  This position paper summarizes a recently developed research program focused
on inference in the context of data centric science and engineering
applications, and forecasts its trajectory forward over the next decade. Often
one endeavours in this context to learn complex systems in order to make more
informed predictions and high stakes decisions under uncertainty. Some key
challenges which must be met in this context are robustness, generalizability,
and interpretability. The Bayesian framework addresses these three challenges
elegantly, while bringing with it a fourth, undesirable feature: it is
typically far more expensive than its deterministic counterparts. In the 21st
century, and increasingly over the past decade, a growing number of methods
have emerged which allow one to leverage cheap low-fidelity models in order to
precondition algorithms for performing inference with more expensive models and
make Bayesian inference tractable in the context of high-dimensional and
expensive models. Notable examples are multilevel Monte Carlo (MLMC),
multi-index Monte Carlo (MIMC), and their randomized counterparts (rMLMC),
which are able to provably achieve a dimension-independent (including
$\infty-$dimension) canonical complexity rate with respect to mean squared
error (MSE) of $1/$MSE. Some parallelizability is typically lost in an
inference context, but recently this has been largely recovered via novel
double randomization approaches. Such an approach delivers i.i.d. samples of
quantities of interest which are unbiased with respect to the infinite
resolution target distribution. Over the coming decade, this family of
algorithms has the potential to transform data centric science and engineering,
as well as classical machine learning applications such as deep learning, by
scaling up and scaling out fully Bayesian inference.
