What are the differences in meaning and usage between posteriori and posterior?
Particularly in probability, statistics and logic, when should I use which?
For example, why are "max a posteriori" and "posterior distributions"?
A posteriori is something "relating to or denoting reasoning or knowledge that proceeds from observations or experiences to the deduction of probable causes."
It constrasts with a priori that is, instead, something "relating to or denoting reasoning or knowledge that proceeds from theoretical deduction rather than from observation or experience."
The first expressions that you asked for, then, is used when you assert or state something after having observed the subject of the matter. You can read more about these two terms in their wikipedia page.
Posterior means the back part of a body, or something coming after.
Note: The emphasized parts are taken from the NOAD.
In Bayesian statistics, you begin with an estimated probability distribution, called the prior or the prior distribution. Then you do your number-crunching, and come out with a (presumably) better estimate of the probability distribution, called the posterior distribution.
As far as I know, this is not related to the philosophical term a posteriori.
In philosophy and logic there is a distinction between two types of knowledge:
I'm not sure how it relates to 'max a posteriori' or 'posterior distributions', but this is a useful distinction for logic.
(examples are from wikipedia)
A Posteriori means "posterior to"; posterior has several meanings, but conveys the idea of "coming afterward" (thus its common use in anatomy to refer to parts of the body on the backside of a corresponding part). Knowing when to use which in various subjects requires understanding the context in which you want to use each; they are both adjectives but convey slightly different connotations.