Maria Athanasiou

In this case, the most common method used is the discretisation of the continuous parent node; as a result, the BN designer faces all the difficulties in order to construct the conditional probability table. This is due to the fact that the BN theory so far does not provide us with a method that allows us to propagate evidence in such topologies. Still though, attempts to use the logistic function in the case of a discrete variable with continuous parent has been carried out by Murphy [1]. If we assume $X$ is a discrete variable with values $x_{1} . . . x_{n}$ and $Z = Z_{1} . . . Z_{k}$ are its continuous parents with values $z_{1} . . . z_{k}$, the logistic function for representing the conditional probability of the discrete child node given its parents is defined as: \[ P(X=x_{i}|Z)= \frac { e^{{g_{i}+{\sideset{}{_{n=1}^{k}}\sum}s}_{i,n}z_{n}}} {{\sideset{}{_{j=1}^{m}}\sum} e^{{g_{j}+{\sideset{}{_{n=1}^{k}}\sum}s}_{j,n}z_{n}}} \] where $s_{i,n}$ determines the ste

In this case, the conditional probability of the child node given its parents is typically a conditional Gaussian distribution for each of the states of the parent node. The MTE-based approximation is also applicable in this type of dependency. In such cases, as well as when both parent and child nodes are continuous, MTE can deliver better approximations, especially in the case where the desired distribution is very different in shape to the Gaussian distribution. Lastly, a novel method for representing conditional distributions between continuous nodes is introduced by the author in this thesis . Back to: Representing Distributions in Bayesian Networks

In contrast with discrete variables, when a variable $X$ is continuous and $D_{1}, · · · , D_{n}$ are real values there is no representation that can depict all the possible densities. A very powerful way for representing these variables, which is widely used in literature, is the choice of multivariate continuous distributions, namely the use of Gaussian distributions. These variables can be represented in a BN by using linear Gaussian conditional densities. In this representation, the conditional density of the continuous variable given its parent is given by the following formula [1] [2]: \[ P(X|D_{1},...,D_{n})=N(\alpha_{o} + \sum_{i} \alpha_{i} \cdot u_{i}, \sigma^{2}) \] That means that $X$ is normally distributed and if all the variables in a network are represented as linear Gaussian distributions then the joint distribution is a multivariate Gaussian [3]. If the variables cannot be considered to follow a Gaussian mixture model, the process of discretisation is applie

Assume $X$ is a discrete variable and $D_{1}. . . D_{k}$ are the discrete values from a finite set of its parents node, then the $P(X|D_{1}. . . D_{k})$ can be represented as a table (this is called C onditional P robability T able - CPT), that states the probability of values for $X$ for each joint assessment to $D_{1}. . .D_{k}$. For example, if all variables are binary (these are the variables where YES or NO is enough for an answer) the table will be constructed for $2^{k}$ distributions. For the case, where a single child node has multiple parents a multidimensional table must be constructed. For example, if a binary child node has 10 discrete parent nodes and each of these is represented by 4 discrete values then we will have to construct a table for 2,097,152 distributions. Constructing a multidimensional table as described above, and most importantly acquiring all these numeric information in order to create the network, is a drawback for the BN theory, as mos

Two types of variables can be used for the description of all nodes in a Bayesian Network. These types are: • Discrete variables : This type is appropriate for nodes where the answer can be selected from a small set of discrete choices. For each possible choice, a confidence measure is attached to the node. • Continuous variables : This type suits nodes where continuous domain measurements are to be taken. For such nodes, a conditional probability distribution is attached to the node so that the  continuous measurement received at the input can be converted into a real number representing a  probability measurement of the node. A Bayesian Network that contains both discrete and continuous nodes is called Hybrid or Mixed Bayesian Network.

A Bayesian Network may contains both discrete and continuous variables. This BN is also called Hybrid Bayesian Network. The figure below illustrates an example of a simple Hybrid Bayesian Network. The discrete variables in the network are depicted as circles and the continuous variables as rectangles. For any directed graphical model the conditional distribution of each node given its parents nodes must be defined. Many strategies have been proposed in the literature, in order to tackle the problem of representing the conditional distribution of any type of dependency between the nodes in a BN. Each of these strategies are discussed below. From this Bayesian Network example the following types of dependencies between the nodes of the network can be identified. Discrete child node with discrete parent node (B --> C). Continuous child node with continuous parent node (A --> E). Continuous child node with discrete parent node (C --> D). Discrete child node

The introduction of Bayesian Networks (BNs) - also known as Belief Nets or Belief Networks - for medical diagnosis dated back in the 80s. BNs have been used as a formalism for representing and reasoning in problems involving uncertainty via the use of graphical models; within such structures, probability theory is adopted as a basic framework [1]. The Bayesian Network (BN) formalism offers a natural way to represent the uncertainty involved in medicine when attempting to deliver a diagnosis [2]. This is due to the fact that the dependencies between signs or symptoms and possible diagnoses, as well as the probabilistic interaction among the data, can be easily described in a Bayesian Network (BN). As the formalism offers this natural representation, any probabilistic statement that may concern both individual and combinations of variables can be computed from a properly structured BN. An example of the BN formalism for medical diagnosis is presented in this thesis . [1] J. Pe