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We consider parametric estimation and tests for multi-dimensional diffusion processes with a small dispersion parameter $\varepsilon$ from discrete observations. For parametric estimation of diffusion processes, the main target is to estimate the drift parameter and the diffusion parameter. In this paper, we propose two types of adaptive estimators for both parameters and show their asymptotic properties under $\varepsilon\to0$, $n\to\infty$ and the balance condition that $(\varepsilon n^\rho)^{-1} =O(1)$ for some $\rho>0$. Using these adaptive estimators, we also introduce consistent adaptive testing methods and prove that test statistics for adaptive tests have asymptotic distributions under null hypothesis. In simulation studies, we examine and compare asymptotic behaviors of the two kinds of adaptive estimators and test statistics. Moreover, we treat the SIR model which describes a simple epidemic spread for a biological application.

Bayesian approaches are appealing for constrained inference problems by allowing a probabilistic characterization of uncertainty, while providing a computational machinery for incorporating complex constraints in hierarchical models. However, the usual Bayesian strategy of placing a prior on the constrained space and conducting posterior computation with Markov chain Monte Carlo algorithms is often intractable. An alternative is to conduct inference for a less constrained posterior and project samples to the constrained space through a minimal distance mapping. We formalize and provide a unifying framework for such posterior projections. For theoretical tractability, we initially focus on constrained parameter spaces corresponding to closed and convex subsets of the original space. We then consider non-convex Stiefel manifolds. We provide a general formulation of projected posteriors in a Bayesian decision-theoretic framework. We show that asymptotic properties of the unconstrained posterior are transferred to the projected posterior, leading to asymptotically correct credible intervals. We demonstrate numerically that projected posteriors can have better performance that competitor approaches in real data examples.

Probabilistic context-free grammars (PCFGs) and dynamic Bayesian networks (DBNs) are widely used sequence models with complementary strengths and limitations. While PCFGs allow for nested hierarchical dependencies (tree structures), their latent variables (non-terminal symbols) have to be discrete. In contrast, DBNs allow for continuous latent variables, but the dependencies are strictly sequential (chain structure). Therefore, neither can be applied if the latent variables are assumed to be continuous and also to have a nested hierarchical dependency structure. In this paper, we present Recursive Bayesian Networks (RBNs), which generalise and unify PCFGs and DBNs, combining their strengths and containing both as special cases. RBNs define a joint distribution over tree-structured Bayesian networks with discrete or continuous latent variables. The main challenge lies in performing joint inference over the exponential number of possible structures and the continuous variables. We provide two solutions: 1) For arbitrary RBNs, we generalise inside and outside probabilities from PCFGs to the mixed discrete-continuous case, which allows for maximum posterior estimates of the continuous latent variables via gradient descent, while marginalising over network structures. 2) For Gaussian RBNs, we additionally derive an analytic approximation, allowing for robust parameter optimisation and Bayesian inference. The capacity and diverse applications of RBNs are illustrated on two examples: In a quantitative evaluation on synthetic data, we demonstrate and discuss the advantage of RBNs for segmentation and tree induction from noisy sequences, compared to change point detection and hierarchical clustering. In an application to musical data, we approach the unsolved problem of hierarchical music analysis from the raw note level and compare our results to expert annotations.

Probabilistic counters are well known tools often used for space-efficient set cardinality estimation. In this paper we investigate probabilistic counters from the perspective of preserving privacy. We use standard, rigid differential privacy notion. The intuition is that the probabilistic counters do not reveal too much information about individuals, but provide only general information about the population. Thus they can be used safely without violating privacy of individuals. It turned out however that providing a precise, formal analysis of privacy parameters of probabilistic counters is surprisingly difficult and needs advanced techniques and a very careful approach. We demonstrate also that probabilistic counters can be used as a privacy protecion mechanism without any extra randomization. That is, the inherit randomization from the protocol is sufficient for protecting privacy, even if the probabilistic counter is used many times. In particular we present a specific privacy-preserving data aggregation protocol based on a probabilistic counter. Our results can be used for example in performing distributed surveys.

