Recursive Bayesian Networks: Generalising and Unifying Probabilistic Context-Free Grammars and Dynamic Bayesian Networks. (arXiv:2111.01853v1 [cs.LG])

Probabilistic context-free grammars (PCFGs) and dynamic Bayesian networks
(DBNs) are widely used sequence models with complementary strengths and
limitations. While PCFGs allow for nested hierarchical dependencies (tree
structures), their latent variables (non-terminal symbols) have to be discrete.
In contrast, DBNs allow for continuous latent variables, but the dependencies
are strictly sequential (chain structure). Therefore, neither can be applied if
the latent variables are assumed to be continuous and also to have a nested
hierarchical dependency structure. 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. RBNs define a joint
distribution over tree-structured Bayesian networks with discrete or continuous
latent variables. The main challenge lies in performing joint inference over
the exponential number of possible structures and the continuous variables. 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. 2) For Gaussian RBNs, we
additionally derive an analytic approximation, allowing for robust parameter
optimisation and Bayesian inference. 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. 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.



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