Probabilistic Structural Equation Model (PSEM)
Overview
A Probabilistic Structural Equation Model (PSEM) is a Bayesian network-based generalization of traditional Structural Equation Models (SEMs). It allows for modeling complex systems where relationships between variables are probabilistic rather than deterministic, and it can incorporate both observed data and prior knowledge.
Traditional Structural Equation Models
- A traditional SEM combines path models with latent variables.
- It represents relationships between variables as equations (often linear), assuming fixed functional forms.
- It is typically estimated using covariance-based methods.
- SEMs are limited in handling nonlinear, non-normal, or incomplete data.
- They typically require large sample sizes and pre-specified model structures.
- They do not handle probabilistic inference (e.g., belief updating, diagnosis).
Probabilistic Structural Equation Models in BayesiaLab
In BayesiaLab, Probabilistic Structural Equation Models (PSEMs) serve a similar purpose to traditional SEMs but are built on the foundation of Bayesian networks rather than systems of equations. Each PSEM is implemented as a directed acyclic graph, where arcs represent relationships between variables, including those between latent (unobserved) factors and manifest (observed) variables. Every node is associated with a Conditional Probability Table (CPT), which can be learned from data or defined by expert input.
Unlike traditional SEMs, which often require substantial statistical expertise and involve many manual steps, PSEMs in BayesiaLab are designed to be more accessible, particularly to subject matter experts without formal training in statistics. The modeling process is also significantly more efficient, often reducing development time by several orders of magnitude.
Comparing SEMs and PSEMs in Detail
| Feature | Traditional SEM | PSEM in BayesiaLab |
|---|---|---|
| Model structure | Equations and path diagrams | Bayesian network |
| Type of relationships | Deterministic (mathematical equations) | Probabilistic (conditional dependencies) |
| Inference capabilities | Limited | Full probabilistic inference (belief updating) |
| Missing data handling | Requires preprocessing or imputation | Can be handled dynamically in BayesiaLab |
| Learning from data | Requires predefined structure | Structure and parameters can be learned from data |
PSEM Example
The following graph of a PSEM was developed in the context of a recent webinar, Integrating Hedonic and Analytic Data for Product Optimization with Bayesian Networks and GenAI
