Comparisons of Maximum Likelihood Estimates and Bayesian Estimates for the Discretized Discovery Process Model

A Bayesian approach using Markov chain Monte Carlo algorithms has been developed to analyze Smith’s discretized version of the discovery process model. It avoids the problems involved in the maximum likelihood method by effectively making use of the information from the prior distribution and that from the discovery sequence according to posterior probabilities. All statistical inferences about the parameters of the model and total resources can be quantified by drawing samples directly from the joint posterior distribution. In addition, statistical errors of the samples can be easily assessed and the convergence properties can be monitored during the sampling. Because the information contained in a discovery sequence is not enough to estimate all parameters, especially the number of fields, geologically justified prior information is crucial to the estimation. The Bayesian approach allows the analyst to specify his subjective estimates of the required parameters and his degree of uncertainty about the estimates in a clearly identified fashion throughout the analysis. As an example, this approach is applied to the same data of the North Sea on which Smith demonstrated his maximum likelihood method. For this case, the Bayesian approach has really improved the overly pessimistic results and downward bias of the maximum likelihood procedure.

Performance of Bayesian Propensity Score Adjustment for Estimating Causal Effects in Small Clinical Trials

Propensity score (PS) adjustment can control confounding effects and reduce bias when estimating treatment effects in non-randomized trials or observational studies. PS methods are becoming increasingly used to estimate causal effects, including when the sample size is small compared to the number of confounders. With numerous confounders, quasi-complete separation can easily occur in logistic regression used for estimating the PS, but this has not been addressed. We focused on a Bayesian PS method to address the limitations of quasi-complete separation faced by small trials. Bayesian methods are useful because they estimate the PS and causal effects simultaneously while considering the uncertainty of the PS by modelling it as a latent variable. In this study, we conducted simulations to evaluate the performance of Bayesian simultaneous PS estimation by considering the specification of prior distributions for model comparison. We propose a method to improve predictive performance with discrete outcomes in small trials. We found that the specification of prior distributions assigned to logistic regression coefficients was more important in the second step than in the first step, even when there was a quasi-complete separation in the first step. Assigning Cauchy (0, 2.5) to coefficients improved the predictive performance for estimating causal effects and improving the balancing properties of the confounder.

Research on aiming methods for small sample size shooting tests of two-dimensional trajectory correction fuse

The longitudinal dispersion of the projectile in shooting tests of two-dimensional trajectory corrections fused with fixed canards is extremely large that it sometimes exceeds the correction ability of the correction fuse actuator.The impact point easily deviates from the target,and thus the correction result cannot be readily evaluated.However,the cost of shooting tests is considerably high to conduct many tests for data collection.To address this issue,this study proposes an aiming method for shooting tests based on small sample size.The proposed method uses the Bootstrap method to expand the test data;repeatedly iterates and corrects the position of the simulated theoretical impact points through an improved compatibility test method;and dynamically adjusts the weight of the prior distribution of simulation results based on Kullback-Leibler divergence,which to some extent avoids the real data being"submerged"by the simulation data and achieves the fusion Bayesian estimation of the dispersion center.The experimental results show that when the simulation accuracy is sufficiently high,the proposed method yields a smaller mean-square deviation in estimating the dispersion center and higher shooting accuracy than those of the three comparison methods,which is more conducive to reflecting the effect of the control algorithm and facilitating test personnel to iterate their proposed structures and algorithms.;in addition,this study provides a knowledge base for further comprehensive studies in the future.

Modified unscented particle filter for nonlinear Bayesian tracking

A modified unscented particle filtering scheme for nonlinear tracking is proposed, in view of the potential drawbacks （such as, particle impoverishment and numerical sensitivity in calculating the prior） of the conventional unscented particle filter （UPF） confronted in practice. Specifically, a different derivation of the importance weight is presented in detail. The proposed method can avoid the calculation of the prior and reduce the effects of the impoverishment problem caused by sampling from the proposal distribution, Simulations have been performed using two illustrative examples and results have been provided to demonstrate the validity of the modified UPF as well as its improved performance over the conventional one.

