  In addition to the evaluation of high-order loop contributions, the precision
and predictive power of perturbative QCD (pQCD) predictions depends on two
important issues: (1) how to achieve a reliable, convergent fixed-order series,
and (2) how to reliably estimate the contributions of unknown higher-order
terms. The recursive use of renormalization group equation, together with the
Principle of Maximum Conformality (PMC), eliminates the renormalization
scheme-and-scale ambiguities of the conventional pQCD series. The result is a
conformal, scale-invariant series of finite order which also satisfies all of
the principles of the renormalization group. In this paper we propose a novel
Bayesian-based approach to estimate the size of the unknown higher order
contributions based on an optimized analysis of probability distributions. We
show that by using the PMC conformal series, in combination with the Bayesian
analysis, one can consistently achieve high degree of reliability estimates for
the unknown high order terms. Thus the predictive power of pQCD can be greatly
improved. We illustrate this procedure for two pQCD observables: $R_{e^+e^-}$
and $R_\tau$, which are each known up to four loops in pQCD. Numerical analyses
confirm that by using the scale-independent and more convergent PMC conformal
series, one can achieve reliable Bayesian probability estimates for the unknown
higher-order contributions.
