If you've ever searched for evaluation metrics to assess model accuracy, chances are that you found many different options to choose from. Accuracy is in some sense the holy grail of prediction so it's not at all surprising that the machine learning community spends a lot time thinking about it. In a world where more and more high-stake decisions are being automated, model accuracy is in fact a very valid concern. But does this recipe for model evaluation seem like a sound and complete approach to automated decision-making? Some would argue that we need to pay more attention to model uncertainty. No matter how many times you have cross-validated your model, the loss metric that it is being optimized against as well as its parameters and predictions remain inherently random variables.

Deep Neural Networks (DNNs) have performed admirably in classification tasks. However, the characterization of their classification uncertainties, required for certain applications, has been lacking. In this work, we investigate the issue by assessing DNNs' ability to estimate conditional probabilities and propose a framework for systematic uncertainty characterization. Denoting the input sample as x and the category as y, the classification task of assigning a category y to a given input x can be reduced to the task of estimating the conditional probabilities p(y|x), as approximated by the DNN at its last layer using the softmax function. Since softmax yields a vector whose elements all fall in the interval (0, 1) and sum to 1, it suggests a probabilistic interpretation to the DNN's outcome. Using synthetic and real-world datasets, we look into the impact of various factors, e.g., probability density f(x) and inter-categorical sparsity, on the precision of DNNs' estimations of p(y|x), and find that the likelihood probability density and the inter-categorical sparsity have greater impacts than the prior probability to DNNs' classification uncertainty.

After the user provides a few examples of desired outcomes, MURAL automatically infers a reward function that takes into account these examples and the agent's uncertainty for each state. Although reinforcement learning has shown success in domains such as robotics, chip placement and playing video games, it is usually intractable in its most general form. In particular, deciding when and how to visit new states in the hopes of learning more about the environment can be challenging, especially when the reward signal is uninformative. These questions of reward specification and exploration are closely connected -- the more directed and "well shaped" a reward function is, the easier the problem of exploration becomes. The answer to the question of how to explore most effectively is likely to be closely informed by the particular choice of how we specify rewards.

We consider the problem of inferring the conditional independence graph (CIG) of a sparse, high-dimensional stationary multivariate Gaussian time series. A sparse-group lasso-based frequency-domain formulation of the problem based on frequency-domain sufficient statistic for the observed time series is presented. We investigate an alternating direction method of multipliers (ADMM) approach for optimization of the sparse-group lasso penalized log-likelihood. We provide sufficient conditions for convergence in the Frobenius norm of the inverse PSD estimators to the true value, jointly across all frequencies, where the number of frequencies are allowed to increase with sample size. This results also yields a rate of convergence. We also empirically investigate selection of the tuning parameters based on Bayesian information criterion, and illustrate our approach using numerical examples utilizing both synthetic and real data.

We consider the problem of simulating diffusion bridges, i.e. diffusion processes that are conditioned to initialize and terminate at two given states. Diffusion bridge simulation has applications in diverse scientific fields and plays a crucial role for statistical inference of discretely-observed diffusions. This is known to be a challenging problem that has received much attention in the last two decades. In this work, we first show that the time-reversed diffusion bridge process can be simulated if one can time-reverse the unconditioned diffusion process. We introduce a variational formulation to learn this time-reversal that relies on a score matching method to circumvent intractability. We then consider another iteration of our proposed methodology to approximate the Doob's $h$-transform defining the diffusion bridge process. As our approach is generally applicable under mild assumptions on the underlying diffusion process, it can easily be used to improve the proposal bridge process within existing methods and frameworks. We discuss algorithmic considerations and extensions, and present some numerical results.

This formula was devised and penned by respected Thomas Bayes, renowned statistician.It is an arithmetical formula for determining conditional probability. Conditional probability is the likelihood of an outcome occurring, based on a previous outcome occurring. This might be a bit brain-teasing as you are working backwards. Bayes' theorem may be derived from the definition of conditional probability, P(Do not launch Stock price increases) 0.4 0.30 0.12 Thus, there is a 57% probability that the company's share price will increase. Bayes' Theorem has several forms.

Hamiltonian Monte Carlo (HMC) methods are widely used to draw samples from unnormalized target densities due to high efficiency and favorable scalability with respect to increasing space dimensions. However, HMC struggles when the target distribution is multimodal, because the maximum increase in the potential energy function (i.e., the negative log density function) along the simulated path is bounded by the initial kinetic energy, which follows a half of the $\chi_d^2$ distribution, where d is the space dimension. In this paper, we develop a Hamiltonian Monte Carlo method where the constructed paths can travel across high potential energy barriers. This method does not require the modes of the target distribution to be known in advance. Our approach enables frequent jumps between the isolated modes of the target density by continuously varying the mass of the simulated particle while the Hamiltonian path is constructed. Thus, this method can be considered as a combination of HMC and the tempered transitions method. Compared to other tempering methods, our method has a distinctive advantage in the Gibbs sampler settings, where the target distribution changes at each step. We develop a practical tuning strategy for our method and demonstrate that it can construct globally mixing Markov chains targeting high-dimensional, multimodal distributions, using mixtures of normals and a sensor network localization problem.

Meta-, multi-task, and federated learning can be all viewed as solving similar tasks, drawn from an unknown distribution that reflects task similarities. In this work, we provide a unified view of all these problems, as learning to act in a hierarchical Bayesian bandit. We analyze a natural hierarchical Thompson sampling algorithm (hierTS) that can be applied to any problem in this class. Our regret bounds hold under many instances of such problems, including when the tasks are solved sequentially or in parallel; and capture the structure of the problems, such that the regret decreases with the width of the task prior. Our proofs rely on novel total variance decompositions, which can be applied to other graphical model structures. Finally, our theory is complemented by experiments, which show that the hierarchical structure helps with knowledge sharing among the tasks. This confirms that hierarchical Bayesian bandits are a universal and statistically-efficient tool for learning to act with similar bandit tasks.

Multimodal regression is a fundamental task, which integrates the information from different sources to improve the performance of follow-up applications. However, existing methods mainly focus on improving the performance and often ignore the confidence of prediction for diverse situations. In this study, we are devoted to trustworthy multimodal regression which is critical in cost-sensitive domains. To this end, we introduce a novel Mixture of Normal-Inverse Gamma distributions (MoNIG) algorithm, which efficiently estimates uncertainty in principle for adaptive integration of different modalities and produces a trustworthy regression result. Our model can be dynamically aware of uncertainty for each modality, and also robust for corrupted modalities. Furthermore, the proposed MoNIG ensures explicitly representation of (modality-specific/global) epistemic and aleatoric uncertainties, respectively. Experimental results on both synthetic and different real-world data demonstrate the effectiveness and trustworthiness of our method on various multimodal regression tasks (e.g., temperature prediction for superconductivity, relative location prediction for CT slices, and multimodal sentiment analysis).

The two most commonly used criteria for assessing causal model discovery with artificial data are edit-distance and Kullback-Leibler divergence, measured from the true model to the learned model. Both of these metrics maximally reward the true model. However, we argue that they are both insufficiently discriminating in judging the relative merits of false models. Edit distance, for example, fails to distinguish between strong and weak probabilistic dependencies. KL divergence, on the other hand, rewards equally all statistically equivalent models, regardless of their different causal claims. We propose an augmented KL divergence, which we call Causal KL (CKL), which takes into account causal relationships which distinguish between observationally equivalent models. Results are presented for three variants of CKL, showing that Causal KL works well in practice.