Inverse probability and sufficient statistics

1949 ◽  
Vol 45 (2) ◽  
pp. 225-229
Author(s):  
V. S. Huzurbazar
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Statistical Estimation ◽  
Sufficient Statistics ◽  
Inverse Probability ◽  
General Terms ◽  
Fiducial Probability ◽  
Posterior Probability Density ◽  
Posterior Probability Density Function

1. It is an interesting fact that in many problems of statistical estimation the results given by the theory of inverse probability (as modified by Jeffreys) are indistinguishable from those given by the methods of ‘fiducial probability’ or ‘confidence intervals’. The derivation of some of the important inverse distributions by Jeffreys(3) arouses one's curiosity. It seems that when this agreement is noticed there are usually sufficient statistics for parameters in the distribution. The object of this note is to throw some light, in general terms, on the similarity in form between the posterior probability-density function of the parameters and the probability-density function of the distribution when it admits sufficient statistics. For convenience the following notation in Jeffreys's probability logic is used below:P(q | p) is the probability of a proposition q on data p.

A Bayesian Framework of Computing and Updating Wavelet Parameter Distributions for Identifying Early Bearing Faults

2014 ◽  
Author(s):  
Peter W. Tse ◽  
Dong Wang
Keyword(s):  
Probability Density Function ◽  
Wavelet Analysis ◽  
Probability Density ◽  
Density Function ◽  
Posterior Probability ◽  
Fault Features ◽  
Bearing Failures ◽  
Posterior Probability Density ◽  
Posterior Probability Density Function ◽  
Parameter Distributions

Rolling element bearings are widely used in machines to support rotation shafts. Bearing failures may result in machine breakdown. In order to prevent bearing failures, early bearing faults are required to be identified. Wavelet analysis has proven to be an effective method for extracting early bearing fault features. Proper selection of wavelet parameters is crucial to wavelet analysis. In this paper, a Bayesian framework is proposed to compute and update wavelet parameter distributions. First, a smoothness index is used as the objective function because it has specific upper and lower bounds. Second, a general sequential Monte Carlo method is introduced to analytically derive the joint posterior probability density function of wavelet parameters. Last, approximately optimal wavelet parameters are inferred from the joint posterior probability density function. Simulated and real case studies are investigated to demonstrate that the proposed framework is effective in extracting early bearing fault features.

Age Reporting of Very Old Samples

Radiocarbon ◽  
1980 ◽  
Vol 22 (4) ◽  
pp. 1021-1027 ◽  
Author(s):  
Adam Walanus ◽  
Mieczysław F Pazdur
Keyword(s):  
Probability Density Function ◽  
Probability Distribution ◽  
Probability Density ◽  
Density Function ◽  
Statistical Interpretation ◽  
Very Old ◽  
Radiocarbon Age ◽  
Fiducial Probability ◽  
Limiting Value

Problems of the statistical interpretation of radiocarbon age measurements of old samples are discussed, based on the notion of fiducial probability distribution. A probability density function of age has been given. A detailed discussion of different facets of the probability distribution of age has led us to the confirmation of the use of 2σ as the best limiting value between the regions of finite and infinite dates. It has been proposed to make use of the principle of constant probability P = 0.68 in the regions of both finite and infinite ages instead of the criterion N + kσ.

A novel mobile agent-based distributed evidential expectation maximization algorithm for uncertain sensor networks

2020 ◽  
pp. 014233122096958
Author(s):  
Mohiyeddin Mozaffari ◽  
Behrouz Safarinejadian ◽  
Mokhtar Shasadeghi
Keyword(s):  
Sensor Networks ◽  
Probability Density Function ◽  
Probability Density ◽  
Convergence Analysis ◽  
Density Function ◽  
Mobile Agent ◽  
Expectation Maximization ◽  
Convergence Criterion ◽  
Sufficient Statistics ◽  
Agent Based

In this paper, a novel mobile agent-based distributed evidential expectation maximization (MADEEM) algorithm is presented for sensor networks. The proposed algorithm is used for probability density function estimation and data clustering in the presence of uncertainties in sensor measurements. It is assumed that the sensor measurements are statistically modeled by a common Gaussian mixture model. The proposed algorithm maximizes a new generalized likelihood criterion in an iterative and distributed manner. For this purpose, mobile agents compute some local sufficient statistics by using local measurements of each sensor node. After the local computations, the global sufficient statistics are updated. At the end of iterations, the parameters of the probability density function are updated by using the global sufficient statistics. The mentioned process will be continued until the convergence criterion is satisfied. Convergence analysis of the proposed algorithm is also presented in this paper. After the convergence analysis, the simulation results show the promising performance of the proposed algorithm. Finally, the last part of the paper is devoted to the concluding remarks.

scholarly journals An FEM-Based State Estimation Approach to Nonlinear Hybrid Positioning Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Yu-Xin Zhao ◽  
Li-Juan Chen ◽  
Yan Ma
Keyword(s):  
Probability Density Function ◽  
State Estimation ◽  
Probability Density ◽  
Density Function ◽  
Differential Model ◽  
Positioning Systems ◽  
Forward Equation ◽  
Hybrid Positioning ◽  
Posterior Probability Density ◽  
Low Dimensional

For hybrid positioning systems (HPSs), the estimator design is a crucial and important problem. In this paper, a finite-element-method- (FEM-) based state estimation approach is proposed to HPS. As the weak solution of hybrid stochastic differential model is denoted by the Kolmogorov's forward equation, this paper constructs its interpolating point through the classical fourth-order Runge-Kutta method. Then, it approaches the solution with biquadratic interpolation function to obtain a prior probability density function of the state. A posterior probability density function is gained through Bayesian formula finally. In theory, the proposed scheme has more advantages in the performance of complexity and convergence for low-dimensional systems. By taking an illustrative example, numerical experiment results show that the new state estimator is feasible and has good performance than PF and UKF.

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