This is the third of 10 chapters on "Spatial Modeling II" from the user manual of CrimeStat IV, a spatial statistics package that can analyze crime incident location data.
This chapter, "Estimating Complex Models with Markov Chain Monte Carlo Simulation," examines the Markov Chain Monte Carlo (MCMC) method for estimating complex models. It is applied to the family of Poisson models for modeling count data (See Chapter 15 of the user manual). In explaining the MCMC approach, the chapter uses a "hill climbing" analogy in which a climber wishes to identify and climb the highest peak in a mountain range, but a cloud cover has obscured all but the bases of the mountains. What the climber needs is a map of the mountain range that not only identifies the highest peak but also the most advantageous route to the highest peak. In the statistical world, such a complex problem requires a "map," which comes from a Bayesian approach to the problem and the alternative search strategy comes from a sampling approach. This is presented as the essential logic underlying the MCMC method. Bayesian probability is first discussed, followed by an explanation of Bayesian inference. Markov chain sequences are then addressed as an alternative search strategy. The five conceptual steps of the MCMC algorithm are outlined. An example of the MCMC Poisson-Gamma model in estimating Houston burglaries is compared with the use of the MLE Poisson-Gamma model. Another example compares the use of the MCMC Normal with the use of the MLE Normal. Risk analysis with MCMC modeling is discussed in another section of the chapter. The chapter concludes with discussions of issues in MCMC modeling and improving the performance of the MCMC algorithm. References and extensive tables