I am reading a manuscript that provides the "maximum posterior probability" in a Bayesian context as a statistical summary of a parameter.

Is the term "maximum posterior probability" equivalent to (i.e., "exchangeable" with) "posterior mode"?


The answer to the (now edited) question is no, there is no difference: maximum posterior probability for a given value $\hat{\theta}_{MAP}$ (the so-called maximum a-posteriori estimate of $\theta$) is equivalent to the mode of the posterior distribution of your parameter. Mathematically, you can express this in the context of parameter estimation as

$$\hat{\theta}_{MAP}=\text{argmax}_{\theta}\ \ p(\theta|D,I),$$

where $p(\theta|D,I)$ is the posterior distribution of $\theta$ given the data $D$ and any information $I$ (e.g. your prior). As some people stated on the comments of the other answers this is not equivalent to the maximum likelihood estimate (MLE) in general (this is the case if you have an uniform prior, as long as this prior is defined on a suitable interval around the MLE), because


and you wish to maximize $p(D|\theta,I)p(\theta|I)$ in order to obtain the MAP estimate. On the other hand, the MLE maximizes $p(D|\theta,I)$ only.


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