Attention-based neural networks have achieved state-of-the-art results on a wide range of tasks. Most such models use deterministic attention while stochastic attention is less explored due to the optimization difficulties or complicated model design. This paper introduces Bayesian attention belief networks, which construct a decoder network by modeling unnormalized attention weights with a hierarchy of gamma distributions, and an encoder network by stacking Weibull distributions with a deterministic-upward-stochastic-downward structure to approximate the posterior. The resulting auto-encoding networks can be optimized in a differentiable way with a variational lower bound. It is simple to convert any models with deterministic attention, including pretrained ones, to the proposed Bayesian attention belief networks. On a variety of language understanding tasks, we show that our method outperforms deterministic attention and state-of-the-art stochastic attention in accuracy, uncertainty estimation, generalization across domains, and robustness to adversarial attacks. We further demonstrate the general applicability of our method on neural machine translation and visual question answering, showing great potential of incorporating our method into various attention-related tasks.

Path-based relational reasoning over knowledge graphs has become increasingly popular due to a variety of downstream applications such as question answering in dialogue systems, fact prediction, and recommender systems. In recent years, reinforcement learning (RL) has provided solutions that are more interpretable and explainable than other deep learning models. However, these solutions still face several challenges, including large action space for the RL agent and accurate representation of entity neighborhood structure. We address these problems by introducing a type-enhanced RL agent that uses the local neighborhood information for efficient path-based reasoning over knowledge graphs. Our solution uses graph neural network (GNN) for encoding the neighborhood information and utilizes entity types to prune the action space. Experiments on real-world dataset show that our method outperforms state-of-the-art RL methods and discovers more novel paths during the training procedure.

Knowledge graph reasoning, which aims at predicting the missing facts through reasoning with the observed facts, is critical to many applications. Such a problem has been widely explored by traditional logic rule-based approaches and recent knowledge graph embedding methods. A principled logic rule-based approach is the Markov Logic Network (MLN), which is able to leverage domain knowledge with first-order logic and meanwhile handle their uncertainty. However, the inference of MLNs is usually very difficult due to the complicated graph structures. Different from MLNs, knowledge graph embedding methods (e.g. TransE, DistMult) learn effective entity and relation embeddings for reasoning, which are much more effective and efficient. However, they are unable to leverage domain knowledge. In this paper, we propose the probabilistic Logic Neural Network (pLogicNet), which combines the advantages of both methods. A pLogicNet defines the joint distribution of all possible triplets by using a Markov logic network with first-order logic, which can be efficiently optimized with the variational EM algorithm. In the E-step, a knowledge graph embedding model is used for inferring the missing triplets, while in the M-step, the weights of logic rules are updated based on both the observed and predicted triplets. Experiments on multiple knowledge graphs prove the effectiveness of pLogicNet over many competitive baselines.

Both generative adversarial network models and variational autoencoders have been widely used to approximate probability distributions of datasets. Although they both use parametrized distributions to approximate the underlying data distribution, whose exact inference is intractable, their behaviors are very different. In this report, we summarize our experiment results that compare these two categories of models in terms of fidelity and mode collapse. We provide a hypothesis to explain their different behaviors and propose a new model based on this hypothesis. We further tested our proposed model on MNIST dataset and CelebA dataset.

This paper addresses the problem of formally verifying desirable properties of neural networks, i.e., obtaining provable guarantees that neural networks satisfy specifications relating their inputs and outputs (robustness to bounded norm adversarial perturbations, for example). Most previous work on this topic was limited in its applicability by the size of the network, network architecture and the complexity of properties to be verified. In contrast, our framework applies to a general class of activation functions and specifications on neural network inputs and outputs. We formulate verification as an optimization problem (seeking to find the largest violation of the specification) and solve a Lagrangian relaxation of the optimization problem to obtain an upper bound on the worst case violation of the specification being verified. Our approach is anytime i.e. it can be stopped at any time and a valid bound on the maximum violation can be obtained. We develop specialized verification algorithms with provable tightness guarantees under special assumptions and demonstrate the practical significance of our general verification approach on a variety of verification tasks.

Amortized inference has led to efficient approximate inference for large datasets. The quality of posterior inference is largely determined by two factors: a) the ability of the variational distribution to model the true posterior and b) the capacity of the recognition network to generalize inference over all datapoints. We analyze approximate inference in variational autoencoders in terms of these factors. We find that suboptimal inference is often due to amortizing inference rather than the limited complexity of the approximating distribution. We show that this is due partly to the generator learning to accommodate the choice of approximation. Furthermore, we show that the parameters used to increase the expressiveness of the approximation play a role in generalizing inference rather than simply improving the complexity of the approximation.