  We examine a parameterized complexity class for randomized computation where
only the error bound and not the full runtime is allowed to depend more than
polynomially on the parameter, based on a proposal by Kwisthout in [15,16]. We
prove that this class, for which we propose the shorthand name PPPT, has a
robust definition and is in fact equal to the intersection of the classes
paraBPP and PP. This result is accompanied by a Cook-style proof of
completeness for the corresponding promise class (under a suitable notion of
reduction) for parameterized approximation versions of the inference problem in
Bayesian networks, which is known to be PP-complete. With these definitions and
results in place, we proceed by showing how it follows from this that
derandomization is equivalent to efficient deterministic approximation methods
for the inference problem. Furthermore, we observe as a straightforward
application of a result due to Drucker in [8] that these problems cannot have
polynomial size randomized kernels unless the polynomial hierarchy collapses to
the third level. We conclude by indicating potential avenues for further
exploration and application of this framework.