Nonlinear Bayesian Estimation： From Kalman Filtering to a Broader Horizon

This article presents an up-to-date tutorial review of nonlinear Bayesian estimation. State estimation for nonlinear systems has been a challenge encountered in a wide range of engineering fields, attracting decades of research effort. To date,one of the most promising and popular approaches is to view and address the problem from a Bayesian probabilistic perspective,which enables estimation of the unknown state variables by tracking their probabilistic distribution or statistics(e.g., mean and covariance) conditioned on a system's measurement data.This article offers a systematic introduction to the Bayesian state estimation framework and reviews various Kalman filtering(KF)techniques, progressively from the standard KF for linear systems to extended KF, unscented KF and ensemble KF for nonlinear systems. It also overviews other prominent or emerging Bayesian estimation methods including Gaussian filtering, Gaussian-sum filtering, particle filtering and moving horizon estimation and extends the discussion of state estimation to more complicated problems such as simultaneous state and parameter/input estimation.

Single channel signal component separation using Bayesian estimation

A Bayesian estimation method to separate multicomponent signals with single channel observation is presented in this paper. By using the basis function projection, the component separation becomes a problem of limited parameter estimation. Then, a Bayesian model for estimating parameters is set up. The reversible jump MCMC （Monte Carlo Markov Chain） algorithmis adopted to perform the Bayesian computation. The method can jointly estimate the parameters of each component and the component number. Simulation results demonstrate that the method has low SNR threshold and better performance.

Bayesian seismic multi-scale inversion in complex Laplace mixed domains

Seismic inversion performed in the time or frequency domain cannot always recover the long-wavelength background of subsurface parameters due to the lack of low-frequency seismic records. Since the low-frequency response becomes much richer in the Laplace mixed domains, one novel Bayesian impedance inversion approach in the complex Laplace mixed domains is established in this study to solve the model dependency problem. The derivation of a Laplace mixed-domain formula of the Robinson convolution is the first step in our work. With this formula, the Laplace seismic spectrum, the wavelet spectrum and time-domain reflectivity are joined together. Next, to improve inversion stability, the object inversion function accompanied by the initial constraint of the linear increment model is launched under a Bayesian framework. The likelihood function and prior probability distribution can be combined together by Bayesian formula to calculate the posterior probability distribution of subsurface parameters. By achieving the optimal solution corresponding to maximum posterior probability distribution, the low-frequency background of subsurface parameters can be obtained successfully. Then, with the regularization constraint of estimated low frequency in the Laplace mixed domains, multi-scale Bayesian inversion inthe pure frequency domain is exploited to obtain the absolute model parameters. The effectiveness, anti-noise capability and lateral continuity of Laplace mixed-domain inversion are illustrated by synthetic tests. Furthermore,one field case in the east of China is discussed carefully with different input frequency components and different inversion algorithms. This provides adequate proof to illustrate the reliability improvement in low-frequency estimation and resolution enhancement of subsurface parameters, in comparison with conventional Bayesian inversion in the frequency domain.

Target threat estimation based on discrete dynamic Bayesian networks with small samples

The accuracy of target threat estimation has a great impact on command decision-making.The Bayesian network,as an effective way to deal with the problem of uncertainty,can be used to track the change of the target threat level.Unfortunately,the traditional discrete dynamic Bayesian network(DDBN)has the problems of poor parameter learning and poor reasoning accuracy in a small sample environment with partial prior information missing.Considering the finiteness and discreteness of DDBN parameters,a fuzzy k-nearest neighbor(KNN)algorithm based on correlation of feature quantities(CF-FKNN)is proposed for DDBN parameter learning.Firstly,the correlation between feature quantities is calculated,and then the KNN algorithm with fuzzy weight is introduced to fill the missing data.On this basis,a reasonable DDBN structure is constructed by using expert experience to complete DDBN parameter learning and reasoning.Simulation results show that the CF-FKNN algorithm can accurately fill in the data when the samples are seriously missing,and improve the effect of DDBN parameter learning in the case of serious sample missing.With the proposed method,the final target threat assessment results are reasonable,which meets the needs of engineering applications.

E-Bayesian estimation for competing risk model under progressively hybrid censoring

This paper considers the Bayesian and expected Bayesian（E-Bayesian） estimations of the parameter and reliability function for competing risk model from Gompertz distribution under Type-I progressively hybrid censoring scheme（PHCS）. The estimations are obtained based on Gamma conjugate prior for the parameter under squared error（SE） and Linex loss functions. The simulation results are provided for the comparison purpose and one data set is analyzed.

