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 consider the problem of designing minimax estimators for estimating the parameters of a probability distribution. Unlike classical approaches such as the MLE and minimum distance estimators, we consider an algorithmic approach for constructing such estimators. We view the problem of designing minimax estimators as finding a mixed strategy Nash equilibrium of a zero-sum game. By leveraging recent results in online learning with non-convex losses, we provide a general algorithm for finding a mixed-strategy Nash equilibrium of general non-convex non-concave zero-sum games. Our algorithm requires access to two subroutines: (a) one which outputs a Bayes estimator corresponding to a given prior probability distribution, and (b) one which computes the worst-case risk of any given estimator. Given access to these two subroutines, we show that our algorithm outputs both a minimax estimator and a least favorable prior. To demonstrate the power of this approach, we use it to construct provably minimax estimators for classical problems such as estimation in the finite Gaussian sequence model, and linear regression.

Estimating parameters of mixture model has wide applications Determining the number of components in a finite mixture model ranging from classification problems to estimating of complex distributions. is crucial in many application areas such as financial data [16, 31, 35], Most of the current literature on estimating the parameters biomedical studies [17, 36] and low-frequency accident occurrence of the mixture densities are based on iterative Expectation Maximization prediction [27, 32]. Existing literature have witnessed numerous (EM) type algorithms which require the use of either computational methods, and in particular Markov Chain Monte taking expectations over the latent label variables or generating Carlo methods [7, 14, 33] and EM algorithms [20-22] have been samples from the conditional distribution of such latent labels using used with a lot of success. However, either these methods are computationally the Bayes rule. Moreover, when the number of components is demanding and/or these methods are developed under unknown, the problem becomes computationally more demanding the assumption of data being generated from mixtures of densities due to well-known label switching issues [28]. In this paper, we from the exponential family, in part because the family of exponential propose a robust and quick approach based on change-point methods distribution has a sufficient statistic of constant dimension (i.e., to determine the number of mixture components that works the dimension of the sufficient statistic remains fixed for any sample for almost any location-scale families even when the components size) and so the updates of the data augmentation type algorithm are heavy tailed (e.g., Cauchy). We present several numerical illustrations involve their smaller dimensional sufficient statistics [11, 12, 24].

Outlier detection is an important data mining task with numerous practical applications such as intrusion detection, credit card fraud detection, and video surveillance. However, given a specific complicated task with big data, the process of building a powerful deep learning based system for outlier detection still highly relies on human expertise and laboring trials. Although Neural Architecture Search (NAS) has shown its promise in discovering effective deep architectures in various domains, such as image classification, object detection, and semantic segmentation, contemporary NAS methods are not suitable for outlier detection due to the lack of intrinsic search space, unstable search process, and low sample efficiency. To bridge the gap, in this paper, we propose AutoOD, an automated outlier detection framework, which aims to search for an optimal neural network model within a predefined search space. Specifically, we firstly design a curiosity-guided search strategy to overcome the curse of local optimality. A controller, which acts as a search agent, is encouraged to take actions to maximize the information gain about the controller's internal belief. We further introduce an experience replay mechanism based on self-imitation learning to improve the sample efficiency. Experimental results on various real-world benchmark datasets demonstrate that the deep model identified by AutoOD achieves the best performance, comparing with existing handcrafted models and traditional search methods.

Matrix square roots and their inverses arise frequently in machine learning, e.g., when sampling from high-dimensional Gaussians $\mathcal{N}(\mathbf 0, \mathbf K)$ or whitening a vector $\mathbf b$ against covariance matrix $\mathbf K$. While existing methods typically require $O(N^3)$ computation, we introduce a highly-efficient quadratic-time algorithm for computing $\mathbf K^{1/2} \mathbf b$, $\mathbf K^{-1/2} \mathbf b$, and their derivatives through matrix-vector multiplication (MVMs). Our method combines Krylov subspace methods with a rational approximation and typically achieves $4$ decimal places of accuracy with fewer than $100$ MVMs. Moreover, the backward pass requires little additional computation. We demonstrate our method's applicability on matrices as large as $50,\!000 \times 50,\!000$ - well beyond traditional methods - with little approximation error. Applying this increased scalability to variational Gaussian processes, Bayesian optimization, and Gibbs sampling results in more powerful models with higher accuracy.

Classifiers are often tested on relatively small data sets, which should lead to uncertain performance metrics. Nevertheless, these metrics are usually taken at face value. We present an approach to quantify the uncertainty of classification performance metrics, based on a probability model of the confusion matrix. Application of our approach to classifiers from the scientific literature and a classification competition shows that uncertainties can be surprisingly large and limit performance evaluation. In fact, some published classifiers are likely to be misleading. The application of our approach is simple and requires only the confusion matrix. It is agnostic of the underlying classifier. Our method can also be used for the estimation of sample sizes that achieve a desired precision of a performance metric.

In the Logistic Regression for Machine Learning using Python blog, I have introduced the basic idea of the logistic function. We have discussed the cost function. And in the iterative method, we focus on the Gradient descent optimization method. Now so in this section, we are going to introduce the Maximum Likelihood cost function. And we would like to maximize this cost function.

Dependability assurance of systems embedding machine learning(ML) components---so called learning-enabled systems (LESs)---is a key step for their use in safety-critical applications. In emerging standardization and guidance efforts, there is a growing consensus in the value of using assurance cases for that purpose. This paper develops a quantitative notion of assurance that an LES is dependable, as a core component of its assurance case, also extending our prior work that applied to ML components. Specifically, we characterize LES assurance in the form of assurance measures: a probabilistic quantification of confidence that an LES possesses system-level properties associated with functional capabilities and dependability attributes. We illustrate the utility of assurance measures by application to a real world autonomous aviation system, also describing their role both in i) guiding high-level, runtime risk mitigation decisions and ii) as a core component of the associated dynamic assurance case.

We study probabilistic safety for Bayesian Neural Networks (BNNs) under adversarial input perturbations. Given a compact set of input points, $T \subseteq \mathbb{R}^m$, we study the probability w.r.t. the BNN posterior that all the points in $T$ are mapped to the same region $S$ in the output space. In particular, this can be used to evaluate the probability that a network sampled from the BNN is vulnerable to adversarial attacks. We rely on relaxation techniques from non-convex optimization to develop a method for computing a lower bound on probabilistic safety for BNNs, deriving explicit procedures for the case of interval and linear function propagation techniques. We apply our methods to BNNs trained on a regression task, airborne collision avoidance, and MNIST, empirically showing that our approach allows one to certify probabilistic safety of BNNs with millions of parameters.