 an d X 2 are n o t ad jacen t X1 X2 X 2 is n o t an an cesto r o f X 1 X1 X2 N o set d -sep arates X 2 an d X 1 X1 X2 X 1 is a cau se o f X 2 X1 X2 T h ere is a laten t co m m o n cau se o f X 1 an d X 2 26 Constraint-based Search 1) Adjacency 2) Orientation 27 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 Z Y X _||_ Z | Y X _||_ Z | Y Non-Collider Collider X Y Z X Y Z X Y Z X Y Z X Y Z Search: Orientation PAGs Y Unshielded X Y Z X _||_ Z | Y X _||_ Z | Y Non-Collider Collider X Y Z X Y Z Search: Orientation Away from Collider T est C o n d itio n s X1 X3 * 1 ) X 1 - X 2 ad jacen t, an d in to X 2 . 2 ) X 2 - X 3 ad jacen t 3 ) X 1 - X 3 n o t ad jacen t X2 T est X1 X3 | X2 Y es No X1 * X3 X2 X1 * X3 X2 Caus al Graph Independcies X1 X1 X3 X4 X2 X2 X1 X4 {X3} X2 X4 {X3} X1 Begin w ith: X3 X4 X2 From X1 X1 X3 X2 X4 X2 From X1 X4 X1 {X3} X3 X4 X2 From X1 X2 X4 {X3} X3 X2 X4 Search: Orientation After Orientation Phase X1 P a tte rn X3 X1 X3 X4 X4 X2 X2 X1 X1 X1 || X2 PA G X3 X4 X2 X3 X4 X2 X1 X1 X1 || X4 | X3 X2 || X4 | X3 X3 X3 X2 X4 X2 X4 Interesting Cases M1 X1 L X2 M2 X Y Z Y1 Y2 L1 L1 L2 Z1 X Z2 Y M3 34 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 35 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 36 Variants 1) CPC, CFCI 2) Lingam 37 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 X1 + ... + a1n Xn = e1 ... an1 X1 + ... + ann Xn = en Assume e1,...,en are i.i.d. Try to maximize the non-Gaussianity of w1 X1 + ... + 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 ij 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 nonGaussian) 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. 1. 2. 3. 4. 5. 6. 7. 8. 9. Simulate data from a “complicated” DAG using a SEM IM. Choose the Search from Simulated Data item 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. 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 nonGaussian, this algorithm applies. 2. Runs PC, then orients the unoriented edges (if possible) using nonGaussianity. 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. Simulation Studies with Tetrad 50