Stability of noisy Metropolis-Hastings

Area of Science:

• Computational statistics
• Markov chain Monte Carlo methods

Background:

• Pseudo-marginal Markov chain Monte Carlo (MCMC) methods are popular for sampling from complex distributions.
• These methods are exact but can suffer from poor mixing and slow convergence.
• The noisy algorithm offers an alternative with potentially better mixing but sacrifices exactness.

Purpose of the Study:

• To further characterize the noisy algorithm in Markov chain Monte Carlo.
• To analyze fundamental stability properties such as positive recurrence and geometric ergodicity.
• To establish conditions for inheriting properties from standard Metropolis-Hastings chains.

Main Methods:

• Theoretical analysis of Markov chain properties.
• Investigation of stability and convergence criteria.
• Comparison with standard Metropolis-Hastings algorithms.

Main Results:

• Sufficient conditions for the noisy algorithm to inherit geometric ergodicity from a Metropolis-Hastings chain are provided.
• The convergence of the noisy algorithm's invariant distribution towards the true target distribution is analyzed.
• Fundamental stability properties of the noisy algorithm are characterized.

Conclusions:

• The noisy algorithm presents a viable alternative to pseudo-marginal MCMC when mixing is a concern.
• Understanding its stability and convergence is crucial for practical application.
• This work contributes to the theoretical foundation of advanced MCMC techniques.