Bayesian Estimation for the Order of INAR(q)Model

In this paper, we consider the problem of determining the order ofINAR（Q） model on the basis of the Bayesian estimation theory. The Bayesian es-timator for the order is given with respect to a squared-error loss function. The consistency of the estimator is discussed. The results of a simulation study for the estimation method are presented.

Comparison and combination of EAKF and SIR-PF in the Bayesian filter framework

Bayesian estimation theory provides a general approach for the state estimate of linear or nonlinear and Gaussian or non-Gaussian systems. In this study, we first explore two Bayesian-based methods： ensemble adjustment Kalman filter（EAKF） and sequential importance resampling particle filter（SIR-PF）, using a well-known nonlinear and non-Gaussian model（Lorenz ＇63 model）. The EAKF, which is a deterministic scheme of the ensemble Kalman filter（En KF）, performs better than the classical（stochastic） En KF in a general framework. Comparison between the SIR-PF and the EAKF reveals that the former outperforms the latter if ensemble size is so large that can avoid the filter degeneracy, and vice versa. The impact of the probability density functions and effective ensemble sizes on assimilation performances are also explored. On the basis of comparisons between the SIR-PF and the EAKF, a mixture filter, called ensemble adjustment Kalman particle filter（EAKPF）, is proposed to combine their both merits. Similar to the ensemble Kalman particle filter, which combines the stochastic En KF and SIR-PF analysis schemes with a tuning parameter, the new mixture filter essentially provides a continuous interpolation between the EAKF and SIR-PF. The same Lorenz ＇63 model is used as a testbed, showing that the EAKPF is able to overcome filter degeneracy while maintaining the non-Gaussian nature, and performs better than the EAKF given limited ensemble size.

Speech Signal Detection Based on Bayesian Estimation by Observing Air-Conducted Speech under Existence of Surrounding Noise with the Aid of Bone-Conducted Speech

In order to apply speech recognition systems to actual circumstances such as inspection and maintenance operations in industrial factories to recording and reporting routines at construction sites, etc. where hand-writing is difficult, some countermeasure methods for surrounding noise are indispensable. In this study, a signal detection method to remove the noise for actual speech signals is proposed by using Bayesian estimation with the aid of bone-conducted speech. More specifically, by introducing Bayes’ theorem based on the observation of air-conducted speech contaminated by surrounding background noise, a new type of algorithm for noise removal is theoretically derived. In the proposed speech detection method, bone-conducted speech is utilized in order to obtain precise estimation for speech signals. The effectiveness of the proposed method is experimentally confirmed by applying it to air- and bone-conducted speeches measured in real environment under the existence of surrounding background noise.

Dynamic Bayesian estimation of displacement parameters of continuous curve box based on Novozhilov theory

The finite strip controlling equation of pinned curve box was deduced on basis of Novozhilov theory and with flexibility method, and the problem of continuous curve box was resolved. Dynamic Bayesian error function of displacement parameters of continuous curve box was found. The corresponding formulas of dynamic Bayesian expectation and variance were derived. After the method of solving the automatic search of step length was put forward, the optimization estimation computing formulas were also obtained by adapting conjugate gradient method. Then the steps of dynamic Bayesian estimation were given in detail. Through analysis of a Classic example, the criterion of judging the precision of the known information is gained as well as some other important conclusions about dynamic Bayesian stochastic estimation of displacement parameters of continuous curve box.

Entropy Bayesian Analysis for the Generalized Inverse Exponential Distribution Based on URRSS

This paper deals with the Bayesian estimation of Shannon entropy for the generalized inverse exponential distribution.Assuming that the observed samples are taken from the upper record ranked set sampling(URRSS)and upper record values(URV)schemes.Formulas of Bayesian estimators are derived depending on a gamma prior distribution considering the squared error,linear exponential and precautionary loss functions,in addition,we obtain Bayesian credible intervals.The random-walk Metropolis-Hastings algorithm is handled to generate Markov chain Monte Carlo samples from the posterior distribution.Then,the behavior of the estimates is examined at various record values.The output of the study shows that the entropy Bayesian estimates under URRSS are more convenient than the other estimates under URV in the majority of the situations.Also,the entropy Bayesian estimates perform well as the number of records increases.The obtained results validate the usefulness and efficiency of the URV method.Real data is analyzed for more clarifying purposes which validate the theoretical results.

