Nonconservative exact small-sample inference for discrete data

Exact small-sample methods for discrete data use probability distributions that do not depend on unknown parameters. However, they are conservative inferentially: the actual error probabilities for tests and confidence intervals are bounded above by the nominal level. This article surveys ways of reducing or even eliminating the conservatism. Fuzzy inference is a recent innovation that enables one to achieve the error probability exactly. We present a simple way of conducting fuzzy inference for discrete one-parameter exponential family distributions. In practice, most scientists would find this approach unsuitable yet might be disappointed by the conservatism of ordinary exact methods. Thus, we recommend using exact small-sample distributions but with inferences based on the mid-P value. This approach can be motivated by fuzzy inference, it is less conservative than standard exact methods, yet usually it does well in terms of achieving desired error probabilities. We illustrate for inferences about the binomial parameter.     

离散T-S 模糊系统的稳定条件

针对应用公共Lyapunov函数方法、模糊Lyapunov函数方法和分段模糊Lyapunov函数方法进行T-S模糊系统稳定性分析的保守性问题,通过定义有效最大交叠规则组,并基于离散型分段模糊Lyapunov函数,提出一个判定开环离散T-S模糊系统稳定性的充分条件.该条件仅需在每个有效最大交叠规则组内分别满足模糊Lyapunov方法中的条件,从而降低上述判定方法的保守性和难度.仿真实例验证了所提出条件的有效性和优越性.     

REVISED HIERARCHICAL ANALYSIS METHOD BASED ON CRISP AND FUZZY ENTRIES

Abstract

In this paper, I continue to study the alternative hierarchical analysis method that was initialed in [Saaty, 1976]. Instead of using pairwise ratio matrices, this method uses pairwise subtraction matrices. By doing so, computational complexity is reduced significantly, and as shown by a statistical experiment described in this paper, the two methods lead basically to the same results. Furthermore, this method provides an optimal solution in the sense of minimizing an appropriate error function. Due to the significant reduction in computational complexity, fuzzy hierarchical analysis based on pairwise subtraction matrices becomes more manageable. Finally, I point out a new application area of the hierarchical analysis: estimating crisp or fuzzy prior probability distributions for Bayesian inference with imprecise probabilities.     

Efficient and robust probabilistic guarantees for real-time tasks

This paper presents a new method for providing probabilistic real-time guarantees to tasks scheduled through resource reservations. Previous work on probabilistic analysis of reservation-based schedulers is extended by improving the efficiency and robustness of the probability computation. Robustness is improved by accounting for a possibly incomplete knowledge of the distribution of the computation times (which is typical in realistic applications). The proposed approach computes a conservative bound for the probability of missing deadlines, based on the knowledge of the probability distributions of the execution times and of the inter-arrival times of the tasks. In this paper, such a bound is computed in realistic situations, comparing it with simulative results and with the exact computation of deadline miss probabilities (without pessimistic bounds). Finally, the impact of the incomplete knowledge of the execution times distribution is evaluated.     

概率生成模型变分推理方法综述

概率生成模型是知识表示的重要方法,在该模型上计算似然函数的概率推理问题一般是难解的.变分推理是重要的确定性近似推理方法,具有较快的收敛速度、坚实的理论基础.尤其随着大数据时代的到来,概率生成模型变分推理方法受到工业界和学术界的极大关注.综述了多种概率生成模型变分推理框架及最新进展,具体包括:首先综述了概率生成模型变分推理一般框架及基于变分推理的生成模型参数学习过程;然后对于条件共轭指数族分布,给出了具有解析优化式的变分推理框架及该框架下可扩展的随机化变分推理;进一步,对于一般概率分布,给出了基于随机梯度的黑盒变分推理框架,并简述了该框架下多种变分推理算法的具体实现;最后分析了结构化变分推理,通过不同方式丰富变分分布提高推理精度并改善近似推理一致性.此外,展望了概率生成模型变分推理的发展趋势.     

Classification error in Bayes multistage recognition task with fuzzy observations

The paper considers the problem of classification error in multistage pattern recognition. This model of classification is based primarily on the Bayes rule and secondarily on the notion of fuzzy numbers. In adopting a probability-fuzzy model two concepts of hierarchical rules are proposed. In the first approach the local criterion that denote the probabilities of misclassification for particular nodes of a tree is considered. In the second approach the global optimal strategy that minimises the mean probability of misclassification on the whole multistage recognition process is considered. A probability of misclassifications is derived for a multiclass hierarchical classifier under the assumption that the features at different nodes of the tree are class-conditionally statistically independent, and we have fuzzy information on object features instead of exact information. Numerical example of this difference concludes the work.     

Three-valued abstraction for probabilistic systems

This paper proposes a novel abstraction technique for fully probabilistic systems. The models of our study are classical discrete-time and continuous-time Markov chains (DTMCs and CTMCs, for short). A DTMC is a Kripke structure in which each transition is equipped with a discrete probability; in a CTMC, in addition, state residence times are governed by negative exponential distributions. Our abstraction technique fits within the realm of three-valued abstraction methods that have been used successfully for traditional model checking. The key ingredients of our technique are a partitioning of the state space combined with an abstraction of transition probabilities by intervals. It is shown that this provides a conservative abstraction for both negative and affirmative verification results for a three-valued semantics of PCTL (Probabilistic Computation Tree Logic). In the continuous-time setting, the key idea is to apply abstraction on uniform CTMCs which are readily obtained from general CTMCs. In a similar way as for the discrete case, this is shown to yield a conservative abstraction for a three-valued semantics of CSL (Continuous Stochastic Logic). Abstract CTMCs can be verified by computing time-bounded reachability probabilities in continuous-time MDPs.     

Subsystem inference representation for fuzzy systems based uponproduct-sum-gravity rule

A new representation which expresses a product-sum-gravity (PSG) inference in terms of additive and multiplicative subsystem inferences of single variable is proposed. The representation yields additional insight into the structure of a fuzzy system and produces an approximate functional characterization of its inferred output. The form of the approximating function is dictated by the choice or polynomial, sinusoidal, or other designs of subsystem inferences. With polynomial inferences, the inferred output approximates a polynomial function the order of which is dependent on the numbers of input membership functions. Explicit expressions for the function and corresponding error of approximation are readily obtained for analysis. Subsystem inferences emulating sinusoidal functions are also discussed. With proper scaling, they produce a set of orthonormal subsystem inferences. The orthonormal set points to a possible “modal” analysis of fuzzy inference and yields solution to an additive decomposable approximation problem. This work also shows that, as the numbers of input membership functions become large, a fuzzy system with PSG inference would converge toward polynomial or Fourier series expansions. The result suggests a new framework to consider fuzzy systems as universal approximators