Introduction

EnhancedBayesianNetworks.jl is a Julia package for building, reducing, and querying enhanced Bayesian networks (eBNs) [1]: Bayesian networks extended with the continuous and functional nodes of structural reliability analysis, and with imprecision — interval probabilities and probability boxes — carried consistently from the inputs through to the inference result.

Why enhanced Bayesian networks

A classical Bayesian network [2] is a directed acyclic graph of discrete random variables, each with a conditional probability table (CPT) given its parents. That is expressive for categorical reasoning, but engineering models rarely stop there: quantities are continuous, and the probabilities that matter — a component's failure probability, say — are not tabulated in advance but computed from a physical model.

An enhanced Bayesian network closes that gap by admitting three kinds of node side by side:

• discrete nodes, whose CPT is known a priori;

• continuous nodes, holding a probability distribution;

• functional nodes, whose conditional table is not given but derived from the parents through one or more UncertaintyQuantification.jl models, evaluated as structural reliability problems [3].

See the Nodes chapter for the full taxonomy.

The system probability distribution

Notation

Throughout the manual, $Y$ denotes a discrete node, $X$ a continuous node, and $Z$ a node of either kind. An uppercase symbol ($Y_i$, $X_i$, $Z_i$) is the node — a random variable — while the matching lowercase symbol ($y_i$, $x_i$, $z_i$) is one of its realizations. Bold marks a collection: a set of nodes ($\mathbf{Y}$, $\mathbf{X}$) or a full joint assignment ($\mathbf{y}$, $\mathbf{x}$). A node's parents, children, and spouses (co-parents) are written $\mathrm{Pa}(\cdot )$, $\mathrm{Ch}(\cdot )$, and $\mathrm{Sp}(\cdot )$.

A Bayesian network encodes a joint distribution that factorizes over the graph. By the local Markov property each variable is conditionally independent of its non-descendants given its parents, so the joint distribution of $n$ discrete variables $Y_1,\dots ,Y_n$ is the product of the per-node conditional tables [2]:

$$p(y_1,\dots ,y_n)=\prod _{i=1}^{n}p(y_i\mid \mathrm{Pa}(Y_i)),$$

where $\mathrm{Pa}(Y_i)$ denotes the parents of $Y_i$.

An enhanced Bayesian network keeps this factorized structure but admits both a set of discrete nodes $\mathbf{Y}=\{Y_1,\dots ,Y_{n_Y}\}$ and a set of continuous nodes $\mathbf{X}=\{X_1,\dots ,X_{n_X}\}$. Its system joint distribution is then the combined measure of the discrete probability mass functions and the continuous probability density functions [1]:

$$p(\mathbf{y}\mid \mathbf{x})f(\mathbf{x})=\prod _{Y_i\in \mathbf{Y}}p(y_i\mid \mathrm{Pa}(Y_i))\prod _{X_i\in \mathbf{X}}f(x_i\mid \mathrm{Pa}(X_i)),$$

where the $p(\cdot )$ are the conditional PMFs of the discrete nodes and the $f(\cdot )$ the conditional PDFs of the continuous ones. Functional nodes are exactly the factors of this product whose conditional table is not given a priori: it is obtained, during reduction, by solving a structural reliability problem (see Reduction & Reliability Analysis).

Precision and imprecision

Every quantity in the model may be precise or imprecise. A discrete CPT entry can be a single probability or an Interval; a continuous node can hold an ordinary distribution or a probability box [4]. Imprecision expresses epistemic uncertainty — what is not known well enough to pin down a single number [5] — and it decides the kind of network you end up with: any surviving imprecision turns a Bayesian network into a credal network [6], a whole convex family of Bayesian networks rather than a single one.

From model to answer

Because a functional or continuous node cannot be queried directly, an eBN is first reduced to a purely discrete network: its continuous nodes are discretized and its functional nodes are evaluated as reliability problems, yielding a BayesianNetwork when everything stays precise or a CredalNetwork when imprecision survives. This reduction is the core operation of the library — see Reduction & Reliability Analysis.

The reduced network is then ready for inference: exact posteriors by variable elimination for a Bayesian network, or lower/upper posterior bounds for a credal one (see Inference). The package also supports learning CPTs from data (Parameter Learning) and drawing the network structure (Plotting).

How this manual is organized

• Getting Started — install the package and run a first model.

• Nodes — the building blocks: discrete, continuous, and functional nodes.

• Networks — Bayesian, credal, and enhanced Bayesian networks.

• Reduction & Reliability Analysis — reducing an eBN, and imprecise reliability.

• Inference — variable elimination and credal inference.

• Parameter Learning — learning CPTs from data.

• Plotting — visualizing a network.

The package builds directly on UncertaintyQuantification.jl [3], whose Model, Parameter, RandomVariable, Interval, and ProbabilityBox types (and its simulation methods) are re-exported and used throughout.