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An Introduction to Causal Modeling and Discovery Using Graphical Models Greg Cooper University of Pittsburgh Overview Introduction Representation Inference Learning Evaluation What Is Causality? Much consideration in philosophy I will treat it as a primitive Roughly, if we manipulate something and something else changes, then the former causally influences the latter. Why is Causation Important? Causal issues arise in most fields including medicine, business, law, economics, and the sciences An intelligent agent is continually considering what to do next in order to change the world (including the agent’s own mind). That is a causal question. Representing Causation Using Causal Bayesian Networks A causal Bayesian network (CBN) represents some entity (e.g., a patient) that we want to model causally Features of the entity are represented by variables/nodes in the CBN Direct causation is represented by arcs An Example of a Causal Bayesian Network Structure History of Smoking (HS) Chronic Bronchitis (CB) Lung Cancer (LC) Fatigue (F) Weight Loss (WL) An Example of the Accompanying Causal Bayesian Network Parameters P(HS = no) = 0.80 P(HS = yes) = 0.20 P(CB = absent | HS = no) = 0.95 P(CB = absent | HS = yes) = 0.75 P(CB = present | HS = no) = 0.05 P(CB = present | HS = yes) = 0.25 P(LC = absent | HS = no) = 0.99995 P(LC = absent | HS = yes) = 0.997 P(LC = present | HS = no) = 0.00005 P(LC = present | HS = yes) = 0.003 Causal Markov Condition A node is independent of its non-effects given just its direct causes. This is the key representational property of causal Bayesian networks. Special case: A node is independent of its distant causes given just its direct causes. General notion: Causality is local Causal Modeling Framework An underlying process generates entities that share the same causal network structure. The entities may have different parameters (probabilities). Each entity independently samples the joint distribution defined by its CBN to generate values (data) for each variable in the CBN model Entity Generator HS1 HS2 HS3 LC1 LC2 LC3 WL1 WL2 WL3 (no, absent, absent) (yes, present, present) (yes, absent, absent) (no, absent, absent) (yes, absent, absent) existing entities entity feature values samples Discovering the Average Causal Bayesian Network HSavg LCavg WLavg Some Key Types of Causal Relationships Direct Causation HS Indirect Causation Confounding HS F HS CB LC Sampling bias WL LC Sampled = true WL Inference Using a Single CBN When Given Evidence in the Form of Observations P(F | CB = present, WL = present, CBN1) History of Smoking (HS) Chronic Bronchitis (CB) Lung Cancer (LC) Fatigue (F) Weight Loss (WL) Inference The Markov Condition implies the following equation: n P( X 1 , X 2 ,..., X n ) P( X i | DirectCauses( X i )) i 1 The above equation specifies the full joint probability distribution over the model variables. From the joint distribution we can derive any conditional probability of interest. Inference Algorithms In the worst case, the brute force algorithm is exponential time in the number of variables in the model Numerous exact inference algorithms have been developed that exploit independences among the variables in the causal Bayesian network. However, in the worst case, these algorithms are exponential time. Inference in causal Bayesian networks is NP-hard (Cooper, AIJ, 1990). Inference Using a Single CBN When Given Evidence in the Form of Manipulations P(F | MCB = present, CBN1) • Let MCB be a new variable that can have the same values as CB (present, absent) plus the value observe. • Add an arc from MCB to CB. • Define the probability distribution of CB given its parents. Inference Using a Single CBN When Given Evidence in the Form of Manipulations P(F | MCB = present, CBN1) MCB History of Smoking (HS) Chronic Bronchitis (CB) Lung Cancer (LC) Fatigue (F) Weight Loss (WL) A Deterministic Manipulation P(F | MCB = present), CBN1) MCB History of Smoking (HS) Chronic Bronchitis (CB) Lung Cancer (LC) Fatigue (F) Weight Loss (WL) Inference Using a Single CBN When Given Evidence in the Form of Observations and Manipulations P(F | MCB = present, WL = present, CBN1) MCB History of Smoking (HS) Chronic Bronchitis (CB) Lung Cancer (LC) Fatigue (F) Weight Loss (WL) Inference Using Multiple CBNs: Model Averaging P( X | manipulate(Y' ), observe( Z' )) P( X | manipulate(Y' ), Z' , CBN i ) P(CBN i | data, K ) i Some Key Reasons for Learning CBNs Scientific discovery among measured variables Example of general: What are the causal relationships among HS, LC, CB, F, and WL? Example of focused: What are the causes of LC from among HS, CB, F, and WL? Scientific discovery of hidden processes Prediction Example: The effect of not smoking on contracting lung cancer Major Methods for Learning CBNs from Data Constraint-based methods Uses tests of independence to find patterns of relationships among variables that support causal relationships Relatively efficient in discovery of causal models with hidden variables See talk by Frederick Eberhardt this morning Score-based methods Bayesian scoring Allows informative prior probabilities