pseudolikelihood
Definition
Pseudolikelihood is an approximation of the exact likelihood function used when direct computation of a model's normalizing constant is computationally intractable. In the context of the Ising model, the normalizing constant requires summing over all possible realizations of the binary variables in the network, a quantity that grows exponentially with the number of variables, making exact maximum likelihood estimation feasible only for small graphs. Two pseudolikelihood variants address this problem: the joint pseudolikelihood (JPL), which replaces the exact likelihood with a product of fully conditional distributions for each variable given all others, and the disjoint pseudolikelihood (DPL), which bypasses the multivariate distribution entirely by separately regressing each variable on all remaining variables via logistic regression. Maximum pseudolikelihood estimators based on both approaches are consistent, yet simulation evidence shows that JPL-based estimation accurately approximates maximum likelihood estimates across a range of sample sizes and network structures, while DPL-based estimation performs well only when sample sizes are large.
Sources: Keetelaar et al. (2024)
Related Terms
- network psychometrics (1 shared article)
- maximum likelihood (1 shared article)
- parameter estimation (1 shared article)
- joint pseudolikelihood (1 shared article)
- Ising model (1 shared article)
- disjoint pseudolikelihood (1 shared article)
Applications
Pseudolikelihood and the Ising Model
The Ising model specifies a joint probability distribution over binary variables whose normalizing constant grows exponentially with the number of variables, rendering exact likelihood estimation impractical for larger graphs. Pseudolikelihood approximations reduce the normalizing constant from one requiring exponentially many terms to one requiring only a linear number, making parameter estimation computationally tractable for realistically sized psychological networks.
Sources: Keetelaar et al. (2024)
Pseudolikelihood and Network Psychometrics
In applied psychometric research, pseudolikelihood methods are used to estimate parameters and select edges in psychological networks, including networks representing symptom co-occurrence, attitude structures, and item responses. The JPL is implemented in the bgms package for Bayesian edge selection, while the DPL underlies IsingFit and MGM, both of which combine nodewise logistic regression with Lasso regularization for network reconstruction.
Sources: Keetelaar et al. (2024)
Pseudolikelihood and Maximum Likelihood Estimation
Maximum pseudolikelihood estimators are consistent alternatives to maximum likelihood estimation when the exact likelihood is computationally prohibitive. Simulation results demonstrate that the JPL-based estimator closely approximates maximum likelihood estimates for both sparse and dense networks, whereas the DPL-based estimator shows higher bias and variance in small to moderate samples, converging toward maximum likelihood performance only at large sample sizes.
Sources: Keetelaar et al. (2024)
