Stochastic partition models divide a multi-dimensional space into a number of rectangular regions, such that the data within each region exhibit certain types of homogeneity. Due to the nature of their partition strategy, existing partition models may create many unnecessary divisions in sparse regions when trying to describe data in dense regions. To avoid this problem we introduce a new parsimonious partition model -- the Rectangular Bounding Process (RBP) -- to efficiently partition multi-dimensional spaces, by employing a bounding strategy to enclose data points within rectangular bounding boxes. Unlike existing approaches, the RBP possesses several attractive theoretical properties that make it a powerful nonparametric partition prior on a hypercube. In particular, the RBP is self-consistent and as such can be directly extended from a finite hypercube to infinite (unbounded) space. We apply the RBP to regression trees and relational models as a flexible partition prior. The experimental results validate the merit of the RBP {in rich yet parsimonious expressiveness} compared to the state-of-the-art methods.

The coefficients are constrained by the prior and end up smaller in the second example. Although the model is not fit here with Bayesian techniques, it has a Bayesian interpretation. The output here does not quite give a distribution over the coefficient (though other packages can), but does give something related: a 95% confidence interval around the coefficient, in addition to its point estimate. By now you may have a taste for Bayesian techniques and what they can do for you, from a few simple examples. Things get more interesting, however, when we see what priors and posteriors can do for a real-world use case. For part 2, please click here.

Illegal wildlife poaching threatens ecosystems and drives endangered species toward extinction. However, efforts for wildlife monitoring and protection in conservation areas are constrained by the limited resources of law enforcement agencies. To aid in wildlife protection, PAWS is an ML pipeline that has been developed as an end-to-end, data-driven approach to combat illegal poaching. PAWS assists park managers by identifying areas at high risk of poaching throughout protected areas based on real-world data and generating optimal patrol routes for deployment in the field. In this paper, we address significant challenges including extreme class imbalance (up to 1:200), bias, and uncertainty in wildlife poaching data to enhance PAWS and apply its methodology to several national parks with diverse characteristics. (i) We use Gaussian processes to quantify predictive uncertainty, which we exploit to increase the robustness of our prescribed patrols. We evaluate our approach on real-world historic poaching data from Murchison Falls and Queen Elizabeth National Parks in Uganda and, for the first time, Srepok Wildlife Sanctuary in Cambodia. (ii) We present the results of large-scale field tests conducted in Murchison Falls and Srepok Wildlife Sanctuary which confirm that the predictive power of PAWS extends promisingly to multiple parks. This paper is part of an effort to expand PAWS to 600 parks around the world through integration with SMART conservation software.

Quantum graphical models (QGMs) extend the classical framework for reasoning about uncertainty by incorporating the quantum mechanical view of probability. Prior work on QGMs has focused on hidden quantum Markov models (HQMMs), which can be formulated using quantum analogues of the sum rule and Bayes rule used in classical graphical models. Despite the focus on developing the QGM framework, there has been little progress in learning these models from data. The existing state-of-the-art approach randomly initializes parameters and iteratively finds unitary transformations that increase the likelihood of the data. While this algorithm demonstrated theoretical strengths of HQMMs over HMMs, it is slow and can only handle a small number of hidden states. In this paper, we tackle the learning problem by solving a constrained optimization problem on the Stiefel manifold using a well-known retraction-based algorithm. We demonstrate that this approach is not only faster and yields better solutions on several datasets, but also scales to larger models that were prohibitively slow to train via the earlier method.

Hamiltonian Monte Carlo is a powerful algorithm for sampling from difficult-to-normalize posterior distributions. However, when the geometry of the posterior is unfavorable, it may take many expensive evaluations of the target distribution and its gradient to converge and mix. We propose neural transport (NeuTra) HMC, a technique for learning to correct this sort of unfavorable geometry using inverse autoregressive flows (IAF), a powerful neural variational inference technique. The IAF is trained to minimize the KL divergence from an isotropic Gaussian to the warped posterior, and then HMC sampling is performed in the warped space. We evaluate NeuTra HMC on a variety of synthetic and real problems, and find that it significantly outperforms vanilla HMC both in time to reach the stationary distribution and asymptotic effective-sample-size rates.

