# Bayes and Causal Relationships

• Ton J. Cleophas
• Aeilko H. Zwinderman
Chapter

## Abstract

The current search for causal relationships with Bayesian structural equation modelings will be addressed in this chapter as an example where Bayesian methodologies successfully helped fostering the deepest enigma of mankind, the proof of causality.

A Bayesian equation looks much like a simple path analysis, and, can, actually, be analyzed similarly. The underneath equations give examples of this (Var = variable):
$$\begin{array}{c}\mathbf{prior}\ \mathbf{likelihood}\ \mathbf{distribution}\times \mathbf{Bayes}\ \mathbf{factor}=\mathbf{posterior}\ \mathbf{likelihood}\ \mathbf{distribution}\\ {}\mathbf{path}\ \mathbf{statistic}\kern0.125em \mathbf{1}\times \mathbf{path}\ \mathbf{statistic}\kern0.125em \mathbf{2}=\mathbf{effect}\ \mathbf{of}\kern0.125em \mathbf{Var}\kern0.125em \mathbf{1}\kern0.125em \mathbf{through}\kern0.125em \mathbf{Var}\kern0.125em \mathbf{2}\kern0.125em \mathbf{on}\kern0.125em \mathbf{Var}\kern0.125em \mathbf{3}.\end{array}}$$
Multistep regressions are the basis of path analysis, and path analysis is the basis of Bayesian networks, using structural equation modeling (SEM). They can be visualized with DAGs (directed acyclic graphs), with arrows meant to indicate causality. If SEMs include
• latent factors (unmeasured variables inferred from measured variables) as predictors, we will call it:

• factor analysis if unsupervised (i.e., no dependent variable);

• partial least squares if supervised;

• discriminant analysis if like the above two but including a grouping variable.

• manifest factors (variables), we will call it:

• multistage regressions which might include flawed predictors, given pejorative names, like problematic predictors, multicollinear predictors, indirect predictors.

Bayesian networks do not necessarily use traditional Bayesian statistics, i.e., posterior and prior odds. Instead they may use the methodologies of more modern Bayesian statistics, i.e., conditional and marginal probability distributions (running from 0 to 1), and likelihood distributions (even running from 0 to ∞).

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© Springer International Publishing AG, part of Springer Nature 2018

## Authors and Affiliations

• Ton J. Cleophas
• 1
• Aeilko H. Zwinderman
• 2
1. 1.Department Medicine Albert Schweitzer HospitalAlbert Schweitzer HospitalSliedrechtThe Netherlands
2. 2.Department Biostatistics and EpidemiologyAcademic Medical Center Department Biostatistics and EpidemiologyAmsterdamThe Netherlands