Is a Bayesian estimate with a "flat prior" the same as a maximum likelihood estimate? In phylogenetics, phylogenetic trees are often constructed using MLE or Bayesian analysis. Oftentimes, a flat prior is used in the Bayesian estimate. As I understand it, a Bayesian estimate is a likelihood estimate that incorporates a prior. My question is, if you use a flat prior, is it any different from simply doing a likelihood analysis?
 A: Summarizing and extending from the comments:  "A Bayesian MAP estimate may coincide with an MLE. However, the posterior distribution has no equivalent from a likelihood perspective".  What do you mean by "A Bayesian estimate"? Often, with Bayes, we will just summarize the data by the posterior distribution (assuming it exists, in this case, sometimes, with a flat prior (not integrating to one) we get a formal posterior which do not integrate to one, so is not really a distribution). Such Bayesian summary do not have a likelihood variant, as usually seen. Some are trying to rectify this, by introducing the concept of a confidence distribution based on the likelihood function, see http://folk.uio.no/tores/Publications_files/Schweder_Hjort_Confidence%20and%20likelihood_SJS2002.pdf (and their forthcoming book). 
But, if you go the way of defining a bayes estimator, you have various ways of doing that! You can choose the MAP estimator, which formally may be the same as the MLE. Or you can choose an estimator based on decision theory, by minimizing some posterior expected loss function. Many possibilities, and none of those has a likelihood equivalent.
