Gelman, Andrew, Professor of statistics and political science and director of the Applied Statistics Center at Columbia University
Professor Gelman refers to Bayesian inference as a "coherent mathematical theory," but does not trust it for scientific applications. Gelman believes that it is too easy to apply subjective beliefs about a given situation to Bayesian theory; because people want to believe their own preconceived notions and reject results statistical results they do not want to agree with. Bayesian methods according Gelman, encourage this kind of thinking.
Gelman takes special issue with political scientists like himself adopting Bayesian methods. Bayesian approaches tend to assume exchangeability of variables. However in political science it is impossible to exchange each of the 50 states, they cannot be used randomly or as samples.
Gelman continues by saying that he is not hostile to mathematics of Bayesianism, but its "philosophical foundations, that the goal of statistics is to make an optimal decision." Gelman believes that statistics are for doing "estimation and hypothesis testing," not to "minimize the average risk." He also faults the Bayesian philosophy of axiomatic reasoning because it implies that random sampling should not be done which Gelman considers to be "strike against the theory right there." He also accuses Bayesians of believing in "the irrelevance of stopping times," which means that stopping an experiment it will not change your inference. Gelman concludes by saying "the p-value does change when you alter the stopping rule, and no amount of philosophical reasoning will get you around that point."