*Medical Imaging: Image Processing*, K.M. Hanson, ed.,
*Proc. SPIE ***3338**, pp. 371-382 (1998)

## Posterior sampling with improved efficiency

Kenneth M. Hanson and Gregory S. Cunningham

*Los Alamos National Laboratory*
### Abstract

The Markov Chain Monte Carlo (MCMC) technique provides a means to generate a random sequence of model realizations that sample the posterior probability distribution of a Bayesian analysis. That sequence may be used to make inferences about the model uncertainties that derive from measurement uncertainties. This paper presents an approach to improving the efficiency of the Metropolis approach to MCMC by incorporating an approximation to the covariance matrix of the posterior distribution. The covariance matrix is approximated using the update formula from the BFGS quasi-Newton optimization algorithm. Examples are given for uncorrelated and correlated multidimensional Gaussian posterior distributions.

**Keywords:** Adaptive Markov Chain Monte Carlo,
statistical efficiency, BFGS optimization, Bayesian analysis,
uncertainty estimation

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