We introduce a novel procedure for obtaining cross-validated predictive estimates for Bayesian hierarchical regression models (BHRMs). Bayesian hierarchical models are popular for their ability to model complex dependence structures and provide probabilistic uncertainty estimates, but can be computationally expensive to run. Cross-validation (CV) is therefore not a common practice to evaluate the predictive performance of BHRMs. Our method circumvents the need to re-run computationally costly estimation methods for each cross-validation fold and makes CV more feasible for large BHRMs. By conditioning on the variance-covariance parameters, we shift the CV problem from probability-based sampling to a simple and familiar optimization problem. In many cases, this produces estimates which are equivalent to full CV. We provide theoretical results and demonstrate its efficacy on publicly available data and in simulations.

Knowledge compilation studies the trade-off between succinctness and efficiency of different representation languages. For many languages, there are known strong lower bounds on the representation size, but recent work shows that, for some languages, one can bypass these bounds using approximate compilation. The idea is to compile an approximation of the knowledge for which the number of errors can be controlled. We focus on circuits in deterministic decomposable negation normal form (d-DNNF), a compilation language suitable in contexts such as probabilistic reasoning, as it supports efficient model counting and probabilistic inference. Moreover, there are known size lower bounds for d-DNNF which by relaxing to approximation one might be able to avoid. In this paper we formalize two notions of approximation: weak approximation which has been studied before in the decision diagram literature and strong approximation which has been used in recent algorithmic results. We then show lower bounds for approximation by d-DNNF, complementing the positive results from the literature.

Its application involves a wide range of area including expert systems[3][4][5], information fusion[6], pattern classfication[7][8][9], risk evaluation [10,11] [12], image recognition [13], classification[14,15] and data mining [16] etc. The original DS theory requires deterministic belie degrees and belief structures. However, in practical situations, evidence coming from multiple sources may be influenced by unexpected extraneous factors. The lack of information, linguistic ambiguity or vagueness and cognitive bias all contribute to the uncertain evidence obtained in practical situations. For example, during risk assessment, expert may be unable to provide a precise assessment if he/she is not 100% sure.

We study the posterior contraction rates of a Bayesian method with Gaussian process priors in nonparametric regression and its plug-in property for differential operators. For a general class of kernels, we establish convergence rates of the posterior measure of the regression function and its derivatives, which are both minimax optimal up to a logarithmic factor for functions in certain classes. Our calculation shows that the rate-optimal estimation of the regression function and its derivatives share the same choice of hyperparameter, indicating that the Bayes procedure remarkably adapts to the order of derivatives and enjoys a generalized plug-in property that extends real-valued functionals to function-valued functionals. This leads to a practically simple method for estimating the regression function and its derivatives, whose finite sample performance is assessed using simulations. Our proof shows that, under certain conditions, to any convergence rate of Bayes estimators there corresponds the same convergence rate of the posterior distributions (i.e., posterior contraction rate), and vice versa. This equivalence holds for a general class of Gaussian processes and covers the regression function and its derivative functionals, under both the $L_2$ and $L_{\infty}$ norms. In addition to connecting these two fundamental large sample properties in Bayesian and non-Bayesian regimes, such equivalence enables a new routine to establish posterior contraction rates by calculating convergence rates of nonparametric point estimators. At the core of our argument is an operator-theoretic framework for kernel ridge regression and equivalent kernel techniques. We derive a range of sharp non-asymptotic bounds that are pivotal in establishing convergence rates of nonparametric point estimators and the equivalence theory, which may be of independent interest.

Distributed inference/estimation in Bayesian framework in the context of sensor networks has recently received much attention due to its broad applicability. The variational Bayesian (VB) algorithm is a technique for approximating intractable integrals arising in Bayesian inference. In this paper, we propose two novel distributed VB algorithms for general Bayesian inference problem, which can be applied to a very general class of conjugate-exponential models. In the first approach, the global natural parameters at each node are optimized using a stochastic natural gradient that utilizes the Riemannian geometry of the approximation space, followed by an information diffusion step for cooperation with the neighbors. In the second method, a constrained optimization formulation for distributed estimation is established in natural parameter space and solved by alternating direction method of multipliers (ADMM). An application of the distributed inference/estimation of a Bayesian Gaussian mixture model is then presented, to evaluate the effectiveness of the proposed algorithms. Simulations on both synthetic and real datasets demonstrate that the proposed algorithms have excellent performance, which are almost as good as the corresponding centralized VB algorithm relying on all data available in a fusion center.

