We employ a new stochastic methodology for the construction of surrogate models for uncertainty quantification (UQ) and sensitivity analysis (SA). It is based on polynomial dimensional decomposition (PDD), as are widely used in solving high-dimensional stochastic problems that arise in various applications. In our approach, the coefficients of the PDD expansion are determined by using a least-squares regression (LSR). Compared to a projection approach, the use of LSR not only avoids the computation of high-dimensional integrals, but also affords an attractive flexibility in choosing the sampling points, which facilitates importance sampling using a calibrated posterior distribution based on a Bayesian approach. LSR can be particularly advantageous in cases where the asymptotic convergence properties of polynomial expansions cannot be realized due to computation expense, focusing effort on efficient finite-resolution sampling. To efficiently include parameter spaces with a moderate number of uncertain parameters (up to 7 in this work), the PDD is coupled with an adaptive ANOVA (analysis of variance) decomposition. This provides an accurate surrogate as the union of several low-dimensional spaces, avoiding the typical computational overhead cost of a high-dimensional expansion. In addition, the PDD representation of the ANOVA component functions is further simplified in an adaptive way according to the relative contribution of the different polynomials to the variance. The overall methodology is demonstrated on plasma-mediated ignition simulations as part of a large predictive science effort in the Center for Exascale Simulation of Plasma-Coupled Combustion (XPACC). The specific configuration we study includes model parameters arising from reaction rates in a global chemical kinetics description, and a laser-induced breakdown ignition seed.