A normalizing flow is an invertible mapping between an arbitrary probability distribution and a standard normal distribution; it can be used for density estimation and statistical inference. Computing the flow follows the change of variables formula and thus requires invertibility of the mapping and an efficient way to compute the determinant of its Jacobian. To satisfy these requirements, normalizing flows typically consist of carefully chosen components. Continuous normalizing flows (CNFs) are mappings obtained by solving a neural ordinary differential equation (ODE). The neural ODE's dynamics can be chosen almost arbitrarily while ensuring invertibility. Moreover, the log-determinant of the flow's Jacobian can be obtained by integrating the trace of the dynamics' Jacobian along the flow. Our proposed OT-Flow approach tackles two critical computational challenges that limit a more widespread use of CNFs. First, OT-Flow leverages optimal transport (OT) theory to regularize the CNF and enforce straight trajectories that are easier to integrate. Second, OT-Flow features exact trace computation with time complexity equal to trace estimators used in existing CNFs. On five high-dimensional density estimation and generative modeling tasks, OT-Flow performs competitively to a state-of-the-art CNF while on average requiring one-fourth of the number of weights with 19x speedup in training time and 28x speedup in inference.

Many democratic countries use district-based elections where there is a "seat" for each district in the governing body. In each district, the party whose candidate gets the maximum number of votes wins the corresponding seat. The result of the election is decided based on the number of seats won by the different parties. The electors (voters) can cast their votes only in the district of their residence. Thus, locations of the electors and boundaries of the districts may severely affect the election result even if the proportion of popular support (number of electors) of different parties remains unchanged. This has led to significant amount of research on whether the districts may be redrawn or electors may be moved to maximize seats for a particular party. In this paper, we frame the spatial distribution of electors in a probabilistic setting, and explore different models to capture the intra-district polarization of electors in favour of a party, or the spatial concentration of supporters of different parties. Our models are inspired by elections in India, where supporters of different parties tend to be concentrated in certain districts. We show with extensive simulations that our model can capture different statistical properties of real elections held in India. We frame parameter estimation problems to fit our models to the observed election results. Since analytical calculation of the likelihood functions are infeasible for our complex models, we use Likelihood-free Inference methods under the Approximate Bayesian Computation framework. Since this approach is highly time-consuming, we explore how supervised regression using Logistic Regression or Deep Neural Networks can be used to speed it up. We also explore how the election results can change by varying the spatial distributions of the voters, even when the proportions of popular support of the parties remain constant.

Bayesian meta-learning enables robust and fast adaptation to new tasks with uncertainty assessment. The key idea behind Bayesian meta-learning is empirical Bayes inference of hierarchical model. In this work, we extend this framework to include a variety of existing methods, before proposing our variant based on gradient-EM algorithm. Our method improves computational efficiency by avoiding back-propagation computation in the meta-update step, which is exhausting for deep neural networks. Furthermore, it provides flexibility to the inner-update optimization procedure by decoupling it from meta-update. Experiments on sinusoidal regression, few-shot image classification, and policy-based reinforcement learning show that our method not only achieves better accuracy with less computation cost, but is also more robust to uncertainty.

The model uncertainty obtained by variational Bayesian inference with Monte Carlo dropout is prone to miscalibration. In this paper, different logit scaling methods are extended to dropout variational inference to recalibrate model uncertainty. Expected uncertainty calibration error (UCE) is presented as a metric to measure miscalibration. The effectiveness of recalibration is evaluated on CIFAR-10/100 and SVHN for recent CNN architectures. Experimental results show that logit scaling considerably reduce miscalibration by means of UCE. Well-calibrated uncertainty enables reliable rejection of uncertain predictions and robust detection of out-of-distribution data.

