SPFlow is a flexible, modular library for building and reasoning with Sum-Product Networks (SPNs) and Probabilistic Circuits (PCs). These are deep generative and discriminative models that enable tractable (polynomial-time) probabilistic inference while maintaining expressive power. SPFlow is built on PyTorch, providing GPU acceleration and seamless integration with modern deep learning workflows.

Key Features:

• Exact probabilistic inference: marginals, conditionals, most probable explanations
• Modular model construction: manual design or automatic structure learning
• Learning algorithms: gradient descent, expectation-maximization, structure learning
• Ready-made probabilistic circuit models in spflow.zoo, including Naive Bayes
• Full support for missing data and various distribution types
• GPU acceleration via PyTorch

Install from PyPI:

pip install spflow

For contributor/development setup (from source), see CONTRIBUTING.md.

Let's start with a tiny DSL-based example that builds a simple circuit, evaluates log-likelihood, and visualizes the structure.

import shutil

import torch
from spflow.dsl import dsl
from spflow.modules.leaves import Normal
from spflow.utils.visualization import visualize

# Define a tiny probabilistic circuit with two weighted Gaussian-product branches.
with dsl():
    terms = 0.4 * Normal(0) * Normal(1) + 0.6 * Normal(0) * Normal(1)

# Materialize the DSL expression into an executable circuit.
pc = terms.build()

# Score a batch of 8 synthetic 2D observations.
ll = pc.log_likelihood(torch.randn(8, 2))
print(ll.shape)

# Plot graph
visualize(pc, output_path="dsl-structure", format="svg")

A more complex circuit with explicit products/sums, likelihood evaluation, and sampling could look like the following.

import shutil

import torch
from spflow.meta import Scope
from spflow.modules.leaves import Categorical, Normal
from spflow.modules.products import Product
from spflow.modules.sums import Sum
from spflow.utils.visualization import visualize

# Fix the RNG seed so the example remains reproducible.
torch.manual_seed(0)

# Feature indices (X, Z1, Z2)
x_idx, z1_idx, z2_idx = 0, 1, 2

# ---- Leaf modules ----
# Left branch will model Z1 together with a mixture over (X, Z2)
leaf_z1_left = Categorical(scope=Scope([z1_idx]), out_channels=2, K=3)
leaf_x_1 = Normal(scope=Scope([x_idx]), out_channels=2)
leaf_z2_1 = Normal(scope=Scope([z2_idx]), out_channels=2)
leaf_x_2 = Normal(scope=Scope([x_idx]), out_channels=2)

# Right branch will model Z2 together with a mixture over (Z1, X)
leaf_z2_right = Normal(scope=Scope([z2_idx]), out_channels=2)
leaf_z1_1 = Categorical(scope=Scope([z1_idx]), out_channels=2, K=3)
leaf_x_3 = Normal(scope=Scope([x_idx]), out_channels=2)
leaf_z1_2 = Categorical(scope=Scope([z1_idx]), out_channels=2, K=3)

# ---- Left branch: Z1 × Sum(X × Z2) ----
# Products combine disjoint scopes (decomposability)
prod_x_z2 = Product(inputs=[leaf_x_1, leaf_z2_1])
prod_z2_x = Product(inputs=[leaf_z2_1, leaf_x_2])

# Sum mixes alternatives with identical scope
sum_x_z2 = Sum(inputs=[prod_x_z2, prod_z2_x], out_channels=2)
prod_z1_sum_xz2 = Product(inputs=[leaf_z1_left, sum_x_z2])

# ---- Right branch: Z2 × Sum(Z1 × X) ----
prod_z1_x_1 = Product(inputs=[leaf_z1_1, leaf_x_3])
prod_z1_x_2 = Product(inputs=[leaf_z1_2, leaf_x_3])
sum_z1_x = Sum(inputs=[prod_z1_x_1, prod_z1_x_2], out_channels=2)
prod_z2_sum_z1x = Product(inputs=[leaf_z2_right, sum_z1_x])

