English.SE, Hi, first time being here. I have had this confusion about the pronunciation of "a priori" and "a posteriori" for a long time, normally I just read the last vowel as /i/, however today my office mate asked me about this for he saw the pronunciation on Merriam-Webster online dictionary shows that the last vowel reads as /ai/, this reminded me that the philosophy professors whose lectures I took before didn't agree on this pronunciation either. Since I learned that Immanuel Kant borrowed these words from Latin, I wonder what is the correct way to pronounce these words in Latin or at least in a Latin'ish way.
The OED gives ay - pr - eye - 'or - eye as the only pronunciation. I (an American English speaker) usually say ah - pr - ee - 'or - ee (which appears in the MW pronunciation you cited), but I hear both. As for Immanuel Kant, he would have likely pronounced it differently than an ancient Roman anyhow. One of the key differences between classical and ecclesiastical Latin (the latter of which Kant certainly would have learned) is pronunciation. So, it's hard to say which one would be more "Latin" (neither is spot on).
As a speaker of British English, I pronounce the final syllable of each the same way I pronounce the first person singular personal pronoun.
These two are among the select (?) group of foreign phrases that have been Anglicised for so long that their pronunciation has lost any connection with Latin; unless in a specifically Roman context, the A sounds like the article, and the I like the pronoun. If you really wanted the Latin, both would be short; but I wouldn't recommend it.