When has a Bayesian approach been critical to addressing a theory, hypothesis or problem? A question was recently posted to an listserve that I subscribe to asking (perhaps cynically?) when a Bayesian approach has been crucial to "getting the job done" in addressing questions in the field of ecology.  I'm wondering, in general, about when a Bayesian approach has been essential to progress in a particular field.  
In ecology, Bayesian methods seem to be most frequently used in applied situations with big, complicated data sets, so I'd be especially interested in circumstances that relate to important or classic theories or hypotheses in a field.  
For example, in ecology, Bayesian methods appear to be the only way to fit complex hierarchical models and obtain accurate estimates of things like the size of a population of animals or the survival rate of an individual in a population of critters.  I'm not familiar with instances when progress was made on burning theoretical questions because a Bayesian approach was used, though this could be because ecological theory is often addressed with reductionistic experiments in an ANOVA-like framework where p-values are the historically valued currency.
 A: In the study of medical devices for approval to use for specific indications the US Food and Drug Adminstration has for at least a decade encouraged the use of Bayesian methods in phase III clinical trials to allow prior information about the device to be incorporated along with the trial data.
A: A number of papers have been written on using Bayesian methods to estimate diagnostic testing parameters (false-positive, false-negative, ...).  The Bayesian method is often preferred due to the fact there are often more parameters than observations.  Unlike other common situations, it is nearly impossible to increase the number of observations.
The article below is a nice overview of the problem:
An Application of a Bayesian Approach in Diagnostic Testing Problems in the Absence of a Gold Standard
A: In response to my own question, an article was just published in the journal Ecology titled "Density estimation in tiger populations: combining information for strong inference" by Gopalaswamy et al.  They used a Bayesian model that combined information from tiger studies with different methodologies to improve the accuracy of their estimation of the density of tigers nature preserve.  On their own the two separate studies indicated that there were ~12 +/- 1.95 tigers/100km2 (posterior mean +/- SD) or 6.7 +/- 2.37 tigers/100km2.  The combined Bayesian model provided an estimate of 8.5 +/- 1.95 tigers/100km2.
