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For a continuous-time Markov chain with finite state space and an observation process with additive Gaussian noise, we consider the problem of designing optimal filters when the measurements of the observation process are available at randomly sampled time instants. We first define the optimal filter in this setting, and derive a recursive expression for it in the form of a continuous-discrete filter. Our main result is oriented at comparing the performance of the proposed filter with the continuous-time counterpart, that is, the classical Wonham filter obtained from continuous observation process. In particular, we show that by taking the sampling process to be a Poisson counter, and increasing the mean sampling rate, the expected value of the posterior conditional distribution of continuous-discrete filter converges to the posterior distribution of a purely continuous Wonham filter.

The rapidly growing amount and diversity of data are confronting us more than ever with the need to make informed predictions under uncertainty. The adverse impacts of climate change and natural hazards also motivate our search for reliable predictions. The range of statistical techniques that geomorphologists use to tackle this challenge has been growing, but rarely involves Bayesian methods. Instead, many geomorphic models rely on estimated averages that largely miss out on the variability of form and process. Yet seemingly fixed estimates of channel heads, sediment rating curves or glacier equilibrium lines, for example, are all prone to uncertainties. Neighbouring scientific disciplines such as physics, hydrology or ecology have readily embraced Bayesian methods to fully capture and better explain such uncertainties, as the necessary computational tools have advanced greatly. The aim of this article is to introduce the Bayesian toolkit to scientists concerned with Earth surface processes and landforms, and to show how geomorphic models might benefit from probabilistic concepts. I briefly review the use of Bayesian reasoning in geomorphology, and outline the corresponding variants of regression and classification in several worked examples. © 2020 The Authors. Earth Surface Processes and Landforms published by John Wiley & Sons Ltd

Establishing an accurate diagnosis is crucial in everyday clinical practice. It forms the starting point for clinical decision-making, for instance regarding treatment options or further testing. In this context, clinicians have to deal with probabilities (instead of certainties) that are often hard to quantify. During the diagnostic process, clinicians move from the probability of disease before testing (prior or pretest probability) to the probability of disease after testing (posterior or posttest probability) based on the results of one or more diagnostic tests. This reasoning in probabilities is reflected by a statistical theorem that has an important application in diagnosis: Bayes' rule. A basic understanding of the use of Bayes' rule in diagnosis is pivotal for clinicians. This rule shows how both the prior probability (also called prevalence) and the measurement properties of diagnostic tests (sensitivity and specificity) are crucial determinants of the posterior probability of disease (predictive value), on the basis of which clinical decisions are made. This article provides a simple explanation of the interpretation and use of Bayes' rule in diagnosis. (C) 2020 The Author(s). Published by Elsevier Inc.

This tutorial uses publicly available data from drug makers and the Food and Drug Administration to guide learners to estimate the confidence intervals of COVID-19 vaccine efficacy rates with a Bayesian framework. Under the classical approach, there is no probability associated with a parameter, and the meaning of confidence intervals can be misconstrued by inexperienced students. With Bayesian statistics, one can find the posterior probability distribution of an unknown parameter, and state the probability of vaccine efficacy rate, which makes the communication of uncertainty more flexible. We use a hypothetical example and a real baseball example to guide readers to learn the beta-binomial model before analyzing the clinical trial data. This note is designed to be accessible for lower-level college students with elementary statistics and elementary algebra skills. © 2021, National Numeracy Network. All rights reserved.

In late November 2020, there was a flurry of media coverage of two companies’ claims of 95% efficacy rates of newly developed COVID-19 vaccines, but information about the confidence interval was not reported. This paper presents a way of teaching the concept of hypothesis testing and the construction of confidence intervals using numbers announced by the drug makers Pfizer and Moderna publicized by the media. Instead of a two-sample test or more complicated statistical models, we use the elementary one-proportion z-test to analyze the data. The method is designed to be accessible for students who have only taken a one-semester elementary statistics course. We will justify the use of a z-distribution as an approximation for the confidence interval of the efficacy rate. Bayes’s rule will be applied to relate the probability of being in the vaccine group among the volunteers who were infected by COVID-19 to the more consequential probability of being infected by COVID-19 given that the person is vaccinated.

