However, TobyTheCamel's point is valid in that there are some parameter spaces where the MLE is going to be much less useful than others.
Even without having to go to high dimensions, if you've got a posterior that looks like a normal distribution, the MLE is going to the you a lot, whereas if it's a multimodal distribution with a lot of mass scattered around, knowing the MLE much less informative.
But this is a complex topic to address in general, so I'm trying to stick to what I see as the intuition behind the original question!
However, TobyTheCamel's point is valid in that there are some parameter spaces where the MLE is going to be much less useful than others.
Even without having to go to high dimensions, if you've got a posterior that looks like a normal distribution, the MLE is going to the you a lot, whereas if it's a multimodal distribution with a lot of mass scattered around, knowing the MLE much less informative.
But this is a complex topic to address in general, so I'm trying to stick to what I see as the intuition behind the original question!