of causal structure and parameters Non-Bayesian scoring Does not allow informative prior probabilities Learning CBNs from Observational Data: A Bayesian Formulation P( X Y | D, K ) P(S i | D, K ) i: Si containsX Y where D is observational data, Si is the structure of CBNi, and K is background knowledge and belief. Learning CBNs from Observational Data When There Are No Hidden Variables P( S i | D, K ) P( S i | K ) P( D | S i , i , K ) P( i | S i , K ) d i P(S j | K ) P(D | S j , j , K ) P( j | S j , K ) d j j where i are the parameters associated with Si and the sum is over all CBNs for which P(Sj | K) > 0. The BD Marginal Likelihood The previous integral has the following closed form solution, when we assume Dirichlet priors (ijk and ij), multinomial likelihoods (Nijk and Nij denote counts), parameter independence, and parameter modularity: ( ijk N ijk ) ( N ) ( ) i 1 j 1 ij ij k 1 ijk n qi ( ij ) ri Searching for Network Structures Greedy search often used Hybrid methods have been explored that constraints and scoring Some algorithms guarantee locating the generating model in the large sample limit (assuming Markov and Faithfulness conditions), as for example the GES algorithm (Chickering, JMLR, 2002) The ability to approximate the generating network is often quite good An excellent discussion and evaluation of several stateof-the-art methods, including a relatively new method (Max-Min Hill Climbing) is at: Tsamardinos, Brown, Aliferis, Machine Learning, 2006. The Complexity of Search Given a complete dataset and no hidden variables, locating the Bayesian network structure that has the highest posterior probability is NP-hard (Chickering, AIS, 1996; Chickering, et al, JMLR, 2004). We Can Learn More from Observational and Experimental Data Together than from Either One Alone H C E We cannot learn the above causal structure from observational or experimental data alone. We need both. Learning CBNs from Observational Data When There Are Hidden Variables P( S i | K ) P ( S i | D, K ) P(D | S i , i , K ) P( i | S i , K ) d i Hi P(S j | K ) P( D | S j , j , K ) P( j | S j , K ) d j j Hj where Hi (Hj) are the hidden variables in Si (Sj) and the sum in the numerator (denominator) is taken over all values of Hi (Hj). Learning CBNs from Observational and Experimental Data: A Bayesian Formulation • For each model variable Xi that is experimentally manipulated in at least one case, introduce a potential parent MXi of Xi . • Xi can have parents as well from among the other {X1, ..., Xi-1, Xi+1, ..., Xn} domain variables in the model. • Priors on the distribution of Xi will include conditioning on MXi,when it is a parent of Xi, as well as conditioning on the other parents of Xi. • Define MXi to have the same values vi1, vi2, ... , viq as Xi, plus a value o (for observe). o When MXi has value vij in a given case, this represents that the experimenter intended to manipulate Xi to have value vij in the case. o When MXi has value observe in a given case, this represents that no attempt was made by the experimenter to manipulate Xi, but rather, Xi was merely observed to have the value recorded for it. • With the above variable additions in place, use the previous Bayesian methods for causal modeling from observational data. An Example Database Containing Observations and Manipulations HS MCB CB LC F WL T obs T F T T F F F T T F F F T T F F F T F T obs F Faithfulness Condition Faithfulness Condition Any independence among variables in the data generating distribution follows from the Markov Condition applied to the data generating causal structure. A simple counter example: H C E Challenges of Bayesian Learning of Causal Networks Major challenges Large search spaces Hidden variables Feedback Assessing parameter and structure priors Modeling complicated distributions The remainder of this talk will summarize several methods for dealing with hidden variables, which is arguably the biggest major challenge today These examples provide only a small sample of previous research Learning Belief Networks in the Presence of Missing Values and Hidden Variables (N. Friedman, ICML, 1997) Assumes a fixed set of measured and hidden variables Uses Expectation Maximization (EM) to “fill in” the values of the hidden variable Uses BIC to score causal network structures with the filled-in data. Greedily finds best structure and then returns to the EM step using this new structure. Some subsequent work Use patterns of induced relationships among the measured variables to suggest where to introduce hidden variables (Elidan, et al., NIPS, 2000) Determining the cardinality of the hidden variables introduced (Elidan & Friedman, UAI, 2001) A Non-Parametric Bayesian Methods for Inferring Hidden Causes (Wood, et al., UAI, 2006) Learns hidden causes of measured variables hidden variables measured variables Assumes binary variables and noisy-OR interactions Uses MCMC to sample the hidden structures Allows in principle an infinite number of hidden variables In practice, the number of optimal hidden variables is constrained by the measured data Bayesian Learning of Measurement and Structural Model (Silva & Scheines, ICML, 2006) Learns