Tests and Scores: By Type

Many Tetrad structure-learning algorithms rely on:

an independence test (used by constraint-based algorithms such as PC, CPC, FCI, RFCI, PCMCI), and
a score (used by score-based and hybrid algorithms such as FGES, BOSS, GRaSP, IMaGES, GFCI, FCIT).

This page explains the wrappers in:

edu.cmu.tetrad.algcomparison.independence
edu.cmu.tetrad.algcomparison.score

which are the selectable “Tests” and “Scores” in the Tetrad GUI.

Independence Tests

Package: edu.cmu.tetrad.algcomparison.independence

These are the available CI tests, grouped by data type and modeling assumptions.

Independence Tests Overview

Name	Appropriate Data Type	Description
FisherZ	Continuous (linear-Gaussian)	Fisher Z partial correlation test. Assumes linear relationships and Gaussian (or approximately Gaussian) residuals. Default for continuous data.
GSquare	Discrete (categorical)	Likelihood-ratio G test for discrete conditional independence. Default choice for purely discrete data.
ChiSquare	Discrete (categorical)	Pearson chi-square test for discrete CI; similar to GSquare but uses chi-square statistic.
BasisFunctionLrt	Mixed continuous/discrete; nonlinear continuous effects	LRT using truncated basis expansions (Legendre, Chebyshev, Hermite, etc.) for the continuous parts of a conditional Gaussian model. Supports nonlinear additive effects through basis expansions while allowing discrete parents.
ConditionalGaussianLrt	Mixed continuous/discrete	Classical CG CI test: continuous variables are linear-Gaussian conditional on configurations of discrete parents. Discrete variables are modeled with multinomial logistic regression.
DegenerateGaussianLrt	Mixed continuous/discrete, possibly rank-deficient	CG CI test that tolerates singular or nearly singular covariance blocks. Used when mixed CG assumptions hold but covariance matrices are degenerate because of collinearity or small sample size.
Kci	Continuous (general nonlinear)	Kernel Conditional Independence test. Fully nonparametric; no linearity or Gaussian assumptions. More computationally expensive.
Mvplrt	Mixed or continuous	Likelihood-ratio CI test from multivariate projection regressions. Compares full vs. restricted projection models.
MSeparationTest	Graph-based (no data)	“Test” that answers CI queries using m-separation in a given graph. Used when treating a known DAG/MAG/PAG as the oracle (simulation studies).
ProbabilisticTest	Discrete, Bayesian	Uses Bayesian sampling over conditional distributions to infer CI probabilistically. Useful for probabilistic or likelihood-based independence assessment.
SemBicTest	Continuous (linear-Gaussian)	Independence test induced by the SemBicScore: for each candidate edge, compares SEM BIC with and without the edge and treats variables as independent when the edge does not improve the score. A purely score-based analogue of FisherZ that pairs naturally with FGES, BOSS, and GRaSP.
PoissonBicTest	Any model class with a compatible BIC-type score (typically continuous linear-Gaussian)	Independence test that wraps the PoissonPriorScore: compares models with and without an edge using BIC plus a Poisson prior on parent counts, and declares independence when adding the edge does not improve the score (ΔBIC < 0). Useful when you want sparsity driven by a Poisson structural prior rather than pure BIC.
Rcit	Continuous (general nonlinear)	Random Conditional Independence test. Fully nonparametric; no linearity or Gaussian assumptions. Less computationally expensive than KCI.

Interfaces not included:
IndependenceWrapper and TakesGraph are interfaces and do not appear as user-selectable tests.

Scores

Package: edu.cmu.tetrad.algcomparison.score

These are the scores used by score-based and hybrid search methods.

Scores Overview

Name	Appropriate Data Type	Description
SemBicScore	Continuous (linear-Gaussian)	Standard DAG BIC for linear SEMs. Default for FGES, BOSS, GRaSP, IMaGES, and hybrids.
DiscreteBicScore	Discrete	BIC for discrete Bayesian networks with multinomial CPDs.
BdeuScore	Discrete	Bayesian Dirichlet Equivalent Uniform (BDeu) score. Fully Bayesian alternative to discrete BIC.
ConditionalGaussianBicScore	Mixed continuous/discrete	CG BIC score: continuous nodes are linear-Gaussian conditional on discrete parent configurations.
DegenerateGaussianBicScore	Mixed continuous/discrete, possibly rank-deficient	CG BIC variant adapted for degenerate covariance blocks, allowing singular/near-singular Gaussian components.
BasisFunctionBicScore	Mixed continuous/discrete; nonlinear continuous effects	BIC score for conditional Gaussian models where continuous variables are represented using basis expansions (polynomials or orthogonal bases). Nonlinear additive models become linear in the basis.
EbicScore	Continuous	Extended BIC for linear Gaussian models. Adds sparsity penalty; useful in high-dimensional settings.
GicScores	Continuous or mixed	Generalized Information Criterion (GIC) family for flexible penalty structures.
MSepScore	Latent blocks / clustered indicators	Score defined on measurement blocks using trek-/m-separation structure. Used for latent-variable search after clustering.
MVPBicScore	Mixed projections	BIC score for mixed-variable projection models. Useful for pipelines using projection-based covariance modeling.
PoissonPriorScore	Structural prior (any model class)	Prior score with a Poisson distribution over edge or parent counts; encodes a sparsity preference on graph structure rather than any particular noise model.
ZhangShenBoundScore	Continuous	Score with additional Zhang–Shen-type penalties for stronger complexity control than BIC.

How Tests and Scores Are Used in Algorithms

Constraint-based algorithms (PC, CPC, RFCI, FCI, PCMCI) take any independence test listed above.
Score-based algorithms (FGES, BOSS, GRaSP, SP, IMaGES) take any score listed above.
Hybrid algorithms (GFCI, GRaSP-FCI, BOSS-FCI, SP-FCI, FCIT) use both:
- a score to propose or prune adjacencies, and
- a CI test for collider orientation or selective FCI-style checking.

In the Tetrad GUI:

Choose an algorithm.
Choose a test and/or score appropriate to the data type (continuous, discrete, mixed, non-Gaussian, nonlinear).
Optionally tune parameters such as alpha, penalty discount, prior equivalent sample size, or truncation level.

This page can later be expanded with individual per-test and per-score documentation pages.