Working with Probabilities — Bayesian Networks

Bayesian networks are based on Bayes Theorem, which gives a mathematical framework for describing the probability of an event that may have been the result of any of two or more causes [37]. The questions is this What is the probability that the event was the result of a particular cause, and how does it change if the cause is changing  

Bayesian networks are statistic models for describing probabilistic dependencies for a set of variables. They trace back to a theorem in the eighteenth century found by Thomas Bayes, who first established a mathematical base for probability inference [38]. Bayes theorem is based on two different states  

This equation describes the probability P for state A existing for a given state B. To calculate the probability, Bayes used the probability of B existing given that A exists, multiplied by the probability that A exists, and normalized by the probability that B exists. This admittedly complicated explanation can be interpreted as follows For an existing state B, what is the probability that state B is caused by state A The importance of this theorem is that probabilities can be derived without specified knowledge about P(A B), if information about P(B A), P A), and P B) is available. 

The essence of the Bayesian approach is to provide a mathematical rule explaining how a hypothesis changes in light of new evidence [39], Back to John for an example John has to reach his bus not only once, bnt every morning. The experience he makes if he leaves his honse at different times each day affects his feeling of what the probability is for reaching his bns. 

In a Bayesian analysis, a set of observations shonld be seen as something that changes opinion. In other words, Bayesian theory allows scientists to combine new data with their existing knowledge or expertise. 

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Figure 1 shows a Bayesian network with its respective table of conditional probabiUty. In this network, it can be seen that variable W (environmental conditions) and variable Y (work load) are the probable causes of variable Z (fatigue). The size of the table of conditional probabilities will depend on the mnnber of parents that each node has, and on the number of levels or states that each one can have. On node X with k states, the number of probabihties to be specified is ... 

Probability techniques can help the system work in uncertain situations or scenarios. With respect to the proposed model, DL Reasoning can provide more than one possible solution, which can enable indecisive behavior of robots. In a nondeterministic world, the deterministic way of seeing the world is often not expressive enough to address real-world problems [16]. Mathematically, a Bayesian Network is a directed acyclic graph in which a set of random variables makes up the nodes in the network. A set of directed links connects pairs of nodes, and each node has a conditional probability table that quantifies the effects of parents on it. 

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