A note on the role of projectivity in likelihood-based inference for random graph models

TL;DR

This paper clarifies misconceptions about the importance of projectivity in likelihood-based inference for random graph models, demonstrating that it is not necessary for consistent estimation.

Contribution

It proves that likelihood-based superpopulation inference does not require projectivity, challenging previous claims about its necessity for estimator consistency.

Findings

Likelihood-based inference is unaffected by lack of projectivity.

Projectivity is not necessary for maximum likelihood estimator consistency.

Clarifies misconceptions about projectivity's role in random graph models.

Abstract

There is widespread confusion about the role of projectivity in likelihood-based inference for random graph models. The confusion is rooted in claims that projectivity, a form of marginalizability, may be necessary for likelihood-based inference and consistency of maximum likelihood estimators. We show that likelihood-based superpopulation inference is not affected by lack of projectivity and that projectivity is not a necessary condition for consistency of maximum likelihood estimators.