We consider an inverse reinforcement learning problem involving us versus an enemy radar equipped with a Bayesian tracker. By observing the emissions of the enemy radar,how can we identify if the radar is cognitive (constrained utility maximizer)? Given the observed sequence of actions taken by the enemy's radar, we consider three problems: (i) Are the enemy radar's actions (waveform choice, beam scheduling) consistent with constrained utility maximization? If so how can we estimate the cognitive radar's utility function that is consistent with its actions. We formulate and solve the problem in terms of the spectra (eigenvalues) of the state and observation noise covariance matrices, and the algebraic Riccati equation. (ii) How to construct a statistical test for detecting a cognitive radar (constrained utility maximization) when we observe the radar's actions in noise or the radar observes our probe signal in noise? We propose a statistical detector with a tight Type-II error bound. (iii) How can we optimally probe (interrogate) the enemy's radar by choosing our state to minimize the Type-II error of detecting if the radar is deploying an economic rational strategy, subject to a constraint on the Type-I detection error? We present a stochastic optimization algorithm to optimize our probe signal. The main analysis framework used in this paper is that of revealed preferences from microeconomics.

An interesting development in automatic visual recognition has been the emergence of tasks where it is not possible to assign ground truth labels to images, yet still feasible to collect annotations that reflect human judgements about them. Such tasks include subjective visual attribute assignment and the labeling of ambiguous scenes. Machine learning-based predictors for these tasks rely on supervised training that models the behavior of the annotators, e.g., what would the average person's judgement be for an image? A key open question for this type of work, especially for applications where inconsistency with human behavior can lead to ethical lapses, is how to evaluate the uncertainty of trained predictors. Given that the real answer is unknowable, we are left with often noisy judgements from human annotators to work with. In order to account for the uncertainty that is present, we propose a relative Bayesian framework for evaluating predictors trained on such data. The framework specifies how to estimate a predictor's uncertainty due to the human labels by approximating a conditional distribution and producing a credible interval for the predictions and their measures of performance. The framework is successfully applied to four image classification tasks that use subjective human judgements: facial beauty assessment using the SCUT-FBP5500 dataset, social attribute assignment using data from TestMyBrain.org,

We present a sparse representation of model uncertainty for Deep Neural Networks (DNNs) where the parameter posterior is approximated with an inverse formulation of the Multivariate Normal Distribution (MND), also known as the information form. The key insight of our work is that the information matrix, i.e. the inverse of the covariance matrix tends to be sparse in its spectrum. Therefore, dimensionality reduction techniques such as low rank approximations (LRA) can be effectively exploited. To achieve this, we develop a novel sparsification algorithm and derive a cost-effective analytical sampler. As a result, we show that the information form can be scalably applied to represent model uncertainty in DNNs. Our exhaustive theoretical analysis and empirical evaluations on various benchmarks show the competitiveness of our approach over the current methods.

While feedback loops are known to play important roles in many complex systems, their existence is ignored in a large part of the causal discovery literature, as systems are typically assumed to be acyclic from the outset. When applying causal discovery algorithms designed for the acyclic setting on data generated by a system that involves feedback, one would not expect to obtain correct results. In this work, we show that---surprisingly---the output of the Fast Causal Inference (FCI) algorithm is correct if it is applied to observational data generated by a system that involves feedback. More specifically, we prove that for observational data generated by a simple and $\sigma$-faithful Structural Causal Model (SCM), FCI is sound and complete, and can be used to consistently estimate (i) the presence and absence of causal relations, (ii) the presence and absence of direct causal relations, (iii) the absence of confounders, and (iv) the absence of specific cycles in the causal graph of the SCM. We extend these results to constraint-based causal discovery algorithms that exploit certain forms of background knowledge, including the causally sufficient setting (e.g., the PC algorithm) and the Joint Causal Inference setting (e.g., the FCI-JCI algorithm).

We present high quality image synthesis results using diffusion probabilistic models, a class of latent variable models inspired by considerations from nonequilibrium thermodynamics. Our best results are obtained by training on a weighted variational bound designed according to a novel connection between diffusion probabilistic models and denoising score matching with Langevin dynamics, and our models naturally admit a progressive lossy decompression scheme that can be interpreted as a generalization of autoregressive decoding. On the unconditional CIFAR10 dataset, we obtain an Inception score of 9.46 and a state-of-the-art FID score of 3.17. On 256x256 LSUN, we obtain sample quality similar to ProgressiveGAN.