One way of dealing with this is to do bayesian algebra with the hyper real or surreal numbers (I'm guessing) but I'm not sure that you can do that mathematically, or that "P(a)=x where x is a hyperreal (or surreal number)" makes any sense.

"Nonstandard analysis" is not a substantive departure from the standard real number system; it is simply an alternative language that some people like for aesthetic reasons, or can sometimes be useful for "bookkeeping". Basically, if nonstandard analysis solves your problem, there was already a solution in terms of standard analysis. Robinson's "hyperreals" essentially just replace the concept of a "limit", and do not represent a fundamentally "new kind of number" in the way that, say, Cantor's transfinite ordinals do.

Conway's surreals are a different story. However, if the sorts of problems you're talking about could be solved simply by saying "oh, just use that other number system over there", they would have been solved long ago (and everybody would probably be using that other number system).