METHODS
Historically, one of the first methods of simulation by computer was Monte Carlo integration (MCI). Its computational algorithm relies on repeated random sampling from distributions of interest to obtain numerical results. For example, MCI is used to calculate the expectation of a distribution function when its estimation through integration is impossible. Its plain logic is to generate samples from a distribution function\(\ \)and approximate the expectation numerically by calculating the average of generated samples. In other words, empirical experimental summation is substituted for analytical (or possibly numerical) integration. This technique is conceptually simple but not always very efficient if the target probability function is not well-behaved. Attempts to deal with this problem led to a gradual evolution first to IS and next in RS; they can be regarded as improved versions of MCI. However, in some more challenging cases they have also proven inadequate to the task. Over the past 30 years or so, MCMC methods have revolutionized statistical computing and permitted ever more complex problems to be handled. Finally, we have included DA which follows a different approach to calculate posterior estimates.