The Valencia International Meetings on Bayesian Statistics, held every four years, provide the forum for researchers to come together to present and discuss frontier developments in the field. The resulting proceedings provide a definitive, up-to-date overview encompassing a wide range of theoretical and applied research. This fourth volume of proceedings is no exception. In particular, it reflects a growing emphasis on computational issues, concerned with making Bayesian methods routinely available to applied practitioners, both statisticians and other specialists whose work depends on careful quantification of uncertainties. The growing interest in Bayesian methods is revealed by the ever-increasing participation in the Valencia meetings. This, in turn, is reflected in the high quality of this work, which contains 30 invited papers by leading authorities and 33 refereed contributed papers, selected from over 150 presented at the meeting.