I'm using prophet model for one of my time series analysis. I learnt that it optimizes the parameters by MAP approach. The fundamental idea of when to use MAP vs MLE is that when we have a strong prior probability. If we fail to have a prior probability then MLE is the parameter estimation technique that is used. Having said that, I would like to learn what and how the prior is estimated in a time series data and what do we use as an evidence for the posterior estimation.
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The simple answer is that Prophet uses MAP because it is a Bayesian model. While one of the advantages of Bayesian models is that if you have some prior information, it can be included in a Bayesian model through the priors, though it is not the only scenario where we use Bayesian models. For example, the Why would someone use a Bayesian approach with a 'noninformative' improper prior instead of the classical approach? thread discusses when and why the Bayesian approach is used while there is no strong prior information. The long story short, it may be desirable because the results may be easier to interpret, it returns the uncertainty estimates out-of-the-box, it may be easier to estimate the parameters (returning some, not necessarily "good" result), it allows for using priors, etc.
Tim
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