# A Note on Bayesian Networks with Latent Root Variables

arXiv: https://arxiv.org/abs/2402.17087 · HTML: https://mlsanity.com/paper/2402.17087/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 1 of its 5 references that are in the map.
- First posted on arXiv 26 Feb 2024.

## Builds on (most cited first, 1 of 1)

- [Causal Expectation-Maximisation](https://mlsanity.com/paper/2011.02912/) (2020)

## Abstract (arXiv)

We characterise the likelihood function computed from a Bayesian network with latent variables as root nodes. We show that the marginal distribution over the remaining, manifest, variables also factorises as a Bayesian network, which we call empirical. A dataset of observations of the manifest variables allows us to quantify the parameters of the empirical Bayesian net. We prove that (i) the likelihood of such a dataset from the original Bayesian network is dominated by the global maximum of the likelihood from the empirical one; and that (ii) such a maximum is attained if and only if the parameters of the Bayesian network are consistent with those of the empirical model.