maximum likelihood
Definition
Maximum likelihood refers to a parameter estimation method that identifies the values most consistent with observed data under a specified probability model. In network psychometrics, maximum likelihood estimation for the Ising model requires evaluating a normalizing constant that is a sum over all possible realizations of the modeled binary variables, a quantity that grows exponentially with the number of variables in the network. For a graph of fifteen variables, this sum already exceeds thirty thousand terms, making exact computation feasible only for small graphs. Because of this computational barrier, pseudolikelihood approximations such as the joint pseudolikelihood and the disjoint pseudolikelihood are used in place of the exact likelihood for larger networks, though their finite-sample performance relative to maximum likelihood had remained largely unexamined before direct simulation comparison.
Sources: Keetelaar et al. (2024)
Related Terms
- network psychometrics (1 shared article)
- pseudolikelihood (1 shared article)
- parameter estimation (1 shared article)
- joint pseudolikelihood (1 shared article)
- Ising model (1 shared article)
- disjoint pseudolikelihood (1 shared article)
Applications
Maximum Likelihood and Maximum Pseudolikelihood Estimation
Maximum pseudolikelihood estimation replaces the intractable exact likelihood of the Ising model with a product of conditional distributions, bypassing direct evaluation of the normalizing constant. Simulation evidence shows that the joint pseudolikelihood estimator closely approximates maximum likelihood estimates and is stable across network conditions, while the disjoint pseudolikelihood performs adequately only at large sample sizes.
Sources: Keetelaar et al. (2024)
Maximum Likelihood and the Ising Model
The Ising model specifies a joint probability distribution over binary variables using pairwise interaction parameters and threshold parameters, and its normalizing constant makes exact maximum likelihood estimation computationally feasible only for graphs of up to approximately fifteen variables. For the ordinal extension of the model, exact likelihood estimation is impossible even for small graphs because the normalizing constant grows far more rapidly than in the binary case.
Sources: Keetelaar et al. (2024)
