Almost always almost right I am looking for a complexity class that relates to APX as BPP relates to P. I have already asked the same question here, but perhaps TCS would be a more fruitful location for answers.

The reason for the question is that in practical problems one often needs to find approximate answers (thus APX) with sufficiently high confidence (thus BPP), which would make the class of problems with bounded probabilistic approximation algorithms potentially a useful model of what's computable in practice.

A possible candidate of such class would be $APX^{BPP}$: problems that admit approximated solutions with bounded probabilistic subroutines; however, I'm not confident that such class would be the appropriate setting for the class probabilistically computable approximations.

Both BPP and APX have been extensively studied. Is that the case for $APX^{BPP}$, or whichever class would be the best to capture the above problems?
For any given objective function, let BotL (best-of-the-list) be the algorithm that evaluates the objective function on a set of inputs and returns an input from that list that had maximal output (from among those inputs), with ties broken arbitrarily. $\:$ Since APX only includes problems

who's objective function can be computed in deterministic polynomial time, BotL can be implemented deterministically in polynomial time. $\:$ Furthermore, the value returned by BotL

is at least as good as any of inputs in the least that BotL was evaluated on. $\:$ In particular,

if any of the inputs in that list is good-enough then BotL's output will be good-enough.

Therefore running BotL on the outputs of a sufficiently large number of independent executions of a base algorithm can amplify the success probability from 1/poly to 1-(1/(2^poly)).

As consequence of the preceding paragraph, the precise

confidence level essentially does not affect the resulting class.

(This situation is highly analogous to RP.)

I was not able to find anything about that on the complexity zoo, although there

might have been talks on it given at the workshop referred to in this paper.