4. CAM — Causal Additive Model

Type: Nonlinear / Additive Noise • Score-based (order search + regression)
Output: DAG
Reference: Bühlmann, P., Peters, J., & Ernest, J. (2014). CAM: Causal additive models, high-dimensional order search and penalized regression. Annals of Statistics.

CAM is a nonlinear causal discovery algorithm for additive nonlinear models (ANMs).
It assumes each variable is generated from a sum of nonlinear functions of its parents plus independent noise.
Tetrad includes a standalone implementation because the original R package (CAM) is no longer maintained on CRAN.

4.1. Key Idea

CAM decomposes causal discovery into two stages:

  1. Order Search (High-Dimensional)

    • Search over variable orderings consistent with an additive SEM.

    • Uses a score based on regression residuals to evaluate the plausibility of each order.

  2. Pruning & Refitting

    • Given a candidate order, regress each variable on its predecessors using generalized additive models (GAMs) or penalized regression.

    • Remove weak parent relationships using a pruning step (e.g., stability selection).

This yields a fully directed acyclic graph (DAG) representing nonlinear causal relationships.

4.2. When to Use CAM

  • You expect nonlinear causal mechanisms, but still additive in structure.

  • Noise terms are independent, though not necessarily Gaussian.

  • You want a DAG, not a CPDAG or PAG.

  • Datasets of moderate size.

  • You prefer a non-Gaussian regression-based ANM method.

CAM is useful for:

  • Nonlinear scientific systems

  • Gene regulatory network discovery

  • Benchmarking nonlinear SEM-based methods

  • Replacing the discontinued R CAM implementation

4.3. Prior Knowledge Support

CAM in Tetrad supports:

  • Required / forbidden edges

  • Tier constraints (temporal or domain-based)

  • Variable exclusion / inclusion rules

Knowledge constrains admissible variable orders and regressions.

4.4. Strengths

  • Handles nonlinear relationships naturally

  • Produces a fully directed DAG

  • Includes pruning for false-parent removal

  • Strong theoretical foundations for additive-noise models

  • Robust to non-Gaussian noise

4.5. Limitations

  • Assumes additive noise

  • Regression step can be computationally expensive

  • Sensitive to hyperparameters

  • Not designed for latent confounding (assumes causal sufficiency)

4.6. Key Parameters in Tetrad

Parameter (camelCase)

Description

maxNumParents

Limit on parent set size during pruning

numBootstrapSamples

Optional stability selection

penalty

Regularization strength for regression

knowledge

Background knowledge constraints

verbose

Display progress and regression diagnostics

4.7. Reference

Bühlmann, P., Peters, J., & Ernest, J. (2014).
CAM: Causal additive models, high-dimensional order search and penalized regression.
The Annals of Statistics, 42(6), 2526–2556.

4.8. Summary

CAM is a nonlinear additive-noise causal discovery algorithm combining order search and penalized regression to produce a fully directed DAG. It serves as Tetrad’s internal replacement for the discontinued R CAM package.