Bayesian estimation for nonlinear stochastic hybrid systems with state dependent transitions

The Bayesian approach is considered as the most general formulation of the state estimation for dynamic systems. However, most of the existing Bayesian estimators of stochastic hybrid systems only focus on the Markov jump system, few liter- ature is related to the estimation problem of nonlinear stochastic hybrid systems with state dependent transitions. According to this problem, a new methodology which relaxes quite a restrictive as- sumption that the mode transition process must satisfy Markov properties is proposed. In this method, a general approach is presented to model the state dependent transitions, the state and output spaces are discreted into cell space which handles the nonlinearities and computationally intensive problem offline. Then maximum a posterior estimation is obtained by using the Bayesian theory. The efficacy of the estimator is illustrated by a simulated example .

Research on Bayesian Method of Direction Finding

In this paper, Bayesian technique of direction finding based on two different priorities is described. Some useful formulas are deduced. The performance of the method and the influence of the priors on direction finding are demonstrated by computer simulations.

Weibull-Bayesian Estimation Based on Maximum Ranked Set Sampling with Unequal Samples

A modification of ranked set sampling (RSS) called maximum ranked set sampling with unequal sample (MRSSU) is considered for the Bayesian estimation of scale parameter α of the Weibull distribution. Under this method, we use Linex loss function, conjugate and Jeffreys prior distributions to derive the Bayesian estimate of α. In order to measure the efficiency of the obtained Bayesian estimates with respect to the Bayesian estimates of simple random sampling (SRS), we compute the bias, mean squared error (MSE) and asymptotic relative efficiency of the obtained Bayesian estimates using simulation. It is shown that the proposed estimates are found to be more efficient than the corresponding one based on SRS.

Constructing Statistical Intervals for Small Area Estimates Based on Generalized Linear Mixed Model in Health Surveys

Generalized Linear Mixed Model (GLMM) has been widely used in small area estimation for health indicators. Bayesian estimation is usually used to construct statistical intervals, however, its computational intensity is a big challenge for large complex surveys. Frequentist approaches, such as bootstrapping, and Monte Carlo (MC) simulation, are also applied but not evaluated in terms of the interval magnitude, width, and the computational time consumed. The 2013 Florida Behavioral Risk Factor Surveillance System data was used as a case study. County-level estimated prevalence of three health-related outcomes was obtained through a GLMM;and their 95% confidence intervals (CIs) were generated from bootstrapping and MC simulation. The intervals were compared to 95% credential intervals through a hierarchial Bayesian model. The results showed that 95% CIs for county-level estimates of each outcome by using MC simulation were similar to the 95% credible intervals generated by Bayesian estimation and were the most computationally efficient. It could be a viable option for constructing statistical intervals for small area estimation in public health practice.

A Noise Suppression Method for Speech Signal by Jointly Using Bayesian Estimation and Fuzzy Theory

Speech recognition systems have been applied to inspection and maintenance operations in industrial factories to recording and reporting routines at construction sites, etc. where hand-writing is difficult. In these actual circumstances, some countermeasure methods for surrounding noise are indispensable. In this study, a new method to remove the noise for actual speech signal was proposed by using Bayesian estimation with the aid of bone-conducted speech and fuzzy theory. More specifically, by introducing Bayes’ theorem based on the observation of air-conducted speech contaminated by surrounding background noise, a new type of algorithm for noise removal was theoretically derived. In the proposed noise suppression method, bone-conducted speech signal with the reduced high-frequency components was regarded as fuzzy observation data, and a stochastic model for the bone-conducted speech was derived by applying the probability measure of fuzzy events. The proposed method was applied to speech signals measured in real environment with low SNR, and better results were obtained than an algorithm based on observation of only air-conducted speech.

Bayesian Analysis of Simple Random Densities

A tractable nonparametric prior over densities is introduced which is closed under sampling and exhibits proper posterior asymptotics.