Description

raftery.diag is a run length control diagnostic based on a criterion of accuracy of estimation of the quantile q. It is intended for use on a short pilot run of a Markov chain. The number of iterations required to estimate the quantile q to within an accuracy of +/- r with probability p is calculated. Separate calculations are performed for each variable within each chain.

If the number of iterations in data is too small, an error message is printed indicating the minimum length of pilot run. The minimum length is the required sample size for a chain with no correlation between consecutive samples. Positive autocorrelation will increase the required sample size above this minimum value. An estimate I (the ‘dependence factor’) of the extent to which autocorrelation inflates the required sample size is also provided. Values of I larger than 5 indicate strong autocorrelation which may be due to a poor choice of starting value, high posterior correlations or ‘stickiness’ of the MCMC algorithm.

The number of ‘burn in’ iterations to be discarded at the beginning of the chain is also calculated.

Usage

Arguments

Value

A list with class raftery.diag. A print method is available for objects of this class. the contents of the list are

Theory

The estimated sample size for variable U is based on the process Z_t = d(U_t <= u) where d is the indicator function and u is the qth quantile of U. The process Z_t is derived from the Markov chain data by marginalization and truncation, but is not itself a Markov chain. However, Z_t may behave as a Markov chain if it is sufficiently thinned out. raftery.diag calculates the smallest value of thinning interval k which makes the thinned chain Z^k_t behave as a Markov chain. The required sample size is calculated from this thinned sequence. Since some data is ‘thrown away’ the sample size estimates are conservative.

The criterion for the number of ‘burn in’ iterations m to be discarded, is that the conditional distribution of Z^k_m given Z_0 should be within converge.eps of the equilibrium distribution of the chain Z^k_t.

Note

raftery.diag is based on the FORTRAN program ‘gibbsit’ written by Steven Lewis, and available from the Statlib archive.

References

Raftery, A.E. and Lewis, S.M. (1992). One long run with diagnostics: Implementation strategies for Markov chain Monte Carlo. Statistical Science, 7, 493-497.

Raftery, A.E. and Lewis, S.M. (1995). The number of iterations, convergence diagnostics and generic Metropolis algorithms. In Practical Markov Chain Monte Carlo (W.R. Gilks, D.J. Spiegelhalter and S. Richardson, eds.). London, U.K.: Chapman and Hall.