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 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:

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

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

The conditional probability distribution of X|Pa is estimated as

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