Lukas Mayer

Michael B. Giles, Mario Hefter, Lukas Mayer et al.

We study the approximation of expectations $\E(f(X))$ for Gaussian random elements $X$ with values in a separable Hilbert space $H$ and Lipschitz continuous functionals $f \colon H \to \R$. We consider restricted Monte Carlo algorithms, which may only use random bits instead of random numbers. We determine the asymptotics (in some cases sharp up to multiplicative constants, in the other cases sharp up to logarithmic factors) of the corresponding $n$-th minimal error in terms of the decay of the eigenvalues of the covariance operator of $X$. It turns out that, within the margins from above, restricted Monte Carlo algorithms are not inferior to arbitrary Monte Carlo algorithms, and suitable random bit multilevel algorithms are optimal. The analysis of this problem leads to a variant of the quantization problem, namely, the optimal approximation of probability measures on $H$ by uniform distributions supported by a given, finite number of points. We determine the asymptotics (up to multiplicative constants) of the error of the best approximation for the one-dimensional standard normal distribution, for Gaussian measures as above, and for scalar autonomous SDEs.

Michael B. Giles, Mario Hefter, Lukas Mayer et al.

We study the approximation of expectations $\E(f(X))$ for solutions $X$ of SDEs and functionals $f \colon C([0,1],\R^r) \to \R$ by means of restricted Monte Carlo algorithms that may only use random bits instead of random numbers. We consider the worst case setting for functionals $f$ from the Lipschitz class w.r.t.\ the supremum norm. We construct a random bit multilevel Euler algorithm and establish upper bounds for its error and cost. Furthermore, we derive matching lower bounds, up to a logarithmic factor, that are valid for all random bit Monte Carlo algorithms, and we show that, for the given quadrature problem, random bit Monte Carlo algorithms are at least almost as powerful as general randomized algorithms.

Michael B. Giles, Mario Hefter, Lukas Mayer et al.

We study the approximation of expectations $\operatorname{E}(f(X))$ for solutions $X$ of stochastic differential equations and functionals $f$ on the path space by means of Monte Carlo algorithms that only use random bits instead of random numbers. We construct an adaptive random bit multilevel algorithm, which is based on the Euler scheme, the Lévy-Ciesielski representation of the Brownian motion, and asymptotically optimal random bit approximations of the standard normal distribution. We numerically compare this algorithm with the adaptive classical multilevel Euler algorithm for a geometric Brownian motion, an Ornstein-Uhlenbeck process, and a Cox-Ingersoll-Ross process.

Mark Steyvers, Heliodoro Tejeda, Aakriti Kumar et al.

As artificial intelligence (AI) systems, particularly large language models (LLMs), become increasingly integrated into decision-making processes, the ability to trust their outputs is crucial. To earn human trust, LLMs must be well calibrated such that they can accurately assess and communicate the likelihood of their predictions being correct. Whereas recent work has focused on LLMs' internal confidence, less is understood about how effectively they convey uncertainty to users. Here we explore the calibration gap, which refers to the difference between human confidence in LLM-generated answers and the models' actual confidence, and the discrimination gap, which reflects how well humans and models can distinguish between correct and incorrect answers. Our experiments with multiple-choice and short-answer questions reveal that users tend to overestimate the accuracy of LLM responses when provided with default explanations. Moreover, longer explanations increased user confidence, even when the extra length did not improve answer accuracy. By adjusting LLM explanations to better reflect the models' internal confidence, both the calibration gap and the discrimination gap narrowed, significantly improving user perception of LLM accuracy. These findings underscore the importance of accurate uncertainty communication and highlight the effect of explanation length in influencing user trust in AI-assisted decision-making environments. Code and Data can be found at https://osf.io/y7pr6/ . Journal publication can be found on Nature Machine Intelligence at https://www.nature.com/articles/s42256-024-00976-7 .