Bayesian Network
Also known as: Bayes network, belief network, probabilistic graphical model, directed graphical model, Bayesian Ağı
A Bayesian network is a probabilistic graphical model, introduced by Judea Pearl in 1988, that encodes a set of variables and their conditional dependencies as a directed acyclic graph (DAG). Each node represents a variable; each directed edge encodes a direct probabilistic influence. By combining Bayes' rule with the graph's conditional independence structure, the model supports reasoning under uncertainty — computing the probability of any variable given observed evidence about others.
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When to use it
A Bayesian network is appropriate when you want to model a system of interacting variables and reason about conditional dependencies among them — for example, in diagnostic systems, causal discovery, or risk modelling. The method requires at least around 100 observations for the conditional probability tables to be estimated reliably; with smaller samples the structure learned from data is unstable and expert elicitation of the DAG is necessary. The variables can be categorical, binary, or discretised continuous. It is not suitable when the causal or dependency structure cannot be determined either from theory or data, because a misspecified DAG gives misleading inference.
Strengths & limitations
- Represents complex conditional dependency structures compactly through a DAG, making assumptions explicit and interpretable.
- Supports both predictive inference (given causes, what are the effects?) and diagnostic inference (given effects, what caused them?).
- Works with categorical, binary, and mixed variable types without requiring normality.
- Allows prior domain knowledge to be encoded directly in the graph structure or the conditional probability tables.
- Requires at least about 100 observations; with small samples the conditional probability tables cannot be estimated reliably and the learned structure is unstable.
- Exact inference is NP-hard in general networks; large networks require approximate inference methods that introduce additional error.
- Structure learning from data is computationally intensive and can produce spurious edges, especially when variables are correlated but not causally linked.
- A misspecified or incorrectly oriented DAG leads to incorrect conditional independence claims and unreliable inference.
Frequently asked
What is a DAG and why must it be acyclic?
A directed acyclic graph is a graph whose edges have a direction and which contains no directed cycles — you cannot start at a node and follow edges back to the same node. The acyclicity requirement ensures that the joint probability factorises consistently: each node's probability can be conditioned on its parents without circular dependencies. Models with cycles (feedback loops) require different formalisms such as dynamic Bayesian networks or cyclic graphical models.
What is a conditional probability table?
A conditional probability table (CPT) for a node lists the probability of each of that node's values for every possible combination of its parents' values. A root node (no parents) has a simple marginal probability table. CPTs are either estimated from data using maximum likelihood or Bayesian estimation, or elicited from domain experts when data are scarce.
How is the graph structure determined?
Structure can come from domain theory (expert elicitation), from data using score-based algorithms (e.g. hill-climbing with BIC score) or constraint-based algorithms (e.g. PC algorithm using conditional independence tests), or from a combination of both. Data-driven structure learning requires sufficient observations — typically at least 100 — because sparse data produces unreliable edge orientations.
What is the difference between a Bayesian network and a Bayesian regression model?
Bayesian regression is a single-equation model that places priors on regression coefficients and uses the posterior to predict one outcome from several predictors. A Bayesian network is a multivariate system: it models the joint distribution of all variables simultaneously through a graph, supporting inference in any direction — predicting downstream effects, diagnosing upstream causes, or computing any conditional distribution the graph supports.
Sources
- Pearl, J. (1988). Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann. ISBN: 978-1558604797
How to cite this page
ScholarGate. (2026, June 1). Bayesian Network. ScholarGate. https://scholargate.app/en/bayesian/bayesian-network
Which method?
Set this method beside its closest kin and read them side by side — the library lays the books on the table; the choice is yours.
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- Structural Equation ModelingResearch Statistics↔ compare