# Marginal likelihood and model selection for Gaussian latent tree and forest models

arXiv: https://arxiv.org/abs/1412.8285 · HTML: https://mlsanity.com/paper/1412.8285/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Dec 2014.

## Built on by (most cited first, 12 of 12)

- [Latent Hierarchical Causal Structure Discovery with Rank Constraints](https://mlsanity.com/paper/2210.01798/) (2022)
- [Identification of Nonlinear Latent Hierarchical Models](https://mlsanity.com/paper/2306.07916/) (2023)
- [CaRiNG: Learning Temporal Causal Representation under Non-Invertible Generation Process](https://mlsanity.com/paper/2401.14535/) (2024)
- [Learning Discrete Concepts in Latent Hierarchical Models](https://mlsanity.com/paper/2406.00519/) (2024)
- [Causal Temporal Representation Learning with Nonstationary Sparse Transition](https://mlsanity.com/paper/2409.03142/) (2024)
- [Generalized Independent Noise Condition for Estimating Causal Structure with Latent Variables](https://mlsanity.com/paper/2308.06718/) (2023)
- [Bayesian Generalization Error in Linear Neural Networks with Concept Bottleneck Structure and Multitask Formulation](https://mlsanity.com/paper/2303.09154/) (2023)
- [Optimal estimation of Gaussian (poly)trees](https://mlsanity.com/paper/2402.06380/) (2024)
- [Differentiable Causal Discovery For Latent Hierarchical Causal Models](https://mlsanity.com/paper/2411.19556/) (2024)
- [Learning under Singularity: An Information Criterion improving WBIC and sBIC](https://mlsanity.com/paper/2402.12762/) (2024)
- [Upper Bound of Bayesian Generalization Error in Partial Concept Bottleneck Model (CBM): Partial CBM outperforms naive CBM](https://mlsanity.com/paper/2403.09206/) (2024)
- [Upper Bounds for Local Learning Coefficients of Three-Layer Neural Networks](https://mlsanity.com/paper/2603.12785/) (2026)

## Abstract (arXiv)

Gaussian latent tree models, or more generally, Gaussian latent forest models have Fisher-information matrices that become singular along interesting submodels, namely, models that correspond to subforests. For these singularities, we compute the real log-canonical thresholds (also known as stochastic complexities or learning coefficients) that quantify the large-sample behavior of the marginal likelihood in Bayesian inference. This provides the information needed for a recently introduced generalization of the Bayesian information criterion. Our mathematical developments treat the general setting of Laplace integrals whose phase functions are sums of squared differences between monomials and constants. We clarify how in this case real log-canonical thresholds can be computed using polyhedral geometry, and we show how to apply the general theory to the Laplace integrals associated with Gaussian latent tree and forest models. In simulations and a data example, we demonstrate how the mathematical knowledge can be applied in model selection.