# ---- Root: mixture over the two branches ----
root = Sum(inputs=[prod_z1_sum_xz2, prod_z2_sum_z1x], out_channels=1)

# Likelihood evaluation expects data shaped (N, D)
# Build each feature separately so categorical dimensions get valid integer values.
num_rows = 32
data_x = torch.randn(num_rows)
data_z1 = torch.randint(low=0, high=3, size=(num_rows,), dtype=torch.int64).to(torch.float32)
data_z2 = torch.randn(num_rows)
data = torch.stack([data_x, data_z1, data_z2], dim=1)

# Evaluate the circuit on the batch and then draw unconditional samples from it.
ll = root.log_likelihood(data)

# Unconditional sampling
samples = root.sample(num_samples=5)

print(f"root.out_shape={root.out_shape}")
print(f"data.shape={data.shape}")
print(f"ll.shape={ll.shape}")
print(f"samples.shape={samples.shape}")

# Plot graph
visualize(root, output_path="structure", format="svg")

More examples can be found in the Guides.

• Guides Index: Landing page for end-to-end tutorials and workflow-oriented documentation
• User Guide: Comprehensive notebook with examples covering model construction, training, inference, and advanced use cases
• Developer Guide: Developer-focused notebook for extending and working on SPFlow
• APC MNIST Guide: Notebook walkthrough for APC training on MNIST
• sklearn Guide: Guide to the optional scikit-learn compatible wrappers
• Contributing Guide: Contributor workflow, coding standards, PR process, and commit conventions
• Versioning Guide: Semantic versioning and deprecation policy
• Release Guide: Maintainer runbook for creating and publishing releases

SPFlow 1.1.0 builds on the PyTorch-based rewrite introduced in 1.0.0. The current release features:

• Modern PyTorch architecture for GPU acceleration
• A Naive Bayes model in spflow.zoo for density estimation and classification
• Additional performance improvements across learning, leaf, einsum, and RAT modules
• Enhanced modular design

See the CHANGELOG for detailed version history and recent changes.

We welcome contributions! Please see CONTRIBUTING.md for contribution guidelines.

If you find SPFlow useful please cite us in your work:

@misc{Molina2019SPFlow,
  Author = {Alejandro Molina and Antonio Vergari and Karl Stelzner and Robert Peharz and Pranav Subramani and Nicola Di Mauro and Pascal Poupart and Kristian Kersting},
  Title = {SPFlow: An Easy and Extensible Library for Deep Probabilistic Learning using Sum-Product Networks},
  Year = {2019},
  Eprint = {arXiv:1901.03704},
}

• Steven Braun - TU Darmstadt
• Arseny Skryagin - TU Darmstadt
• Alejandro Molina - TU Darmstadt
• Antonio Vergari - University of Edinburgh
• Karl Stelzner - TU Darmstadt
• Robert Peharz - TU Graz
• Nicola Di Mauro - University of Bari Aldo Moro
• Kristian Kersting - TU Darmstadt

• Philipp Deibert - TU Darmstadt
• Kevin Huy Nguyen - TU Darmstadt
• Bennet Wittelsbach - TU Darmstadt
• Felix Divo - TU Darmstadt
• Moritz Kulessa - TU Darmstadt
• Claas Voelcker - TU Darmstadt
• Simon Roesler - Karlsruhe Institute of Technology
• Alexander L. Hayes - Indiana University, Bloomington
• Alexander Zeikowsky - TU Darmstadt

See the full list of contributors on GitHub.

This project is licensed under the Apache License, Version 2.0 - see the LICENSE file for details.

• Parts of SPFlow as well as its motivating research have been supported by the Germany Science Foundation (DFG) - AIPHES, GRK 1994, and CAML, KE 1686/3-1 as part of SPP 1999- and the Federal Ministry of Education and Research (BMBF) - InDaS, 01IS17063B.

• This project received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Grant Agreement No. 797223 (HYBSPN).