the following type of models hidden variables measured variables Assumes continuous variables, mixture of Gaussian distributions, and linear interactions Mixed Ancestral Graphs* A MAG(G) is a graphical object that contains only the observed variables, causal arcs, and a new relationship for representing hidden confounding. There exist methods for scoring linear MAGS (Richardson & Spirtes Ancestral Graph Markov Models, Annals of Statistics, 2002) SES SES SEX PE L1 CP SEX PE L2 IQ Latent Variable DAG IQ Corresponding MAG * This slide was adapted from a slide provided by Peter Spirtes. CP A Theoretical Study of Y Structures for Causal Discovery (Mani, Spirtes, Cooper, UAI, 2006) Learn a Bayesian network structure on the measured variables Identify patterns in the structure that suggest causal relationships A B C D E F The “Y” structure shown in green supports that D is an unconfounded cause of F. Causal Discovery Using Subsets of Variables Search for an estimate M of the Markov blanket of a variable X (e.g., Aliferis, et al., AMIA, 2002) X is independent of other variables in the generating causal network model, conditioned on the variables in X’s Markov blanket Within M search for patterns among the variables that suggest a causal relationship to X (e.g., Mani, doctoral dissertation, Un. of Pittsburgh, 2006) Causal Identifiability Generally depends upon Markov Condition Faithfulness Condition Informative structural relationships among the measured variables Example of the “Y structure”: A B C E Evaluation of Causal Discovery In evaluating a classifier, the correct answer in any instance is just the value of some variable of interest, which typically is explicitly in the data set. This make evaluation relatively straightforward. In evaluating the output of a causal discovery algorithm, the answer is not in the dataset. In general we need some outside knowledge to confirm that the causal output is correct. This makes evaluation relatively difficult. Thus, causal discovery algorithms have not been thoroughly evaluated. Methods for Evaluating Causal Discovery Algorithms Simulated data Real data with expert judgments of causation Real data with previously validated causal relationships Real data with follow up experiments An Example of an Evaluation Using Simulated Data (Mani, poster here) Generated 20,000 observational data samples from each of five CBNs that were manually constructed Applied the BLCD algorithm, which considers many 4variable subsets of all the variables and applies Bayesian scoring. It is based on the causal properties of “Y” structures. Results Precision: 83% Recall: 27% An Example of an Evaluation Using Previously Validated Causal Relationships (Yoo, et al., PSB, 2002) ILVS is a Bayesian method that considers pairwise relationships among a set of variables It works best when given both observational and experimental data ILVS was applied to a previously collected DNA microarray dataset on 9 genes that control galactose metabolism in yeast (Ideker, et al., Science, 2001) The causal relationships among the genes have been extensively studied and reported in the literature. ILVS predicted 12 of 27 known causal relationships among the genes (44% recall) and of those 12 eight were correct (67% precision) Yoo has explored numerous extensions to ILVS An Example of an Evaluation Using Real Data with Follow Up Experiments (Sachs, et al., Science, 2005) Experimentally manipulated human immune system cells Used flow cytometry to measure the effects on 11 proteins and phospholipids on a large number of individual cells Used a Bayesian method for causally learning from observational and experimental data Derived 17 causal relationships with high probability 15 highly supported by the literature (precision = 15/17 = 88%) The other two were confirmed experimentally by the authors (precision = 17/17 = 100%) Three causal relationships were missed (“recall” = 17 /20 = 85%) A Possible Approach to Combining Causal Discovery and Feature Selection 1. 2. 3. 4. 5. 6. Use prior knowledge and statistical associations to develop overlapping groups of features (variables) Derive causal probabilistic relationships within groups Have the causal groups constrain each other Determine additional groups of features that might constrain causal relationships further Either go to step 2 or step 6 Model average within and across groups to derive approximate model-averaged causal relationships David Danks 2002. Learning the Causal Structure of Overlapping Variable Sets. In S. Lange, K. Satoh, & C.H. Smith, eds. Discovery Science: Proceedings of the 5th International Conference. Berlin: Springer-Verlag. pp. 178-191. Some Suggestions for Further Information Books Conferences Glymour, Cooper (eds), Computation, Causation, and Discovery (MIT Press, 1999) Pearl, Causality: Models, Reasoning, and Inference (Cambridge University Press, 2000) Spirtes, Glymour, Scheines, Causation, Prediction, and Search (MIT Press, 2001) Neapolitan, Learning Bayesian Networks (Prentice Hall, 2003) UAI, ICML, NIPS, AAAI, IJCAI Journals JMLR, Machine Learning Acknowledgement Thanks to Peter Spirtes for his comments on an outline of this talk