# Enhanced Bayesian Network

The enhanced Bayesian network (EBN) is a computational extension to the traditional Bayesian network (BN) that uses Structural Reliability Methods (SRM) to implement continuous random variables for Risk analysis and Engineering reliability studies of different engineering structures [1]. The SRMs are used to work out the conditional probability tables of BNs containing both discrete and continuous random variables resulting in the reduction of the network to a traditional BN [2].

## Mathematical Approach

a) Enhanced Bayesian network with continuous and discrete nodes. b) Reduced Bayesian network with only discrete nodes.

The nodes in the EBN of Figure~\ref{EBN1}, correspond to independent random ariables. Here $C_1$ is a node representing a continuous random variable while $X_2$ and $X_3$ are discrete nodes. Then, the joint probability function describing the EBN will be given as follows.

$f(C_1)P(X_2,X_3)= f(C_1)P(X_2)P(X_3|C_1,X_2)$

where $f(C_1)$ corresponds to the probability density function of $C_1$, $P(X_2,X_3)$ is the probability mass function of variables $X_2$ and $X_3$. The last term of the equation ($P(X_3|C_1,X_2)$) corresponds to the conditional probability of $X_3$ given its parents $C_1$ and $X_2$. The reduced BN in Figure 1, is obtained by integrating equation above over the domain of the continuous variable, in this case $C_1$

$P(X_2,X_3)=p(X_2)\int_{C_1} P(X_3|X_2,C_1)f(C_1) dC_1$

Then, the conditional probability table for variables $X_2$ and $X_3$ must be computed as:

$P(X_3|X_2)=p(X_2)\int_{\Omega_{X_2}^{x_2},c_1} f(C_1) dC_1$

here $\Omega_{X_2}^{x_2}$ represents the domain of the event when $X_3$ is in state $x_3$ for any state $x_2$ of $X_2$. This equation corresponds to that of a typical reliability problem so it can be solved by using SRM. The currently available methods in OpenCossan are Monte Carlo simulation and Advanced Line Sampling.

## Implementation

To successfully reduce an Enhanced Bayesian network to a typical Bayesian network some rules must be kept in mind:

1. Make sure that the reliability problem on the discrete node whose parents are continuous is correctly defined, i.e., loads are correctly subtracting the capacities.

2. Interval probability nodes must have spouses of at least a continuous node containing a distribution over which the SRM can sample.

## Tutorial

echodemo TutorialEnhancedBayesianNetwork


## References

1. Straub, and Der Kiureghian, A. 2010. “Bayesian Network Enhanced with Structural Reliability Methods: Methodology.” Journal of Engineering Mechanics 136 (10):1248--1258. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000173.
2. Tolo, Silvia, Edoardo Patelli, and Michael Beer. n.d. “Robust Vulnerability Analysis of Nuclear Facilities Subject to External Hazards.” Stochastic Environmental Research and Risk Assessment 31 (10). Springer Berlin Heidelberg: 2733-- 2756. https://doi.org/10.1007/s00477-016-1360-1.