5 Bayesian inference for extremes Throughout this short course, the method of maximum likelihood has provided a general and Summary: 5 Bayesian inference for extremes Throughout this short course, the method of maximum likelihood has provided a general and flexible technique for parameter estimation. Given a (generic) parameter vector within a family , the likelihood function is the probability (density) of the observed data as a function of . Values of that have high likelihood correspond to models which give high probability to the observed data. The principle of maximum likelihood estimation is to adopt the model with greatest likelihood; of all the models under consideration, this is the one that assigns the highest probability to the observed data. Other inferential procedures, such as "method of moments", provide viable alternatives to maximum likelihood estimation; moments­based techniques choose optimally by equating model­based and empirical moments, and solving for to obtain parameter estimates. These, and other procedures (such as probability weighted moments, L­moments and ranked set estimation), are discussed in detail in, amongst other places, Kotz and Nadarajah (2000). 5.1 General theory Bayesian techniques offer an alternative way to draw inferences from the likelihood func- tion, which many practitioners often prefer. As in the non­Bayesian setting, we assume data x = (x1, . . . , xn) to be realisations of a random variable whose density falls within a parametric family F = {f(x; ) : }. However, parameters of a distribution are now treated as ran- dom variables, for which we specify prior distributions ­ distributions of the parameters prior to the inclusion of data. The specification of these prior distributions enables us to supplement Collections: Mathematics