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)โˆ‘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(โ‹…)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=xiโˆฃPa=pai)=N(X=xi,Pa=pai)โˆ‘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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