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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=xi∣Pa=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})} }}
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Maximum Likelihood Estimation with Priors
Last modified 1mo ago