I have a conceptual confusion about the use of the expected value of a distribution.
Often, we want to estimate the most likely value of something. For example, I have X= ten observations. I know X was drawn from the poisson distribution, and I know that this poisson distribution itself has a lambda that was drawn from the lognormal distribution with mu = 4, std = .05.
So given my observations, and my knowledge about the underlying distributions, I want to estimate lambda. So Bayes theorem comes in handy, with
posterior = lambda given X
prior = lognormal distribution
likelihood = likelihood function for a poisson distribution
This can be solved (e.g., with rejection sampling) to discover that our Bayes estimator for lambda is ~4.28.
What we did, in intuitive/oversimplified terms, is take the "center of mass" of our posterior distribution, and found it was ~4.28.
My question is: if we're trying to find the most likely value of lambda, why are we taking the "center of mass" of the posterior distribution, rather than the highest point on our posterior distribution?
In this example these two are extremely similar, but you could imagine changing some assumptions in this problem, and getting a posterior distribution that is (for instance) bimodal. If this were the case, the highest point of this bimodal distribution seems like it corresponds nicely to the "most likely value of lambda," whereas the center of mass of this posterior distribution could be an extremely unlikely value of lambda (e.g., a value in the valley between the two peaks of the distribution).
Any help is appreciated!
TL;DR: When estimating the 'most likely value' for a parameter, why do we ever want to do so by finding the 'center of mass' (expected value) from that parameter's distribution? Shouldn't we always want the 'highest point' (MLE) on that distribution?