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Localtab Group 

Localtab 

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title  Minimum Description Length 

 The Minimum Description Length (MDL) score is a twocomponent score that has been traditionally used in the Artificial Intelligence community for estimating the number of bits required to represent a model and the data given this model. For structural learning of Bayesian networks, the model is the Bayesian network (graph plus probability tables), whereas the number of bits for representing the data given the Bayesian network is inversely proportional to the probability of the observations returned by the model. Image Added where: 
Image Added represents the BayesiaLab Structural Coefficient (the default value is 1), a parameter that allows changing the weight of the MDL structural part (the lower its value, the greater the complexity of the resulting networks),  DL(B) the number of bits to represent the Bayesian network B (graph and probabilities), and
 DL(DB) the number of bits to represent the dataset D given the Bayesian network B.

Localtab 

title  Bayesian Network Part 

 Localtab Group 

Localtab 

 The number of bits to represent a Bayesian network is equal to the number of bits to represent the structure plus the number of bits to represent the probability distributions. Image Addedwhere G refers to the Graphical structure, and P to the set of Probability tables. 
Localtab 

 The coding of the structure implies the identification of each node plus its parents. Image Addedwhere n is the number of random variables (nodes): Image Added Image Added is the set of the random variables that are parents of Image Added in the graph G  and is
and Image Added is the number of parents of random variable Image Added.

Localtab 

 The number of bits to represent the probability distributions is proportional to the number of cells of the conditional probability tables. Image Addedwhere  val(X) represents Image Added is the number of states of random variable XImage Added,
 p is the probability recorded in associated with the cell.
As this probability is not known prior to learning the network, we are using the following classical heuristic: Image Added where N is the size number of observations in the datasetdata set. 


Localtab 

 The number of bits for representing the data given the Bayesian network is inversely proportional to the probability of the observations returned by the model. Image Added Image Added Image Added where  Image Added is the ndimensional observation described in row j, and
 Image Added is the joint probability of this observation returned by the Bayesian network B.
The chain rule allows rewriting this equation with: Image Added Image Added 

For each candidate Bayesian network, which is generated during the search and evaluated with the MDL score, the corresponding parameters are computed using Maximum Likelihood Estimation.
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