Bayesian networks for discrete multivariate data: an algebraic approach to inference

In this paper we demonstrate how Gröbner bases and other algebraic techniques can be used to explore the geometry of the probability space of Bayesian networks with hidden variables. These techniques employ a parametrisation of Bayesian network by moments rather than conditional probabilities. We show that whilst Gröbner bases help to explain the local geometry of these spaces a complimentary analysis, modelling the positivity of probabilities, enhances and completes the geometrical picture. We report some recent geometrical results in this area and discuss a possible general methodology for the analyses of such problems.     

The computation of the logarithmic cohomology for plane curves

We will give algorithms of computing bases of logarithmic cohomology groups for square-free polynomials in two variables.     

An algebraic geometry algorithm for scheduling in presence of setups and correlated demands

We study here a problem of schedulingn job types onm parallel machines, when setups are required and the demands for the products are correlated random variables. We model this problem as a chance constrained integer program.Methods of solution currently available—in integer programming and stochastic programming—are not sufficient to solve this model exactly. We develop and introduce here a new approach, based on a geometric interpretation of some recent results in Gröbner basis theory, to provide a solution method applicable to a general class of chance constrained integer programming problems.Out algorithm is conceptually simple and easy to implement. Starting from a (possibly) infeasible solution, we move from one lattice point to another in a monotone manner regularly querying a membership oracle for feasibility until the optimal solution is found. We illustrate this methodology by solving a problem based on a real system.Corresponding author.     

泊松逆高斯回归模型的贝叶斯统计推断

本文研究泊松逆高斯回归模型的贝叶斯统计推断.基于应用Gibbs抽样,Metropolis-Hastings算法以及Multiple-Try Metropolis算法等MCMC统计方法计算模型未知参数和潜变量的联合贝叶斯估计,并引入两个拟合优度统计量来评价提出的泊松逆高斯回归模型的合理性.若干模拟研究与一个实证分析说明方...     

Propagation effects of model-calculated probability values in Bayesian networks

Probabilistic causal interaction models have become quite popular among Bayesian-network engineers as elicitation of all probabilities required often proves the main bottleneck in building a real-world network with domain experts. The best-known interaction models are the noisy-OR model and its generalisations. These models in essence are parameterised conditional probability tables for which just a limited number of parameter probabilities are required. The models assume specific properties of intercausal interaction and cannot be applied uncritically. Given their clear engineering advantages however, they are subject to ill-considered use. This paper demonstrates that such ill-considered use can result in poorly calibrated output probabilities from a Bayesian network. By studying, in an analytical way, the propagation effects of noisy-OR calculated probability values, we identify conditions under which use of the model can be harmful for a network's performance. These conditions demonstrate that use of the noisy-OR model for mere pragmatic reasons is sometimes warranted, even when the model's underlying assumptions are not met in reality.     

Facets and volume of Gorenstein Fano polytopes

It is known that every integral convex polytope is unimodularly equivalent to a face of some Gorenstein Fano polytope. It is then reasonable to ask whether every normal polytope is unimodularly equivalent to a face of some normal Gorenstein Fano polytope. In the present paper, it is shown that, by giving new classes of normal Gorenstein Fano polytopes, each order polytope as well as each chain polytope of dimension d is unimodularly equivalent to a facet of some normal Gorenstein Fano polytopes of dimension . Furthermore, investigation on combinatorial properties, especially, Ehrhart polynomials and volume of these new polytopes will be achieved. Finally, some curious examples of Gorenstein Fano polytopes will be discovered.     

The role of local partial independence in learning of Bayesian networks

Bayesian networks are one of the most widely used tools for modeling multivariate systems. It has been demonstrated that more expressive models, which can capture additional structure in each conditional probability table (CPT), may enjoy improved predictive performance over traditional Bayesian networks despite having fewer parameters. Here we investigate this phenomenon for models of various degree of expressiveness on both extensive synthetic and real data. To characterize the regularities within CPTs in terms of independence relations, we introduce the notion of partial conditional independence (PCI) as a generalization of the well-known concept of context-specific independence (CSI). To model the structure of the CPTs, we use different graph-based representations which are convenient from a learning perspective. In addition to the previously studied decision trees and graphs, we introduce the concept of PCI-trees as a natural extension of the CSI-based trees. To identify plausible models we use the Bayesian score in combination with a greedy search algorithm. A comparison against ordinary Bayesian networks shows that models with local structures in general enjoy parametric sparsity and improved out-of-sample predictive performance, however, often it is necessary to regulate the model fit with an appropriate model structure prior to avoid overfitting in the learning process. The tree structures, in particular, lead to high quality models and suggest considerable potential for further exploration.     

The rate of approximation of Gaussian radial basis neural networks in continuous function space

There have been many studies on the dense theorem of approximation by radial basis feedforword neural networks, and some approximation problems by Gaussian radial basis feedforward neural networks(GRBFNs)in some special function space have also been investigated. This paper considers the approximation by the GRBFNs in continuous function space. It is proved that the rate of approximation by GRNFNs with n~d neurons to any continuous function f defined on a compact subset K(R~d)can be controlled by ω(f, n~(-1/2)), where ω(f, t)is the modulus of continuity of the function f .     