Excellent variational approximations to Gaussian process posteriors have been developed which avoid the $\mathcal{O}\left(N^3\right)$ scaling with dataset size $N$. They reduce the computational cost to $\mathcal{O}\left(NM^2\right)$, with $M\ll N$ being the number of inducing variables, which summarise the process. While the computational cost seems to be linear in $N$, the true complexity of the algorithm depends on how $M$ must increase to ensure a certain quality of approximation. We address this by characterising the behavior of an upper bound on the KL divergence to the posterior. We show that with high probability the KL divergence can be made arbitrarily small by growing $M$ more slowly than $N$. A particular case of interest is that for regression with normally distributed inputs in D-dimensions with the popular Squared Exponential kernel, $M=\mathcal{O}(\log^D N)$ is sufficient. Our results show that as datasets grow, Gaussian process posteriors can truly be approximated cheaply, and provide a concrete rule for how to increase $M$ in continual learning scenarios.

The Era of Big Data has forced researchers to explore new distributed solutions for building fuzzy classifiers, which often introduce approximation errors or make strong assumptions to reduce computational and memory requirements. As a result, Big Data classifiers might be expected to be inferior to those designed for standard classification tasks (Small Data) in terms of accuracy and model complexity. To our knowledge, however, there is no empirical evidence to confirm such a conjecture yet. Here, we investigate the extent to which state-of-the-art fuzzy classifiers for Big Data sacrifice performance in favor of scalability. To this end, we carry out an empirical study that compares these classifiers with some of the best performing algorithms for Small Data. Assuming the latter were generally designed for maximizing performance without considering scalability issues, the results of this study provide some intuition around the tradeoff between performance and scalability achieved by current Big Data solutions. Our findings show that, although slightly inferior, Big Data classifiers are gradually catching up with state-of-the-art classifiers for Small data, suggesting that a unified learning algorithm for Big and Small Data might be possible.

This paper develops a path planner that minimizes risk (e.g. motion execution) while maximizing accumulated reward (e.g., quality of sensor viewpoint) motivated by visual assistance or tracking scenarios in unstructured or confined environments. In these scenarios, the robot should maintain the best viewpoint as it moves to the goal. However, in unstructured or confined environments, some paths may increase the risk of collision; therefore there is a tradeoff between risk and reward. Conventional state-dependent risk or probabilistic uncertainty modeling do not consider path-level risk or is difficult to acquire. This risk-reward planner explicitly represents risk as a function of motion plans, i.e., paths. Without manual assignment of the negative impact to the planner caused by risk, this planner takes in a pre-established viewpoint quality map and plans target location and path leading to it simultaneously, in order to maximize overall reward along the entire path while minimizing risk. Exact and approximate algorithms are presented, whose solution is further demonstrated on a physical tethered aerial vehicle. Other than the visual assistance problem, the proposed framework also provides a new planning paradigm to address minimum-risk planning under dynamical risk and absence of substructure optimality and to balance the trade-off between reward and risk.

Three-way decision theory, which trisects the universe with less risks or costs, is considered as a powerful mathematical tool for handling uncertainty in incomplete and imprecise information tables, and provides an effective tool for conflict analysis decision making in real-time situations. In this paper, we propose the concepts of the agreement, disagreement and neutral subsets of a strategy with two evaluation functions, which establish the three-way decisions-based conflict analysis models(TWDCAMs) for trisecting the universe of agents, and employ a pair of two-way decisions models to interpret the mechanism of the three-way decision rules for an agent. Subsequently, we develop the concepts of the agreement, disagreement and neutral strategies of an agent group with two evaluation functions, which build the TWDCAMs for trisecting the universe of issues, and take a couple of two-way decisions models to explain the mechanism of the three-way decision rules for an issue. Finally, we reconstruct Fan, Qi and Wei's conflict analysis models(FQWCAMs) and Sun, Ma and Zhao's conflict analysis models(SMZCAMs) with two evaluation functions, and interpret FQWCAMs and SMZCAMs with a pair of two-day decisions models, which illustrates that FQWCAMs and SMZCAMs are special cases of TWDCAMs.

We propose an input design method for a general class of parametric probabilistic models, including nonlinear dynamical systems with process noise. The goal of the procedure is to select inputs such that the parameter posterior distribution concentrates about the true value of the parameters; however, exact computation of the posterior is intractable. By representing (samples from) the posterior as trajectories from a certain Hamiltonian system, we transform the input design task into an optimal control problem. The method is illustrated via numerical examples, including MRI pulse sequence design.