  Advancements in computer hardware have made it possible to utilize low- and
mixed-precision arithmetic for enhanced computational efficiency. In practical
predictive modeling, however, it is vital to quantify uncertainty due to
rounding along other sources like measurement, sampling, and numerical
discretization. Traditional deterministic rounding uncertainty analysis (DBEA)
assumes that the rounding errors equal the unit roundoff $u$. However, despite
providing strong guarantees, DBEA severely overestimates rounding uncertainty.
This work presents a novel probabilistic rounding uncertainty analysis called
VIBEA. By treating rounding errors as i.i.d. random variables and leveraging
concentration inequalities, VIBEA provides high-confidence estimates for
rounding uncertainty using higher-order rounding error statistics. The
presented framework is valid for all problem sizes $n$, unlike DBEA, which
necessitates $nu<1$. Further, it can account for the potential cancellation of
rounding errors, resulting in rounding uncertainty estimates that grow slowly
with $n$. We show that for $n>n_c(u)$, VIBEA produces tighter estimates for
rounding uncertainty than DBEA. We also show that VIBEA improves existing
probabilistic rounding uncertainty analysis techniques for $n\ge3$ by using
higher-order rounding error statistics. We conduct numerical experiments on
random vector dot products, a linear system solution, and a stochastic boundary
value problem. We show that quantifying rounding uncertainty along with
traditional sources (numerical discretization, sampling, parameters) enables a
more efficient allocation of computational resources, thereby balancing
computational efficiency with predictive accuracy. This study is a step towards
a comprehensive mixed-precision approach that improves model reliability and
enables budgeting of computational resources in predictive modeling and
decision-making.
