Efficient strategy for the Markov chain Monte Carlo in high-dimension with heavy-tailed target probability distribution

Abstract

The purpose of this paper is to introduce a new Markov chain Monte Carlo method and exhibit its efficiency by simulation and high-dimensional asymptotic theory. Key fact is that our algorithm has a reversible proposal transition kernel, which is designed to have a heavy-tailed invariant probability distribution. The high-dimensional asymptotic theory is studied for a class of heavy-tailed target probability distribution. As the number of dimension of the state space goes to infinity, we will show that our algorithm has a much better convergence rate than that of the preconditioned Crank Nicolson (pCN) algorithm and the random-walk Metropolis (RWM) algorithm. We also show that our algorithm is at least as good as the pCN algorithm and better than the RWM algorithm for light-tailed target probability distribution.

Hybrid estimation for ergodic diffusion processes based on noisy discrete observations

“…Moreover, the initial Bayes estimator of β are derived from the reduced data with q 2 = 1/2, τ 2 = 2.0, k η 2 ,τ 2 ,n = 10 4 , ∆ τ 2 ,n = 2 × 10 −3 , T η 2 ,n = k η 2 ,τ 2 ,n ∆ τ 2 ,n = 20. Furthermore, the initial Bayes type estimators of α and β are calculated with MpCN method proposed by Kamatani (2018) for 10 3 and 10 6 Markov chains and 10 2 and 10 5 burn-in iterations, respectively. Table 6.…”

Estimation of risk contributions with MCMC

“…However, these proposal distributions often fail to perform well when the target distribution π is heavy-tailed. To overcome this problem, the mixed preconditioned Crank-Nicolson (MpCN) proposal distribution is proposed by Kamatani (2014). This proposal distribution updates the candidate according to the following process:…”

“…Throughout this paper, ρ is set to be 0.8 as a default choice in Kamatani (2014). Ideally, µ and Σ are set to be µ = E[X] and Σ = Var[X], while in practice, they can be replaced by their rough estimates since moments of X are typically unknown.…”

“…Ideally, µ and Σ are set to be µ = E[X] and Σ = Var[X], while in practice, they can be replaced by their rough estimates since moments of X are typically unknown. Note that the original MpCN proposed in Kamatani (2014) is the standardized version (that is, µ = 0 and Σ = I d , where I d is an identity matrix). The acceptance probability (15) of the MpCN proposal distribution can be written as…”

Adaptive Bayes Estimators and Hybrid Estimators for Small Diffusion Processes Based on Sampled Data

“…We calculate the mean and the standard deviation of each estimator from 1000 independent sample paths based on the true model. Optim in R is used to calculate the adaptive ML type estimatorθ and the hybrid estimatorθ, and the initial Bayes type estimatorsα andβ are calculated with one of the MCMC method, the mixed preconditioned Crank-Nicolson (MpCN) method studied in Kamatani (2014) for 5 × 10 4 Markov chains and 5 × 10 3 burn-in iterations. The initial value (α…”