arXiv subjects

Juozas Vaicenavicius

Publications and source records attributed to Juozas Vaicenavicius.

8 recordsLinked to original sources

arXiv2021

Self-driving car safety quantification via component-level analysis

In this paper, we present a rigorous modular statistical approach for arguing safety or its insufficiency of an autonomous vehicle through a concrete illustrative example. The methodology relies on making appropriate quantitative studies of the performance of constituent components. We explain the importance of sufficient and necessary conditions at the component level for the overall safety of the vehicle as well as the cost-saving benefits of the approach. A simple concrete automated braking example studied illustrates how separate perception system and operational design domain statistical analyses can be used to prove or disprove safety at the vehicle level.

stat.AP↗

arXiv2019

Bayesian sequential least-squares estimation for the drift of a Wiener process

Given a Wiener process with unknown and unobservable drift, we try to estimate this drift as effectively but also as quickly as possible, in the presence of a quadratic penalty for the estimation error and of a fixed, positive cost per unit of observation time. In a Bayesian framework, where the unobservable drift is assumed to have a known "prior" distribution, this question reduces to choosing judiciously a stopping time for an appropriate diffusion process in natural scale. We establish structural properties of the solution for the corresponding problem of optimal stopping. In particular, we show that, regardless of the prior distribution, the continuation region is monotonically shrinking in time; and provide conditions on the prior distribution guaranteeing a one-sided stopping region. Finally, we illustrate the theoretical results through a detailed study of some concrete prior distributions.

math.ST↗

arXiv2019

Evaluating model calibration in classification

Probabilistic classifiers output a probability distribution on target classes rather than just a class prediction. Besides providing a clear separation of prediction and decision making, the main advantage of probabilistic models is their ability to represent uncertainty about predictions. In safety-critical applications, it is pivotal for a model to possess an adequate sense of uncertainty, which for probabilistic classifiers translates into outputting probability distributions that are consistent with the empirical frequencies observed from realized outcomes. A classifier with such a property is called calibrated. In this work, we develop a general theoretical calibration evaluation framework grounded in probability theory, and point out subtleties present in model calibration evaluation that lead to refined interpretations of existing evaluation techniques. Lastly, we propose new ways to quantify and visualize miscalibration in probabilistic classification, including novel multidimensional reliability diagrams.

cs.LG↗

arXiv2019

Optimal stopping of a Brownian bridge with an unknown pinning point

The problem of stopping a Brownian bridge with an unknown pinning point to maximise the expected value at the stopping time is studied. A few general properties, such as continuity and various bounds of the value function, are established. However, structural properties of the optimal stopping region are shown to crucially depend on the prior, and we provide a general condition for a one-sided stopping region. Moreover, a detailed analysis is conducted in the cases of the two-point and the mixed Gaussian priors, revealing a rich structure present in the problem.

math.PR↗

arXiv2019

Monotonicity and robustness in Wiener disorder detection

We study the problem of detecting a drift change of a Brownian motion under various extensions of the classical case. Specifically, we consider the case of a random post-change drift and examine monotonicity properties of the solution with respect to different model parameters. Moreover, robustness properties -- effects of misspecification of the underlying model -- are explored.

math.ST↗

arXiv2019

Asset liquidation under drift uncertainty and regime-switching volatility

Optimal liquidation of an asset with unknown constant drift and stochastic regime-switching volatility is studied. The uncertainty about the drift is represented by an arbitrary probability distribution; the stochastic volatility is modelled by $m$-state Markov chain. Using filtering theory, an equivalent reformulation of the original problem as a four-dimensional optimal stopping problem is found and then analysed by constructing approximating sequences of three-dimensional optimal stopping problems. An optimal liquidation strategy and various structural properties of the problem are determined. Analysis of the two-point prior case is presented in detail, building on which, an outline of the extension to the general prior case is given.

q-fin.MF↗

arXiv2015

Bayesian sequential testing of the drift of a Brownian motion

We study a classical Bayesian statistics problem of sequentially testing the sign of the drift of an arithmetic Brownian motion with the $0$-$1$ loss function and a constant cost of observation per unit of time for general prior distributions. The statistical problem is reformulated as an optimal stopping problem with the current conditional probability that the drift is non-negative as the underlying process. The volatility of this conditional probability process is shown to be non-increasing in time, which enables us to prove monotonicity and continuity of the optimal stopping boundaries as well as to characterize them completely in the finite-horizon case as the unique continuous solution to a pair of integral equations. In the infinite-horizon case, the boundaries are shown to solve another pair of integral equations and a convergent approximation scheme for the boundaries is provided. Also, we describe the dependence between the prior distribution and the long-term asymptotic behaviour of the boundaries.

math.PR↗

arXiv2015

Optimal liquidation of an asset under drift uncertainty

We study a problem of finding an optimal stopping strategy to liquidate an asset with unknown drift. Taking a Bayesian approach, we model the initial beliefs of an individual about the drift parameter by allowing an arbitrary probability distribution to characterise the uncertainty about the drift parameter. Filtering theory is used to describe the evolution of the posterior beliefs about the drift once the price process is being observed. An optimal stopping time is determined as the first passage time of the posterior mean below a monotone boundary, which can be characterised as the unique solution to a non-linear integral equation. We also study monotonicity properties with respect to the prior distribution and the asset volatility.

q-fin.MF↗