Bayesian and frequentist optimization and intervals I realize the methodology pursued by the Frequentist and Bayesian camps generally differ. However, one method of estimation that they do share is optimization of a certain function:

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*Frequentists maximize the likelihood function, giving the Max. Likelihood (ML) estimator.

*Bayesians maximize the posterior function, giving the Max A-Posteriori (MAP) estimator.

Both functions will typically have been constructed using Baye's rule/theorem, which is universally agreed upon, and which might have been applied once (in "batch mode") or multiple times iteratively.
Similarly, both Frequentists and Bayesians will deduce their interval (confidence/credibility) from this function.
So if the prior is uninformative (assuming we can formulate such a prior), there should be no distinction between the "results" obtained by Bayesians and Frequentists, even though the interpretation of said results will be different.
If this is right, then the only practical difference between Bayesians and Frequentists is the prior. Is this true?

Edit:
Actually, the optimization bit of my question is a bit misleading, as it is only a specific example of differences between Bayesian and Frequentist thinking. My question could be posed simply as the difference simply between the likelihood function and the posterior. For example, would frequentists ever use MCMC to calculate the likelihood function?
Edit, 10 years on:
This was a confused question from a confused student. To be fair, the topic tends to confuse the uninitiated. Thanks to everyone who participated in the discussions.
 A: The maximum aposteriori (MAP) approach isn't really fully Bayesian, ideally inference should involve marginalising over the whole posterior.  Optimisation is the root of all evil in statistics; it is difficult to over-fit if you don't optimise! ;o)  So the practical difference between Bayesian and frequentist runs rather more deeply if you opt for a fully Bayesian solution, although there will often be a prior for which the result is numerically the same as the frequentist approach.
However, the credible interval and confidence interval are answers to different questions, and shouldn't be considered interchangable, even if they happen to be numerically the same.  Treating a frequentist confidence interval as if it was a Bayesian credible interval can lead to problems of interpretation.
UPDATE "My question could be posed simply as the difference simply between the likelihood function and the posterior."
No, the definitions of probability are different, this means that even if the solutions are numerically identical, this doesn't imply they mean the same thing.  This is a practical issue as well as a conceptual one, as the correct intepretation of the result depends on the definition of probability.
Tongue in cheek, if the answer to the question was "yes", it would imply that frequentists were merely Bayesians that always used flat (often improper) priors for everything.  I doubt many frequentists would agree with that! ;o)
A: I would agree that roughly speaking you are right.  Priors noninformative or not will lead to different solutions.  The solutions will converge when the data dominates the prior.  Also Jeffreys needed improper priors in some cases to match Bayesian results with frequentist results.  The real difference and the controversy is philosophical.  Frequentists want objectivity.  The prior brings in subjective opinion.  Bayesians following the teachings of Di Finetti believe that probability is subjective.  For a true Bayesian priors should be informative.  The other point also related to the differing concepts of probability is that probability can be assigned to an unknown parameter according to Bayesians while frequentist think strictly in term of probability spaces as given in the theory developed by Kolmogorov and von Mises. For frequentists it is only the random avriables that you can define on the probaility space that have probabilities associated with their outcomes.  So the probability of getting a head on a coin toss is 1/2 because repeated flipping leads to a relative frequence of heads that converges to 1/2 as the sample size approaches infinity.
For frequentists Bayes' theorem applies to events which are measureable sets in a probabilty space. The Bayesians apply it to parameters as if the parameter was a random variable.  That is the frequentists objection to Bayesian methods. The Bayesians object to the frequenttist approach because it lacks a property called coherence.  I will not go into that here but you can look up the definiton on the internet or read Dennis Lindley's books.
