Maximum Likelihood Estimation

Context

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

Example

Let’s consider this simple network:

Maximum Likelihood Estimation

The marginal probability distribution of $Pa$ is estimated as:

$\displaystyle \widehat{P}(Pa = p{a_i}) = \frac{{N(Pa = p{a_i})}}{{\sum\nolimits_j {N(Pa = p{a_j})} }}$

where $N\left( \cdot \right)$ represents the number of occurrences of the specified configuration in the dataset.

The conditional probability distribution of $X \mid Pa$ is estimated as

$\displaystyle \widehat{P}(X = {x_i} \mid Pa = p{a_i}) = \frac{{N(X = {x_i},Pa = p{a_i})}}{{\sum\nolimits_j {N(X = {x_j},Pa = p{a_j})} }}$