Inception PCs strictly subsume monotone and squared PCs, and are strictly more expressive than both. We show this leads to improved downstream modeling performance when normalizing for FLOPS:

To overcome these limitations, we propose Inception PCs, a novel tractable probabilistic model representing a deep *sum-of-square-of-sums*.

Inception PCs explicitly introduce two types of latent variables into the circuit for the mixtures encoded at sum nodes.

We show that the reverse also holds (!!) - some tractable distributions expressed as monotone circuits cannot be compactly expressed as a square.

Circuits are generative models that use sum-product computation graphs to model probability densities. But how do we ensure the non-negativity of the output?

Check out our poster "On the Relationship between Monotone and Squared Probabilistic Circuits" at AAAI 2025 **today**: 12:30pm-14:30pm #841.

Along the way we also show a bunch of other cool results, like:
- More efficient algorithms for causal inference on circuits
- New circuit properties
- Separation/hardness results

Building upon the prior PC atlas (proceedings.neurips.cc/paper_files/... ), our algebraic atlas provides a comprehensive approach for deriving **efficient algorithms** and **tractability conditions** for arbitrary compositional queries.

Try our atlas the next time you come across a new query!

Just as circuits serve as a unifying representation of models, we show how you can express many queries as compositions of just a few basic operations: aggregation (marginalization, max, etc.), product, and elementwise mappings.

Circuits are a unifying representation of probability distributions as a computation graph of sums and products. Here we consider the more general algebraic circuits, where sum/product is replaced with a semiring operation (think e.g. OR and AND for Boolean circuits).