Spherical Hamiltonian Monte Carlo for Constrained Target Distributions.

Area of Science:

• Machine Learning
• Statistical Modeling
• Computational Statistics

Background:

• Statistical models with constrained probability distributions are common in machine learning.
• Bayesian inference in these models presents challenges for standard sampling algorithms due to boundary conditions.

Purpose of the Study:

• To propose a novel Markov Chain Monte Carlo (MCMC) method for efficient Bayesian inference with constrained probability distributions.
• To develop a general framework for handling boundary conditions in statistical models.

Main Methods:

• Mapping the constrained parameter domain to a sphere.
• Augmenting the mapped domain to a higher-dimensional sphere to implicitly handle constraints.
• Utilizing split dynamics with geodesic flow for computational efficiency.

Main Results:

• The proposed MCMC method effectively handles boundary conditions in constrained domains.
• Demonstrated efficiency and natural framework for various models like truncated Gaussian, Bayesian Lasso, and copula models.
• Successfully applied to identify synchrony in multi-neuron data.

Conclusions:

• The novel MCMC method offers a general and computationally efficient solution for Bayesian inference with constrained distributions.
• This approach simplifies handling boundary conditions, making complex models more tractable.
• The method shows broad applicability across diverse statistical and machine learning problems.