Fully Bayesian Analysis of Switching Gaussian State Space Models

In the present paper we study switching state space models from a Bayesian point of view. We discuss various MCMC methods for Bayesian estimation, among them unconstrained Gibbs sampling, constrained sampling and permutation sampling. We address in detail the problem of unidentifiability, and discuss potential information available from an unidentified model. Furthermore the paper discusses issues in model selection such as selecting the number of states or testing for the presence of Markov switching heterogeneity. The model likelihoods of all possible hypotheses are estimated by using the method of bridge sampling. We conclude the paper with applications to simulated data as well as to modelling the U.S./U.K. real exchange rate.     

Gröbner bases of balanced polyominoes

We introduce balanced polyominoes and show that their ideal of inner minors is a prime ideal and has a squarefree Gröbner basis with respect to any monomial order, and we show that any row or column convex and any tree‐like polyomino is simple and balanced.     

Gröbner bases for the Hilbert ideal and coinvariants of the dihedral group D2p

We consider a finite dimensional representation of the dihedral group D_2p over a field of characteristic two where p is an odd integer and study the corresponding Hilbert ideal I_H. We show that I_H has a universal Gröbner basis consisting of invariants and monomials only. We provide sharp bounds for the degree of an element in this basis and in a minimal generating set for I_H. We also compute the top degree of coinvariants when p is prime.     

Toric rings generated by special stable sets of monomials

In this paper we consider some subalgebras of the d-th Veronese subring of a polynomial ring, generated by stable subsets of monomials. We prove that these algebras are Koszul, showing that the presentation ideals have Gröbner bases of quadrics with respect to suitable term orders. Since the initial monomials of the elements of these Gröbner bases are square- free, it follows by a result of STURMFELS [S, 13.15], that the algebras under consideration are normal, and thus Cohen-Macaulay.     

Two optimal strategies for active learning of causal models from interventional data

From observational data alone, a causal DAG is only identifiable up to Markov equivalence. Interventional data generally improves identifiability; however, the gain of an intervention strongly depends on the intervention target, that is, the intervened variables. We present active learning (that is, optimal experimental design) strategies calculating optimal interventions for two different learning goals. The first one is a greedy approach using single-vertex interventions that maximizes the number of edges that can be oriented after each intervention. The second one yields in polynomial time a minimum set of targets of arbitrary size that guarantees full identifiability. This second approach proves a conjecture of Eberhardt (2008) [1] indicating the number of unbounded intervention targets which is sufficient and in the worst case necessary for full identifiability. In a simulation study, we compare our two active learning approaches to random interventions and an existing approach, and analyze the influence of estimation errors on the overall performance of active learning.     

中国经济随机前沿模型效率的贝叶斯统计推断

基于扩展的随机生产前沿模型,研究了区域生产效率的差异和其影响因素的作用效果,应用贝叶斯统计方法对中国各省份2010-2017的年度数据(不包含港澳台地区,下同)进行了实证研究.研究发现:生产效率总体呈逐渐下降的趋势,地区间生产效率有一定的差异,高等教育规模对生产效率具有显著的直接影响.人力资本能有效促进东部和中部地区的经济增长,西部地区主要依靠资本促进经济增长.环境污染对中部地区的经济增长具有一定的负向作用.     

Bayesian source detection and parameter estimation of a plume model based on sensor network measurements

We consider a network of sensors that measure the intensities of a complex plume composed of multiple absorption–diffusion source components. We address the problem of estimating the plume parameters, including the spatial and temporal source origins and the parameters of the diffusion model for each source, based on a sequence of sensor measurements. The approach not only leads to multiple‐source detection, but also the characterization and prediction of the combined plume in space and time. The parameter estimation is formulated as a Bayesian inference problem, and the solution is obtained using a Markov chain Monte Carlo algorithm. The approach is applied to a simulation study, which shows that an accurate parameter estimation is achievable. Copyright © 2010 John Wiley & Sons, Ltd.     

基于高斯混合和BP神经网络的卵巢癌质谱数据三分类模型

癌症的早期诊断可以显著提高癌症患者的存活率,三分类问题就是将未知样本与已知样本进行匹配度检测,预测样本是健康状态,良性发展状态,还是癌症状态.针对复杂难分的卵巢癌蛋白质质谱数据,提出了一种基于高斯混合模型和BP神经网络的三分类预测模型.首先,去除原数据中的冗余,对其进行方差排序及交集筛选提取特征集合一,再利用高斯混合模型处理求得参数作为特征集合二,最后使用BP神经网络进行样本三分类,准确率达到72.9%.结果表明:模型可以作为卵巢癌质谱数据三分类的可选择工具.     

Bayesian Inference via Filtering for a Class of Counting Processes: Application to the Micromovement of Asset Price

This paper highlights recent developments in a rich class of counting process models for the micromovement of asset price and in the Bayesian inference (estimation and model selection) via filtering for the class of models. A specific micromovement model built upon linear Brownian motion with jumping stochastic volatility is used to demonstrate the procedure to develop a micromovement model with specific tick-level sample characteristics. The model is further used to demonstrate the procedure to implement Bayes estimation via filtering, namely, to construct a recursive algorithm for computing the trade-by-trade Bayes parameter estimates, especially for the stochastic volatility. The consistency of the recursive algorithm model is proven. Simulation and real-data examples are provided as well as a brief example of Bayesian model selection via filtering.     

Convergence for a class of delayed recurrent neural networks without M-matrix condition

In this paper, we investigate the asymptotic behavior of solutions to a class of recurrent neural network model with delays. Without assuming M-matrix condition, it is shown that every solution of the network tends to an equilibrium point as t→∞. Our results improve and extend some corresponding ones already known.