Quantifying uncertainties on excursion sets under a Gaussian random field prior

We focus on the problem of estimating and quantifying uncertainties on the excursion set of a function under a limited evaluation budget. We adopt a Bayesian approach where the objective function is assumed to be a realization of a Gaussian random field. In this setting, the posterior distribution on the objective function gives rise to a posterior distribution on excursion sets. Several approaches exist to summarize the distribution of such sets based on random closed set theory. While the recently proposed Vorob'ev approach exploits analytical formulae, further notions of variability require Monte Carlo estimators relying on Gaussian random field conditional simulations. In the present work we propose a method to choose Monte Carlo simulation points and obtain quasi-realizations of the conditional field at fine designs through affine predictors. The points are chosen optimally in the sense that they minimize the posterior expected distance in measure between the excursion set and its reconstruction. The proposed method reduces the computational costs due to Monte Carlo simulations and enables the computation of quasi-realizations on fine designs in large dimensions. We apply this reconstruction approach to obtain realizations of an excursion set on a fine grid which allow us to give a new measure of uncertainty based on the distance transform of the excursion set. Finally we present a safety engineering test case where the simulation method is employed to compute a Monte Carlo estimate of a contour line

Profile extrema for visualizing and quantifying uncertainties on excursion regions. Application to coastal flooding

We consider the problem of describing excursion sets of a real-valued function $f$, i.e. the set of inputs where $f$ is above a fixed threshold. Such regions are hard to visualize if the input space dimension, $d$, is higher than 2. For a given projection matrix from the input space to a lower dimensional (usually $1,2$) subspace, we introduce profile sup (inf) functions that associate to each point in the projection's image the sup (inf) of the function constrained over the pre-image of this point by the considered projection. Plots of profile extrema functions convey a simple, although intrinsically partial, visualization of the set. We consider expensive to evaluate functions where only a very limited number of evaluations, $n$, is available, e.g. $n<100d$, and we surrogate $f$ with a posterior quantity of a Gaussian process (GP) model. We first compute profile extrema functions for the posterior mean given $n$ evaluations of $f$. We quantify the uncertainty on such estimates by studying the distribution of GP profile extrema with posterior quasi-realizations obtained from an approximating process. We control such approximation with a bound inherited from the Borell-TIS inequality. The technique is applied to analytical functions ($d=2,3$) and to a $5$-dimensional coastal flooding test case for a site located on the Atlantic French coast. Here $f$ is a numerical model returning the area of flooded surface in the coastal region given some offshore conditions. Profile extrema functions allowed us to better understand which offshore conditions impact large flooding events

Reducing probes for quality of transmission estimation in optical networks with active learning

Estimating the quality of transmission (QoT) of a lightpath before its establishment is a critical procedure for efficient design and management of optical networks. Recently, supervised machine learning (ML) techniques for QoT estimation have been proposed as an effective alternative to well-established, yet approximated, analytic models that often require the introduction of conservative margins to compensate for model inaccuracies and uncertainties. Unfortunately, to ensure high estimation accuracy, the training set (i.e., the set of historical field data, or "samples," required to train these supervised ML algorithms) must be very large, while in real network deployments, the number of monitored/monitorable lightpaths is limited by several practical considerations. This is especially true for lightpaths with an above-threshold bit error rate (BER) (i.e., malfunctioning or wrongly dimensioned lightpaths), which are infrequently observed during network operation. Samples with above-threshold BERs can be acquired by deploying probe lightpaths, but at the cost of increased operational expenditures and wastage of spectral resources. In this paper, we propose to use active learning to reduce the number of probes needed for ML-based QoT estimation. We build an estimation model based on Gaussian processes, which allows iterative identification of those QoT instances that minimize estimation uncertainty. Numerical results using synthetically generated datasets show that, by using the proposed active learning approach, we can achieve the same performance of standard offline supervised ML methods, but with a remarkable reduction (at least 5% and up to 75%) in the number of training samples

Efficient probabilistic reconciliation of forecasts for real-valued and count time series

Hierarchical time series are common in several applied fields. Forecasts are required to be coherent, that is, to satisfy the constraints given by the hierarchy. The most popular technique to enforce coherence is called reconciliation, which adjusts the base forecasts computed for each time series. However, recent works on probabilistic reconciliation present several limitations. In this paper, we propose a new approach based on conditioning to reconcile any type of forecast distribution. We then introduce a new algorithm, called Bottom-Up Importance Sampling, to efficiently sample from the reconciled distribution. It can be used for any base forecast distribution: discrete, continuous, or in the form of samples, providing a major speedup compared to the current methods. Experiments on several temporal hierarchies show a significant improvement over base probabilistic forecasts.Comment: 25 pages, 4 figure

Probabilistic Reconciliation of Count Time Series

We propose a principled method for the reconciliation of any probabilistic base forecasts. We show how probabilistic reconciliation can be obtained by merging, via Bayes' rule, the information contained in the base forecast for the bottom and the upper time series. We illustrate our method on a toy hierarchy, showing how our framework allows the probabilistic reconciliation of any base forecast. We perform experiment in the reconciliation of temporal hierarchies of count time series, obtaining major improvements compared to probabilistic reconciliation based on the Gaussian or the truncated Gaussian distribution

Bounding Counterfactuals under Selection Bias

Causal analysis may be affected by selection bias, which is defined as the systematic exclusion of data from a certain subpopulation. Previous work in this area focused on the derivation of identifiability conditions. We propose instead a first algorithm to address both identifiable and unidentifiable queries. We prove that, in spite of the missingness induced by the selection bias, the likelihood of the available data is unimodal. This enables us to use the causal expectation-maximisation scheme to obtain the values of causal queries in the identifiable case, and to compute bounds otherwise. Experiments demonstrate the approach to be practically viable. Theoretical convergence characterisations are provided.Comment: Eleventh International Conference on Probabilistic Graphical Models (PGM 2022

Estimating orthant probabilities of high dimensional Gaussian vectors with an application to set estimation

The computation of Gaussian orthant probabilities has been extensively studied for low-dimensional vectors. Here, we focus on the high-dimensional case and we present a two-step procedure relying on both deterministic and stochastic techniques. The proposed estimator relies indeed on splitting the probability into a low-dimensional term and a remainder. While the low-dimensional probability can be estimated by fast and accurate quadrature, the remainder requires Monte Carlo sampling. We further refine the estimation by using a novel asymmetric nested Monte Carlo (anMC) algorithm for the remainder and we highlight cases where this approximation brings substantial efficiency gains. The proposed methods are compared against state-of-the-art techniques in a numerical study, which also calls attention to the advantages and drawbacks of the procedure. Finally, the proposed method is applied to derive conservative estimates of excursion sets of expensive to evaluate deterministic functions under a Gaussian random field prior, without requiring a Markov assumption. Supplementary material for this article is available online