Hidden Chow-Liu Trees (HCLT)
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Hidden Chow-Liu Trees (HCLTs) are latent-variable models where the structure is derived from a Chow-Liu tree over the observed variables, and hidden states are modeled via the channel dimension.

Reference
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HCLTs and their learning algorithms are discussed in:

- `Learning Hidden Chow-Liu Trees <https://arxiv.org/abs/2106.10332>`_ (Liu & Van den Broeck, 2021)
- `Probabilistic Circuits: A Unifying Framework for Tractable Probabilistic Models <https://starai.cs.ucla.edu/papers/ChoiProbCirc20.pdf>`_ (Choi et al., 2020)

Overview
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HCLTs represent a powerful compromise between the simplicity of tree-structured models and the expressiveness of deep circuits. By introducing latent variables at each node of a Chow-Liu tree, they can capture complex dependencies while remaining extremely efficient to learn and evaluate.

Key features:
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- **Structure learning**: Uses the Chow-Liu algorithm to find the optimal tree structure.
- **Top-k Mixtures**: Supports building mixtures over multiple high-scoring trees for increased robustness.
- **Latent states**: Each observed variable is associated with a hidden category that mediates its dependencies.

Implementation
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SPFlow provides automated learners for binary and categorical HCLTs.

Binary HCLT
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.. autofunction:: spflow.zoo.hclt.learn_hclt_binary

Categorical HCLT
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.. autofunction:: spflow.zoo.hclt.learn_hclt_categorical