This paper proposes an active learning-based Gaussian process (AL-GP) metamodelling method to estimate the cumulative as well as complementary cumulative distribution function (CDF/CCDF) for forward uncertainty quantification (UQ) problems. Within the field of UQ, previous studies focused on developing AL-GP approaches for reliability (rare event probability) analysis of expensive black-box solvers. A naive iteration of these algorithms with respect to different CDF/CCDF threshold values would yield a discretized CDF/CCDF. However, this approach inevitably leads to a trade-off between accuracy and computational efficiency since both depend (in opposite way) on the selected discretization. In this study, a specialized error measure and a learning function are developed such that the resulting AL-GP method is able to efficiently estimate the CDF/CCDF for a specified range of interest without an explicit dependency on discretization. Particularly, the proposed AL-GP method is able to simultaneously provide accurate CDF and CCDF estimation in their median-low probability regions. Three numerical examples are introduced to test and verify the proposed method.
Scientific analyses often rely on slow, but accurate forward models for observable data conditioned on known model parameters. While various emulation schemes exist to approximate these slow calculations, these approaches are only safe if the approximations are well understood and controlled. This workshop submission reviews and updates a previously published method, which has been used in cosmological simulations, to (1) train an emulator while simultaneously estimating posterior probabilities with MCMC and (2) explicitly propagate the emulation error into errors on the posterior probabilitie…
KTH Royal Institute of Technology（瑞典皇家理工学院）；INRIA Grenoble Rhone-Alpes（法国国家信息与自动化研究所格勒诺布尔罗纳-阿尔卑斯分部）；University of Tokyo（东京大学）；Cold Spring Harbor Laboratory（冷泉港实验室）
We design a self size-estimating feed-forward network (SSFN) using a joint optimization approach for estimation of number of layers, number of nodes and learning of weight matrices. The learning algorithm has a low computational complexity, preferably within few minutes using a laptop. In addition the algorithm has a limited need for human intervention to tune parameters. SSFN grows from a small-size network to a large-size network, guaranteeing a monotonically non-increasing cost with addition of nodes and layers. The learning approach uses judicious a combination of `lossless flow property' …
Understanding the number of individuals who have been infected with the novel coronavirus SARS-CoV-2, and the extent to which social distancing policies have been effective at limiting its spread, are critical for effective policy going forward. Here we present estimates of the extent to which confirmed cases in the United States undercount the true number of infections, and analyze how effective social distancing measures have been at mitigating or suppressing the virus. Our analysis uses a Bayesian model of COVID-19 fatalities with a likelihood based on an underlying differential equation mo…
The Alan Turing Institute（艾伦·图灵研究所）；University of Warwick（华威大学）
This paper demonstrates the application of Bayesian Artificial Neural Networks to Ordinary Differential Equation (ODE) inverse problems. We consider the case of estimating an unknown chaotic dynamical system transition model from state observation data. Inverse problems for chaotic systems are numerically challenging as small perturbations in model parameters can cause very large changes in estimated forward trajectories. Bayesian Artificial Neural Networks can be used to simultaneously fit a model and estimate model parameter uncertainty. Knowledge of model parameter uncertainty can then be i…
Bayesian cubature (BC) is a popular inferential perspective on the cubature of expensive integrands, wherein the integrand is emulated using a stochastic process model. Several approaches have been put forward to encode sequential adaptation (i.e. dependence on previous integrand evaluations) into this framework. However, these proposals have been limited to either estimating the parameters of a stationary covariance model or focusing computational resources on regions where large values are taken by the integrand. In contrast, many classical adaptive cubature methods focus computational resou…
We present algorithms for estimating the forward reachable set of a dynamical system using only a finite collection of independent and identically distributed samples. The produced estimate is the sublevel set of a function called an empirical inverse Christoffel function: empirical inverse Christoffel functions are known to provide good approximations to the support of probability distributions. In addition to reachability analysis, the same approach can be applied to general problems of estimating the support of a random variable, which has applications in data science towards detection of n…
Estimating a Gibbs density function given a sample is an important problem in computational statistics and statistical learning. Although the well established maximum likelihood method is commonly used, it requires the computation of the partition function (i.e., the normalization of the density). This function can be easily calculated for simple low-dimensional problems but its computation is difficult or even intractable for general densities and high-dimensional problems. In this paper we propose an alternative approach based on Maximum A-Posteriori (MAP) estimators, we name Maximum Recover…
University of Cambridge（剑桥大学）；Microsoft Research（微软研究院）；The Alan Turing Institute（艾伦·图灵研究所）
Existing methods for estimating uncertainty in deep learning tend to require multiple forward passes, making them unsuitable for applications where computational resources are limited. To solve this, we perform probabilistic reasoning over the depth of neural networks. Different depths correspond to subnetworks which share weights and whose predictions are combined via marginalisation, yielding model uncertainty. By exploiting the sequential structure of feed-forward networks, we are able to both evaluate our training objective and make predictions with a single forward pass. We validate our a…
Uber AI Labs（Uber AI 实验室）；Georgia Institute of Technology（佐治亚理工学院）；ML Collective
In this paper, we introduce a novel form of value function, $Q(s, s')$, that expresses the utility of transitioning from a state $s$ to a neighboring state $s'$ and then acting optimally thereafter. In order to derive an optimal policy, we develop a forward dynamics model that learns to make next-state predictions that maximize this value. This formulation decouples actions from values while still learning off-policy. We highlight the benefits of this approach in terms of value function transfer, learning within redundant action spaces, and learning off-policy from state observations generated…
Inverse problems involve making inference about unknown parameters of a physical process using observational data. This paper investigates an important class of inverse problems -- the estimation of the initial condition of a spatio-temporal advection-diffusion process using spatially sparse data streams. Three spatial sampling schemes are considered, including irregular, non-uniform and shifted uniform sampling. The irregular sampling scheme is the general scenario, while computationally efficient solutions are available in the spectral domain for non-uniform and shifted uniform sampling. For…
Manipal Institute of Technology（马德拉斯理工学院）；German Research Center for Artificial Intelligence（德国人工智能研究中心）；Department of Computer Science & Engineering（计算机科学与工程系）；Department of Information Technology（信息技术系）
Estimating epistemic uncertainty of models used in low-latency applications and Out-Of-Distribution samples detection is a challenge due to the computationally demanding nature of uncertainty estimation techniques. Estimating model uncertainty using approximation techniques like Monte Carlo Dropout (MCD), DropConnect (MCDC) requires a large number of forward passes through the network, rendering them inapt for low-latency applications. We propose Select-DC which uses a subset of layers in a neural network to model epistemic uncertainty with MCDC. Through our experiments, we show a significant …
The ILIUM forward modelling algorithm for multivariate parameter estimation and its application to derive stellar parameters from Gaia spectrophotometry
