[Paper Review] Convergence Rates for Hierarchical Gibbs Samplers

Oliver Jovanovski, Neal Madras|arXiv (Cornell University)|Feb 19, 2014

Bayesian Methods and Mixture Models14 references3 citations

TL;DR

This paper establishes geometric convergence rates for hierarchical Gibbs samplers targeting a gamma posterior distribution in Bayesian hierarchical models with depth 3 or 4. Using coupling techniques and stochastic monotonicity, the authors bound the total variation distance to equilibrium, showing convergence within millions of iterations under realistic parameter settings, with explicit quantitative rates derived via auxiliary processes and moment bounds.

ABSTRACT

We establish some results for the rate of convergence in total variation of a Gibbs sampler to its equilibrium distribution. This sampler is motivated by a hierarchical Bayesian inference construction for a gamma random variable. Our results apply to a wide range of parameter values in the case that the hierarchical depth is 3 or 4, and are more restrictive for depth greater than 4. Our method involves showing a relationship between the total variation of two ordered copies of our chain and the maximum of the ratios of their respective co-ordinates. We construct auxiliary stochastic processes to show that this ratio does converge to 1 at a geometric rate.

Motivation & Objective

• To establish rigorous upper bounds on the total variation convergence rate of a hierarchical Gibbs sampler for a gamma posterior distribution.
• To analyze the convergence behavior of a Gibbs sampler with sequential updating of odd and even coordinates in a 4-dimensional state space.
• To extend the analysis to the n=3 case, demonstrating similar convergence properties with simplified dynamics.
• To derive explicit quantitative bounds on convergence speed using coupling and moment-based inequalities.
• To provide practical convergence thresholds for MCMC sampling in hierarchical Bayesian inference with gamma priors.

Proposed method

• Construct a Gibbs sampler that updates coordinates in the order 1,3,2,4 using conditional gamma distributions.
• Use iterated random functions to represent the Markov chain dynamics via i.i.d. gamma-distributed innovations.
• Introduce auxiliary stochastic processes to track the ratio of corresponding coordinates in two coupled chains.
• Apply coupling arguments to show that the ratio of coordinates converges to 1 at a geometric rate, implying geometric ergodicity.
• Use moment bounds and stopping time arguments to control the expected ratio and derive convergence rate constants.
• Leverage stochastic monotonicity and partial fraction decompositions to compute key expectations like E[(γ₂/γ₄ + γ₄/γ₂)(γ₃/(γ₂+γ₄))].

Experimental results

Research questions

• RQ1 What is the rate of convergence in total variation for a hierarchical Gibbs sampler targeting a gamma posterior with depth 4?
• RQ2 How can coupling techniques be used to establish geometric convergence for high-dimensional hierarchical models?
• RQ3 What conditions on the shape parameters ensure finite moments and geometric ergodicity in such samplers?
• RQ4 How does the convergence rate depend on the model parameters, particularly the shape and scale parameters?
• RQ5 Can similar convergence bounds be derived for the depth-3 case, and how do they compare to the depth-4 case?

Key findings

• For n=4, the total variation distance to equilibrium decays geometrically, with dTV(Ut+3, π) ≤ 31,065 × (1 − 3/4356)^(t/40) + (1 + 59/20) × (7/9)^(⌊t/2⌋+3), implying dTV ≤ 10⁻⁵ for t ≥ 1,050,000.
• For n=3, the convergence bound is dTV(Ut+2, π) ≤ 600 × (1 − 78/79)^(t/20) + 6 × (7/9)^(⌊t/2⌋+3), yielding dTV ≤ 10⁻⁵ for t ≥ 14,000.
• The convergence rate is governed by a contraction factor r < 1, derived from moment bounds on gamma ratios and stopping time control.
• The method relies on constructing a coupling where the ratio of coordinates in two chains converges geometrically to 1, ensuring rapid mixing.
• Key constants such as β ≤ 7/9 and r ≤ 1 − 3/4356 are derived from parameter-dependent moment expectations and partial fraction decompositions.
• The analysis confirms that geometric ergodicity holds under the conditions a₁ + a₄ > 1 and a₂ + a₃ > 1, ensuring integrability of key ratios.