📊Causal Inference Unit 10 – Causal Discovery & Learning Algorithms

Key Concepts & Foundations

• Causal inference aims to understand the causal relationships between variables and how interventions affect outcomes
• Directed Acyclic Graphs (DAGs) represent causal relationships between variables with nodes and edges
• Causal Markov Condition assumes that a variable is independent of its non-descendants given its parents in a causal graph
  • Enables factorization of joint probability distribution into product of conditional probabilities
• Causal Faithfulness Condition assumes that statistical independencies in the data imply causal independencies in the graph
• Causal sufficiency assumes that there are no unmeasured confounders affecting the observed variables
• Interventional data obtained by manipulating variables provides stronger evidence for causal relationships than observational data
• Counterfactuals describe potential outcomes under different hypothetical interventions or treatments

Types of Causal Relationships

• Direct causation occurs when one variable directly influences another without any intermediary variables (smoking directly causes lung cancer)
• Indirect causation involves a causal chain where one variable affects another through intermediary variables (education indirectly affects income through job opportunities)
• Common cause refers to a single variable influencing multiple variables creating a spurious association (age affects both gray hair and wrinkles)
• Confounding occurs when a third variable influences both the cause and effect leading to a non-causal association (socioeconomic status confounds the relationship between education and health)
• Mediation happens when the effect of one variable on another is partially or fully transmitted through a third variable (exercise affects cardiovascular health through reducing blood pressure)
• Moderation arises when the strength or direction of a causal relationship depends on the level of a third variable (the effect of stress on health is moderated by coping skills)
• Bidirectional causation involves variables mutually influencing each other through feedback loops (price and demand in economics)

Causal Discovery Algorithms

• Constraint-based algorithms (PC, FCI) use conditional independence tests to identify causal structures consistent with the data
  • PC algorithm assumes causal sufficiency and acyclicity to find a completed partially directed acyclic graph (CPDAG)
  • FCI algorithm relaxes causal sufficiency assumption to handle latent confounders and selection bias
• Score-based algorithms (GES, GIES) search for causal structures that optimize a scoring function balancing model fit and complexity
  • Greedy Equivalence Search (GES) starts with an empty graph and iteratively adds and removes edges to maximize the score
  • Greedy Interventional Equivalence Search (GIES) extends GES to incorporate interventional data for improved causal discovery
• Hybrid algorithms (MMHC, MMPC) combine constraint-based and score-based approaches for more reliable and efficient causal discovery
• Local causal discovery algorithms (HITON-PC, MMPC) focus on identifying the direct causes and effects of a target variable
• Time series causal discovery algorithms (PCMCI, TCDF) exploit temporal information to infer causal relationships in time series data

Causal Learning Techniques

• Bayesian networks learn the structure and parameters of a directed acyclic graph from data using Bayesian inference
  • Bayesian score (BDe, BGe) evaluates the posterior probability of a graph given the data
  • Markov Chain Monte Carlo (MCMC) methods sample from the posterior distribution of graphs
• Structural Equation Models (SEMs) represent causal relationships using a system of linear equations
  • Can estimate causal effects and test hypotheses about causal structures
• Causal forests extend random forests to estimate heterogeneous treatment effects by considering subgroups
• Causal regularization techniques (Causal Lasso, Causal Dantzig) incorporate sparsity-inducing penalties to select causal variables
• Causal representation learning aims to learn representations that capture causal relationships and generalize across domains
• Causal transfer learning leverages causal knowledge from source domains to improve learning in target domains
• Causal reinforcement learning integrates causal reasoning into sequential decision-making to optimize long-term outcomes

Statistical Methods in Causal Discovery

• Conditional independence tests (partial correlation, mutual information) assess the independence between variables given a conditioning set
  • Fisher's z-test for partial correlation and conditional mutual information test are commonly used
• Causal inference from observational data relies on assumptions like exchangeability, positivity, and consistency
  • Propensity score matching and inverse probability weighting aim to balance confounders across treatment groups
• Instrumental variables (IVs) enable causal effect estimation when unmeasured confounding is present
  • Two-stage least squares (2SLS) and generalized method of moments (GMM) are IV estimation techniques
• Difference-in-differences (DID) estimates causal effects by comparing the pre-post treatment changes in the treated and control groups
• Regression discontinuity design (RDD) exploits a threshold-based treatment assignment to estimate local causal effects
• Granger causality tests whether past values of one time series help predict future values of another series beyond its own past
• Convergent cross mapping (CCM) detects causal relationships between time series by assessing the ability to reconstruct the dynamics of one series from the other

Applications & Case Studies

• Genome-wide association studies (GWAS) aim to identify genetic variants causally associated with traits and diseases
  • Mendelian randomization uses genetic variants as instrumental variables to estimate causal effects
• Causal inference in healthcare evaluates the effectiveness of treatments and interventions on patient outcomes
  • Propensity score methods are used to adjust for confounding in observational studies
• Econometric analysis investigates the causal impact of policies and interventions on economic outcomes
  • Difference-in-differences and instrumental variables are commonly employed
• Marketing attribution determines the causal contribution of different marketing channels to customer conversions and revenue
• Causal discovery in climate science uncovers the complex causal relationships between climate variables and extreme events
• Social network analysis studies the causal effects of social influence and contagion on individual behaviors and outcomes
• Causal inference in education assesses the impact of educational interventions and policies on student learning and achievement

Challenges & Limitations

• Unmeasured confounding remains a major challenge in causal inference from observational data
  • Sensitivity analysis can assess the robustness of causal estimates to potential unmeasured confounders
• Selection bias arises when the sample is not representative of the population of interest due to non-random selection
  • Heckman correction and inverse probability weighting can mitigate selection bias
• Measurement error in variables can bias causal estimates and lead to incorrect conclusions
  • Instrumental variables and error-in-variables models can address measurement error
• Causal insufficiency due to latent variables can hinder the identifiability of causal structures
  • Latent variable models and causal discovery algorithms like FCI can handle latent confounders
• Nonlinearity and high-dimensional data pose computational and statistical challenges for causal discovery and inference
• Transportability and external validity of causal findings across different populations and settings can be limited
• Ethical considerations and potential negative consequences of causal interventions need to be carefully examined

Future Directions & Research

• Integration of causal inference with machine learning techniques like deep learning and reinforcement learning
  • Causal representation learning and causal transfer learning are promising approaches
• Causal discovery from heterogeneous data sources including text, images, and networks
  • Multimodal causal discovery algorithms and frameworks need to be developed
• Causal inference under interference and spillover effects when units interact with each other
  • Spatial and network causal inference methods can account for interference
• Causal discovery and inference for complex systems with feedback loops and time-varying causal relationships
  • Dynamic causal models and time-varying causal graphs are active areas of research
• Scalable and efficient causal discovery algorithms for large-scale datasets and high-dimensional settings
• Incorporation of domain knowledge and expert opinions into causal discovery and learning processes
• Causal explanations and interpretability of causal models to enhance trust and adoption in real-world applications
• Development of software packages and tools for causal inference and discovery to facilitate wider adoption and reproducibility