Specifically, we view all labeled entity spans as observed nodes in a constituency tree, and other spans as latent nodes. In this work, we view nested NER as constituency parsing with partially-observed trees and model it with partially-observed TreeCRFs. However, the widely-used sequence labeling framework is difficult to detect entities with nested structures. Named entity recognition (NER) is a well-studied task in natural language processing. Experiments show that our approach achieves the state-of-the-art (SOTA) F1 scores on the ACE2004, ACE2005 dataset, and shows comparable performance to SOTA models on the GENIA dataset. To compute the probability of partial trees with partial marginalization, we propose a variant of the Inside algorithm, the Masked Inside algorithm, that supports different inference operations for different nodes (evaluation for the observed, marginalization for the latent, and rejection for nodes incompatible with the observed) with efficient parallelized implementation, thus significantly speeding up training and inference. With the TreeCRF we achieve a uniform way to jointly model the observed and the latent nodes. In an application to musical data, we approach the unsolved problem of hierarchical music analysis from the raw note level and compare our results to expert annotations. The capacity and diverse applications of RBNs are illustrated on two examples: In a quantitative evaluation on synthetic data, we demonstrate and discuss the advantage of RBNs for segmentation and tree induction from noisy sequences, compared to change point detection and hierarchical clustering. 2) For Gaussian RBNs, we additionally derive an analytic approximation, allowing for robust parameter optimisation and Bayesian inference. We provide two solutions: 1) For arbitrary RBNs, we generalise inside and outside probabilities from PCFGs to the mixed discrete-continuous case, which allows for maximum posterior estimates of the continuous latent variables via gradient descent, while marginalising over network structures. The main challenge lies in performing joint inference over the exponential number of possible structures and the continuous variables. RBNs define a joint distribution over tree-structured Bayesian networks with discrete or continuous latent variables. In this paper, we present Recursive Bayesian Networks (RBNs), which generalise and unify PCFGs and DBNs, combining their strengths and containing both as special cases. Therefore, neither can be applied if the latent variables are assumed to be continuous and also to have a nested hierarchical dependency structure. In contrast, DBNs allow for continuous latent variables, but the dependencies are strictly sequential (chain structure). While PCFGs allow for nested hierarchical dependencies (tree structures), their latent variables (non-terminal symbols) have to be discrete. Probabilistic context-free grammars (PCFGs) and dynamic Bayesian networks (DBNs) are widely used sequence models with complementary strengths and limitations.
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