# Choosing the number of factors in factor analysis with incomplete data via a hierarchical Bayesian information criterion

arXiv: https://arxiv.org/abs/2204.09086 · HTML: https://mlsanity.com/paper/2204.09086/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 46 references that are in the map.
- First posted on arXiv 19 Apr 2022.

## Builds on (most cited first, 5 of 5)

- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Inferring Parameters and Structure of Latent Variable Models by Variational Bayes](https://mlsanity.com/paper/1301.6676/) (2013)
- [Extended Stochastic Gradient MCMC for Large-Scale Bayesian Variable Selection](https://mlsanity.com/paper/2002.02919/) (2020)
- [Generalized infinite factorization models](https://mlsanity.com/paper/2103.10333/) (2021)
- [Mixture of Latent Trait Analyzers for Model-Based Clustering of Categorical Data](https://mlsanity.com/paper/1301.2167/) (2013)

## Abstract (arXiv)

The Bayesian information criterion (BIC), defined as the observed data log likelihood minus a penalty term based on the sample size N, is a popular model selection criterion for factor analysis with complete data. This definition has also been suggested for incomplete data. However, the penalty term based on the `complete' sample size N is the same no matter whether in a complete or incomplete data case. For incomplete data, there are often only Ni<N observations for variable i, which means that using the `complete' sample size N implausibly ignores the amounts of missing information inherent in incomplete data. Given this observation, a novel criterion called hierarchical BIC (HBIC) for factor analysis with incomplete data is proposed. The novelty is that it only uses the actual amounts of observed information, namely Ni's, in the penalty term. Theoretically, it is shown that HBIC is a large sample approximation of variational Bayesian (VB) lower bound, and BIC is a further approximation of HBIC, which means that HBIC shares the theoretical consistency of BIC. Experiments on synthetic and real data sets are conducted to access the finite sample performance of HBIC, BIC, and related criteria with various missing rates. The results show that HBIC and BIC perform similarly when the missing rate is small, but HBIC is more accurate when the missing rate is not small.