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Bayesian networks are acyclic directed graphs that represent factorizations of joint probability distributions. Every joint probability distribution over n random variables can be factorized in n! The joint probability distribution over these four variables can be factorized in 4! Each of these factorizations can be represented by a Bayesian network.
We construct the directed graph of the network by creating a node for each of the factors in the distribution we label each of the nodes with the name of a variable before the conditioning bar and drawing directed arcs between them, always from the variables on the right hand side of the conditioning bar to the variable on the left hand side.
For the first factorization above. These are encoded in conditional probability distribution matrices equivalent to the factors in the factorized form , called conditional probability tables CPTs that are associated with the nodes.
It is worth noting that there will always be nodes in the network with no predecessors. These nodes are characterized by their prior marginal probability distribution. Any probability in the joint probability distribution can be determined from these explicitly represented prior and conditional probabilities. A straightforward representation of the join probability distribution over n binary variables requires us to represent the probability of every combination of states of these variables.
While there are 30 numbers in the four tables above, please note that half of them are implied by other parameters, as the sum of all probabilities in every distribution has to be 1. This allows us to simplify the factorization to the following. Similarly, suppose that we know that Asbestos Exposure and Smoking are independent of each other. This allows us to further simplify the factorization to the following. Finally, supposed that we realize that knowledge of Smoking status and Asbestos Exposure makes Socio-Economic Status irrelevant to the probability of Cancer , i.
We can simplify the factorization further to. Please note that the new network is missing three arcs compared to the original network marked by dimmed arcs in the previous pictures. These three arcs correspond to three independences that we encoded in the simplified factorization. It is a general rule that every independence between a pair of variables results in a missing arc. Conversely, if two nodes are not directly connected by an arc, then there exists such set of variables in the joint probability distribution that makes them conditionally independent.
A possible interpretation of this is that the variables are not directly dependent. If we need to condition on other variables to make them independent, then they are indirectly dependent. The total number of parameters is 16 and the total number of independent parameters is only 8. This reduction in the number of parameters necessary to represent a joint probability distribution through an explicit representation of independences is the key feature of Bayesian networks.
For example, in case of the first factorization. Using independences to simplify the graphical model is a general principle that leads to simple, efficient representations of joint probability distributions. As an example, consider the following Bayesian network, modeling various problems encountered in diagnosing Diesel locomotives, their possible causes, symptoms, and test results.
This network contains 2, nodes, i. The model pictured above is represented by only 6, independent parameters. As we can see, representing joint probability distributions becomes practical and models of the size of the example above are not uncommon. The purely theoretical view that Bayesian networks represent independences and that lack of an arc between any two variables X and Y represents a possibly conditional independence between them, is not intuitive and convenient in practice.
A popular, slightly informal view of Bayesian networks is that they represent causal graphs in which every arc represents a direct causal influence between the variables that it connects. A directed arc from X to Y captures the knowledge that X is a causal factor for Y. While this view is informal and it is easy to construct mathematically correct counter-examples, it is convenient and widely used by almost everybody applying Bayesian networks in practice. There is a well-established assumption, with no convincing counter-examples, that causal graphs will automatically lead to correct patterns of independences.
However, each row sums up to 1, and therefore, we need two independent parameters per row. The whole table needs 8 2 x 4 independent parameters.
Similarly, the Offer table has six entries, but only 1 independent parameter per row is required, which makes 3 1 x 3 independent parameters. Therefore, the independence assumptions in the Bayesian network helps us avoid specifying the joint distribution.
What I cannot understand is why in the final calculation the number of parameters to define interview table is 12 instead of 8? Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group.
Create a free Team What is Teams? Learn more. Asked 2 years, 10 months ago. Active 8 months ago. Viewed 5k times. Have some problems in understanding the number of parameters in Bayes Net How do we calculate the number of parameters in the Bayes net in the preceding diagram?
I can understand that the total number of parameters is 24 without reduction the number independent parameters for each row in interview table is 2 the number independent parameters for each row in job offer table is 1 What I cannot understand is why in the final calculation the number of parameters to define interview table is 12 instead of 8? Can I seek your help in understanding this? Thanks in advance. Improve this question.
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