To handle more complex latent structures such as hierarchical or dynamic latent variables, the conjugated harmonium theory could be extended in several ways:
Hierarchical Structures: For hierarchical latent variables, the theory could incorporate nested exponential family distributions to model dependencies at different levels of the hierarchy. By defining conjugate priors and likelihoods for each level, researchers could develop a hierarchical conjugated harmonium framework.
Dynamic Latent Variables: To handle dynamic latent variables, the theory could be extended to include time-dependent or sequential data. By introducing temporal dependencies and incorporating dynamic models such as hidden Markov models or Kalman filters, researchers could develop a framework for exact inference and learning in dynamic latent variable models.
Non-linear Relationships: Extending the theory to handle non-linear relationships between latent variables and observations could involve incorporating non-linear transformations or neural network architectures within the conjugated harmonium framework. This extension would allow for more flexible modeling of complex data distributions.
By incorporating these extensions, the conjugated harmonium theory could provide a more comprehensive framework for handling diverse and intricate latent structures in LVMs.