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Maximum Likelihood Estimation

Context

  • BayesiaLab estimates the parameters of a Bayesian network using Maximum Likelihood Estimation.
  • The probability of a state
    x0{x_0}
    of a node
    X{X}
    corresponds to the frequency the state
    x0{x_0}
    is observed in the dataset.

Example

Let's consider this simple network:

Maximum Likelihood Estimation

The marginal probability distribution of
PaPa
is estimated as:
P^(Pa=pai)=N(Pa=pai)jN(Pa=paj)\hat P(Pa = p{a_i}) = \frac{{N(Pa = p{a_i})}}{{\sum\nolimits_j {N(Pa = p{a_j})} }}
where
N()N\left( \cdot \right)
represents the number of occurrences of the specified configuration in the dataset.
The conditional probability distribution of X|Pa is estimated as
P^(X=xiPa=pai)=N(X=xi,Pa=pai)jN(X=xj,Pa=paj)\hat P(X = {x_i}|Pa = p{a_i}) = \frac{{N(X = {x_i},Pa = p{a_i})}}{{\sum\nolimits_j {N(X = {x_j},Pa = p{a_j})} }}