A sense of agency (SoA) is the experience of subjective awareness regarding the control of one’s actions. Humans have a natural tendency to generate prediction models of the environment and adapt their models according to changes in the environment. The SoA is associated with the degree of the adaptation of the prediction models, e.g., insufficient adaptation causes low predictability and lowers the SoA over the environment. Thus, identifying the mechanisms behind the adaptation process of a prediction model related to the SoA is essential for understanding the generative process of the SoA. In the first half of the current study, we constructed a mathematical model in which the SoA represents a likelihood value for a given observation (sensory feedback) in a prediction model of the environment and in which the prediction model is updated according to the likelihood value. From our mathematical model, we theoretically derived a testable hypothesis that the prediction model is updated according to a Bayesian rule or a stochastic gradient. In the second half of our study, we focused on the experimental examination of this hypothesis. In our experiment, human subjects were repeatedly asked to observe a moving square on a computer screen and press a button after a beep sound. The button press resulted in an abrupt jump of the moving square on the screen. Experiencing the various stochastic time intervals between the action execution (button-press) and the consequent event (square jumping) caused gradual changes in the subjects’ degree of their SoA. By comparing the above theoretical hypothesis with the experimental results, we concluded that the update (adaptation) rule of the prediction model based on the SoA is better described by a Bayesian update than by a stochastic gradient descent.

Introduction. The applicability of mathematical scaling and normalization in solving various applied problems is analyzed. The best known formulas often used along the theoretical and experimental studies are considered. The purpose of this work is to identify the properties of mathematical scaling and rationing. Materials and Methods. The errors obtained under using the mathematical scaling and normalization formulas are considered via specific computational examples. Based on a comparative evaluation of the ratio of the degree of magnitude of the initial and resulting values (as well as the ratio of the degree of difference of these values), the correctness of the results obtained which significantly effects the final values is estimated. Research Results. The analysis leads to the conclusion that some known mathematical scaling and normalization formulas possess properties that are ignored in theory and practice. Discussion and Conclusions. The results obtained allow avoiding erroneous decisions caused by the use of invalid scaling and normalization formulas under solving problems in theory and practice of economics, administrative management, medicine, and plenty of other fields.

One-sided Bayes factors are obtained for the association in 2 by 2 tables, under three different sampling designs. These Bayes factors are relevant when one has a prior believe about the direction of an association. It is further shown how prior knowledge can be incorporated in determining the support of a positive association against the hypothesis of row-column independence.

Empirical Bayes estimates of the local false discovery rate can reflect uncertainty about the estimated prior by supplementing their Bayesian posterior probabilities with confidence levels as posterior probabilities. This use of coherent fiducial inference with hierarchical models generates set estimators that propagate uncertainty to varying degrees. Some of the set estimates approach estimates from plug-in empirical Bayes methods for high numbers of comparisons and can come close to the usual confidence sets given a sufficiently low number of comparisons.

In this paper we study various reliability properties of a Weibull inverse exponential distribution. The maximum likelihood and Bayes estimates of unknown parameters and reliability characteristics are obtained. Bayes estimates are obtained with respect to the squared error loss function under proper and improper prior situations. We use the Lindley method and the Metropolis–Hastings algorithm to compute the Bayes estimates. Interval estimation is also considered. Asymptotic and highest posterior density intervals of unknown parameters are constructed in this respect. We perform a numerical study to compare the performance of all methods and obtain comments based on this study. We also analyze two real data sets for illustration purposes. Finally a conclusion is presented.

The problem of estimation of the shape parameter in a generalized half-logistic distribution for progressively type-II censored samples is of interest in reliability and survival analysis. In this paper, Bayesian methods of estimation based on quadratic and Linex loss functions are proposed. Closed expressions and approximations are obtained for the Bayes and posterior risks of Bayes estimators. These results allow us to assess the performance of estimators obtained under the previously cited loss functions. An application to a real dataset and a simulation study are included where the importance of the different features involved in the progressive type-II censoring scheme is also shown.

We propose the power-log loss function plotted in Figure 1 for the restricted parameter space (0,1), which satisfies all the six properties listed in Table 1 for a good loss function on (0,1). In particular, the power-log loss function penalizes gross overestimation and gross underestimation equally, is convex in its argument, and attains its global minimum at the true unknown parameter. The power-log loss function on (0,1) is an analog of the power-log loss function on (0,∞), which is the popular Stein's loss function. We then calculate the Bayes rule (estimator) of the parameter in (0,1) under the power-log loss function, the posterior expected power-log loss (PEPLL) at the Bayes estimator, and the integrated risk under the power-log loss (IRPLL) at the Bayes estimator, which is also the Bayes risk under the power-log loss (BRPLL). We also calculate the usual Bayes estimator under the squared error loss, which has been proved to be larger than that under the power-log loss. Next, we analytically...

In this work, the problem of transformation and simultaneous variable selection is thoroughly treated via objective Bayesian approaches by the use of default Bayes factor variants. Four uniparametric families of transformations (Box–Cox, Modulus, Yeo-Johnson and Dual), denoted by T, are evaluated and compared. The subjective prior elicitation for the transformation parameter $$\lambda _T$$ , for each T, is not a straightforward task. Additionally, little prior information for $$\lambda _T$$ is expected to be available, and therefore, an objective method is required. The intrinsic Bayes factors and the fractional Bayes factors allow us to incorporate default improper priors for $$\lambda _T$$ . We study the behaviour of each approach using a simulated reference example as well as two real-life examples.

