andrieuthomsh: andrieuthomsh function

Description

A Robbins-Munro stochastic approximation update is used to adapt the tuning parameter of the proposal kernel. The idea is to update the tuning parameter at each iteration of the sampler: $$h^{(i+1)} = h^{(i)} + \eta^{(i+1)}(\alpha^{(i)} - \alpha_{opt}),$$ where $h^{(i)}$ and $\alpha^{(i)}$ are the tuning parameter and acceptance probability at iteration $i$ and $\alpha_{opt}$ is a target acceptance probability. For Gaussian targets, and in the limit as the dimension of the problem tends to infinity, an appropriate target acceptance probability for MALA algorithms is 0.574. The sequence $\{\eta^{(i)}\}$ is chosen so that $\sum_{i=0}^\infty\eta^{(i)}$ is infinite whilst $\sum_{i=0}^\infty\left(\eta^{(i)}\right)^{1+\epsilon}$ is finite for $\epsilon>0$. These two conditions ensure that any value of $h$ can be reached, but in a way that maintains the ergodic behaviour of the chain. One class of sequences with this property is, $$\eta^{(i)} = \frac{C}{i^\alpha},$$ where $\alpha\in(0,1]$ and $C>0$.The scheme is set via the mcmcpars function.

Usage

andrieuthomsh(inith, alpha, C, targetacceptance = 0.574)

Arguments

inith

initial h

alpha

parameter $\alpha$

parameter $C$

targetacceptance

target acceptance probability

Value

an object of class andrieuthomsh

References

Andrieu C, Thoms J (2008). A tutorial on adaptive MCMC. Statistics and Computing, 18(4), 343-373.
Robbins H, Munro S (1951). A Stochastic Approximation Methods. The Annals of Mathematical Statistics, 22(3), 400-407.
Roberts G, Rosenthal J (2001). Optimal Scaling for Various Metropolis-Hastings Algorithms. Statistical Science, 16(4), 351-367.

Examples

Run this code

# NOT RUN {
andrieuthomsh(inith=1,alpha=0.5,C=1,targetacceptance=0.574)
# }