Ising model
Definition
Ising model is a probabilistic graphical model that specifies a joint probability distribution over binary variables using two types of parameters: pairwise interaction parameters, which capture the conditional dependence between pairs of variables, and threshold parameters, which model the main effect of each variable. It originates in physics but has been applied extensively in psychometrics to represent binary psychological data, including the presence or absence of mental disorder symptoms, evaluations of attitude statements, and correct or incorrect responses to test items. Statistical analysis of the model is complicated by its normalizing constant, a sum over all possible realizations of the modeled variables that grows exponentially with the number of variables, making exact maximum likelihood estimation computationally feasible only for small graphs. For larger networks, approximation methods such as the joint pseudolikelihood and the disjoint pseudolikelihood are used, and simulation evidence shows that the joint pseudolikelihood accurately approximates maximum likelihood estimates while the disjoint pseudolikelihood performs reliably only at large sample sizes.
Sources: Keetelaar et al. (2024)
Related Terms
- network psychometrics (1 shared article)
- maximum likelihood (1 shared article)
- pseudolikelihood (1 shared article)
- parameter estimation (1 shared article)
- joint pseudolikelihood (1 shared article)
- disjoint pseudolikelihood (1 shared article)
Applications
Ising Model and Maximum Likelihood Estimation
Maximum likelihood estimation using the exact likelihood is the most accurate method for fitting the Ising model, but it is computationally feasible only for networks with a limited number of variables, because the normalizing constant contains a number of terms that grows exponentially. Simulation results demonstrate that maximum pseudolikelihood estimators based on the joint pseudolikelihood closely approximate maximum likelihood estimates and offer a stable alternative when the exact likelihood cannot be computed.
Sources: Keetelaar et al. (2024)
Ising Model and Pseudolikelihood Estimation
Two pseudolikelihood approximations are used in place of the exact likelihood when networks are too large for direct computation: the joint pseudolikelihood, which replaces the exact likelihood with a product of fully conditional distributions, and the disjoint pseudolikelihood, which regresses each variable separately on all others using logistic regression. Both estimators are consistent, but they differ in finite-sample performance: the joint pseudolikelihood is more stable across network sizes and densities, whereas the disjoint pseudolikelihood requires large sample sizes and shows an advantage primarily in sparse networks.
Sources: Keetelaar et al. (2024)