In this paper, we consider a novel framework of positive-unlabeled data in which as positive data survival times are observed for subjects who have events during the observation time as positive data and as unlabeled data censoring times are observed but whether the event occurs or not are unknown for some subjects. We consider two cases: (1) when censoring time is observed in positive data, and (2) when it is not observed. For both cases, we developed parametric models, nonparametric models, and machine learning models and the estimation strategies for these models. Simulation studies show that under this data setup, traditional survival analysis may yield severely biased results, while the proposed estimation method can provide valid results.

Creating noise from data is easy; creating data from noise is generative modeling. We present a stochastic differential equation (SDE) that smoothly transforms a complex data distribution to a known prior distribution by slowly injecting noise, and a corresponding reverse-time SDE that transforms the prior distribution back into the data distribution by slowly removing the noise. Crucially, the reverse-time SDE depends only on the time-dependent gradient field (a.k.a., score) of the perturbed data distribution. By leveraging advances in score-based generative modeling, we can accurately estimate these scores with neural networks, and use numerical SDE solvers to generate samples. We show that this framework encapsulates previous approaches in diffusion probabilistic modeling and score-based generative modeling, and allows for new sampling procedures. In particular, we introduce a predictor-corrector framework to correct errors in the evolution of the discretized reverse-time SDE. We also derive an equivalent neural ODE that samples from the same distribution as the SDE, which enables exact likelihood computation, and improved sampling efficiency. In addition, our framework enables conditional generation with an unconditional model, as we demonstrate with experiments on class-conditional generation, image inpainting, and colorization. Combined with multiple architectural improvements, we achieve record-breaking performance for unconditional image generation on CIFAR-10 with an Inception score of 9.89 and FID of 2.20, a competitive likelihood of 3.10 bits/dim, and demonstrate high fidelity generation of $1024 \times 1024$ images for the first time from a score-based generative model.

In this paper, a novel stepwise learning approach based on estimating desired premise parts' outputs by solving a constrained optimization problem is proposed. This learning approach does not require backpropagating the output error to learn the premise parts' parameters. Instead, the near best output values of the rules premise parts are estimated and their parameters are changed to reduce the error between current premise parts' outputs and the estimated desired ones. Therefore, the proposed learning method avoids error backpropagation, which lead to vanishing gradient and consequently getting stuck in a local optimum. The proposed method does not need any initialization method. This learning method is utilized to train a new Takagi-Sugeno-Kang (TSK) Fuzzy Neural Network with correlated fuzzy rules including many parameters in both premise and consequent parts, avoiding getting stuck in a local optimum due to vanishing gradient. To learn the proposed network parameters, first, a constrained optimization problem is introduced and solved to estimate the desired values of premise parts' output values. Next, the error between these values and the current ones is utilized to adapt the premise parts' parameters based on the gradient-descent (GD) approach. Afterward, the error between the desired and network's outputs is used to learn consequent parts' parameters by the GD method. The proposed paradigm is successfully applied to real-world time-series prediction and regression problems. According to experimental results, its performance outperforms other methods with a more parsimonious structure.

There is a growing trend in molecular and synthetic biology of using mechanistic (non machine learning) models to design biomolecular networks. Once designed, these networks need to be validated by experimental results to ensure the theoretical network correctly models the true system. However, these experiments can be expensive and time consuming. We propose a design of experiments approach for validating these networks efficiently. Gaussian processes are used to construct a probabilistic model of the discrepancy between experimental results and the designed response, then a Bayesian optimization strategy used to select the next sample points. We compare different design criteria and develop a stopping criterion based on a metric that quantifies this discrepancy over the whole surface, and its uncertainty. We test our strategy on simulated data from computer models of biochemical processes.

Imagery is frequently used to model, represent and communicate knowledge. In particular, graphs are one of the most powerful tools, being able to represent relations between objects. Causal relations are frequently represented by directed graphs, with nodes denoting causes and links denoting causal influence. A causal graph is a skeletal picture, showing causal associations and impact between entities. Common methods used for graphically representing causal scenarios are neurons, truth tables, causal Bayesian networks, cognitive maps and Petri Nets. Causality is often defined in terms of precedence (the cause precedes the effect), concurrency (often, an effect is provoked simultaneously by two or more causes), circularity (a cause provokes the effect and the effect reinforces the cause) and imprecision (the presence of the cause favors the effect, but not necessarily causes it). We will show that, even though the traditional graphical models are able to represent separately some of the properties aforementioned, they fail trying to illustrate indistinctly all of them. To approach that gap, we will introduce Fuzzy Stochastic Timed Petri Nets as a graphical tool able to represent time, co-occurrence, looping and imprecision in causal flow.