tmg-hmc 0.0.7

This package implements exact HMC sampling for truncated multivariate gaussians with quadratic constraints

tmg_hmc

Exact Hamiltonian Monte Carlo sampling for truncated multivariate Gaussians with quadratic constraints

This package implements the exact HMC algorithm from Pakman and Paninski (2014) for sampling from truncated multivariate Gaussian distributions.

How It Works

The algorithm uses Hamiltonian Monte Carlo with

1. Analytic Hamiltonian Dynamics: Particles follow deterministic Hamiltonian trajectories that are analytically computable
2. Exact Bounces: When a trajectory hits a constraint boundary, the algorithm computes the exact bounce time by solving the quartic equation for the hit time analytically
3. Perfect Acceptance Probability: Unlike standard HMC, there's no integration error to solve the Hamiltonian dynamics. This means the acceptance probability is always 1.

See Pakman & Paninski (2014) for mathematical details.

Features

• ✅ Flexible constraints - Supports linear and quadratic inequality constraints
• ✅ Efficient - Uses optimized compiled C++ hit time calculation for efficient sampling
• ✅ GPU acceleration - Optional PyTorch backend for large-scale problems

Installation

From PyPi

pip install tmg-hmc

From Source

git clone https://github.com/erik-a-bensen/tmg_hmc.git
cd tmg_hmc 
pip install .

Requirements:

• Python 3.10+
• numpy
• scipy
• torch

Build Requirements:

• C++ compiler (g++, clang, or MSVC)
• make

Quick Start

Linearly Constrained Gaussian

Sample a 2D standard normal with the y-component restricted to be positive:

import numpy as np
from tmg_hmc import TMGSampler 

# Define the mean and covariance of the untruncated distribution
mu = np.zeros((2, 1))
Sigma = np.identity(2)
sampler = TMGSampler(mu, Sigma)

# Define the constraint y >= 0
# This corresponds to the constraint: f^T x + c >= 0
# where f = [0, 1] and c = 0
f = np.array([[0], [1]])
sampler.add_constraint(f=f, c=0)

# Sample 100 samples with 100 burn-in iterations
x0 = np.array([[1], [1]])  # Initial point (must satisfy constraints)
samples = sampler.sample(x0, n_samples=100, burn_in=100)

print(f"Sample mean: {samples.mean(axis=1)}")
print(f"Sample covariance:\n{np.cov(samples)}")

Quadratically Constrained Gaussian

Sample from a Gaussian constrained to a circular region:

import numpy as np
from tmg_hmc import TMGSampler

# 2D standard normal
mu = np.zeros((2, 1))
Sigma = np.identity(2)
sampler = TMGSampler(mu, Sigma)

# Add constraint: x^2 + y^2 <= 4 (inside circle of radius 2)
# Quadratic constraint: x^T A x + f^T x + c <= 0
# For x^2 + y^2 - 4 <= 0, we have A = I, f = 0, c = -4
A = np.identity(2)
c = -4
sampler.add_constraint(A=A, c=c)

# Sample
x0 = np.array([[0.5], [0.5]])
samples = sampler.sample(x0, n_samples=1000, burn_in=100)

Multiple Constraints

Combine multiple constraints (e.g., box constraints):

import numpy as np
from tmg_hmc import TMGSampler

mu = np.zeros((2, 1))
Sigma = np.identity(2)
sampler = TMGSampler(mu, Sigma)

# Box constraint: -1 <= x, y <= 1
# x >= -1  =>  [1,0]^T x + 1 >= 0
sampler.add_constraint(f=np.array([[1], [0]]), c=1)
# x <= 1   =>  [-1,0]^T x + 1 >= 0
sampler.add_constraint(f=np.array([[-1], [0]]), c=1)
# y >= -1  =>  [0,1]^T x + 1 >= 0
sampler.add_constraint(f=np.array([[0], [1]]), c=1)
# y <= 1   =>  [0,-1]^T x + 1 >= 0
sampler.add_constraint(f=np.array([[0], [-1]]), c=1)

x0 = np.array([[0], [0]])
samples = sampler.sample(x0, n_samples=1000, burn_in=100)

Examples

TO BE IMPLEMENTED

Testing

TO BE IMPLEMENTED

Documentation

• Full API Reference - Complete documentation of all functions and classes
• Hit-time Calculations - Mathematica solutions for the hit times of each type of constraint.

Contributing

Contributions are welcome! Please feel free to submit a Pull Request. For major changes, please open an issue first to discuss what you would like to change.

License

This project is licensed under the MIT License - see the LICENSE file for details.

Citation

If you use this package in your research, please cite:

Software:

Bensen, E. A. (2025). tmg_hmc: A Python package for Exact HMC Sampling for Truncated Multivariate Gaussians with Linear and Quadratic Constraints. TBD. [To be updated upon acceptance]

Methodology:

Pakman, A., & Paninski, L. (2014). Exact Hamiltonian Monte Carlo for Truncated Multivariate Gaussians. Journal of Computational and Graphical Statistics, 23(2), 518-542. https://doi.org/10.1080/10618600.2013.788448

BibTeX

@article{Bensen2025tmghmc,
  title={tmg\_hmc: A Python package for Exact HMC Sampling for Truncated Multivariate Gaussians with Linear and Quadratic Constraints},
  author={Bensen, Erik A.},
  journal={TBD},
  year={2025},
  note={[To be updated upon acceptance]}
}

@article{PakmanPaninski2014,
  title={Exact Hamiltonian Monte Carlo for Truncated Multivariate Gaussians},
  author={Pakman, Ari and Paninski, Liam},
  journal={Journal of Computational and Graphical Statistics},
  volume={23},
  number={2},
  pages={518--542},
  year={2014},
  publisher={Taylor \& Francis},
  doi={10.1080/10618600.2013.788448}
}

Acknowledgments

This implementation is based on the exact HMC algorithm developed by Ari Pakman and Liam Paninski.