In abduction we take some observations and try to find the hypothesis that best explains them.
In Bayesian terms this sounds like finding the Maximum A Posteriori (MAP) estimate. To a Bayesian, "best" means "most likely." If we combine this with a Minimum Description Length (MDL) prior distribution, then in Bayesian terms abduction seems to be the process of finding the shortest hypothesis that fully explains the data. Abduction is essentially just what Bayesian inference does, no more and no less.
Bayesian inference is also typically taken to be a model of induction. In Bayesian terms, induction is also a matter of finding the MAP estimate, which with a MDL prior would again be the shortest hypothesis that fully explains the data. It appears that abduction, Bayesian inference, and induction are all the same thing.
Is there any example of induction, or any example of abduction, that does not simply boil down to finding a MAP estimate?
What I see is that listed examples of induction typically focus on the formation of scientific theories, like the theory of gravity, whereas examples of abduction tend to focus on particular events, like figuring out where someone was last night. From a Bayesian perspective these two scenarios are treated in basically the same way. A Theory of Gravity is evaluated by how well it explains the data and how concise it is... and A Theory of Where You Were Last Night is also evaluated by how well it explains the data and how concise it is. If the focus on scientific theories vs the focus on particular events is the only real difference between induction and abduction, it seems arbitrary, when we can think about both kinds of things using the same tools (MAP estimation).