Tests with binary outcomes (e.g., positive versus negative) to indicate a binary state of nature (e.g., disease agent present versus absent) are common. These tests are rarely perfect: chances of a false positive and a false negative always exist. Imperfect results cannot be directly used to infer the true state of the nature; information about the method's uncertainty (i.e., the two error rates and our knowledge of the subject) must be properly accounted for before an imperfect result can be made informative. We discuss statistical methods for incorporating the uncertain information under two scenarios, based on the purpose of conducting a test: inference about the subject under test and inference about the population represented by test subjects. The results are applicable to almost all tests. The importance of properly interpreting results from imperfect tests is universal, although how to handle the uncertainty is inevitably case-specific. The statistical considerations not only will change the way we interpret test results, but also how we plan and carry out tests that are known to be imperfect. Using a numerical example, we illustrate the post-test steps necessary for making the imperfect test results meaningful., Conditional probability; False negative; Uncertainty; False positive; Bayes' rule; Statistics; Environmental assessment; Environmental risk assessment; Bioinformatics; Epidemiology

The development of unconventional plays tends to unfold in many stages, each of which involves incremental investment and a reduction in the geological uncertainty of the reservoir. These two characteristics yield a large decision space where future decisions are optimised based on the near-continuous arrival of new information. This managerial flexibility can be exploited by operators during the development of unconventional plays. We introduce a methodology that demonstrates how value can be created by a staged and partial development of a shale play that would have been unprofitable if fully developed. Compared to existing methods the novel methodology is more consistent with the characteristics of how plays are currently developed as existing methods assume that upon a successful appraisal stage a play is developed in its entirety in a single development phase. As more data become available after each development phase of the play, the potential of the remaining undrilled locations is updated using Bayes's rule. The method is couched in geostatistical principles, combined with an algorithm that allows for a continuous optimisation of drilling targets. An example of a shale gas project has been investigated that consists of 225 possible drilling targets each containing 10 well locations. A maximum of 200 wells can be produced by drilling 20 of the 225 targets. The mean performance of the well population is uncertain. The scenario with the highest mean well performance yields a value of −160 MM USD and the expected project value across all scenarios of mean well performance equals −920 MM USD, given that all 200 wells are drilled at randomly chosen drilling targets. In the model presented in this study the resource can be developed in up to 19 stages upon completion of an appraisal programme. After each development stage an assessment is made where, and if, the next batch of 10 wells should be drilled. This strategy of stage-wise development yields an expected value of 49.2 MM USD. The spatial dependency of well performance enables the algorithm to restrict the development of the play to the most prolific areas. The appraisal programme provides a view on the variability of well performance across the play. A trade off exists between the size, and the consequential accuracy, of the appraisal programme and the cost of appraising. The example illustrates that the expected project value increases from 33.3 MM USD for an appraisal programme in which two locations were appraised, to a maximum of 49.2 MM USD after the appraisal of four locations, and subsequently decreases to 29.0 MM USD after having appraised eight locations. The assumptions around the variability of Estimated Ultimate Recovery (EUR) used in our example are informed by data from 10,000 horizontal wells located in the Mississippian Barnett Shale in the Fort Worth basin in Texas.

Bayes linear kinematics and Bayes linear Bayes graphical models provide an extension of Bayes linear methods so that full conditional updates may be combined with Bayes linear belief adjustment. In this paper we investigate the application of this approach to survival analysis with time-dependent covariate effects, a more complicated problem than previous applications. We use a piecewise-constant hazard function with a prior in which covariate effects are correlated over time. The need for computationally intensive methods is avoided and the relatively simple structure facilitates interpretation. Our approach eliminates the problem of non-commutativity which was observed in earlier work by Gamerman. We apply the technique to data on survival times for leukemia patients. We demonstrate the use Bayes linear kinematics in a more complicated problem.We use a realistically complicated case study in semi-parametric survival analysis.The previously-observed problem of non-commutativity is overcome.Attention is given to the specification of prior beliefs and its effects.

In Bayesian model selection or testing problems, default priors are typically improper; that is, the resulting Bayes factor is not well defined. To circumvent this problem, two methodologies, namely, intrinsic and fractional Bayes factors are proposed and developed. Further, these two Bayes factors are asymptotically equivalent to the ordinary Bayes factors computed with proper priors called intrinsic priors. However, it seems that there are some necessary conditions to satisfy asymptotic equivalence. Such conditions are derived and justified in this article and illustrative examples are provided. Simulations are performed to demonstrate the results.

We consider Khamis' (1960) Laguerre expansion with gamma weight function as a class of “near-gamma” priors (K-prior) to obtain the Bayes predictor of a finite population mean under the Poisson regression superpopulation model using Zellner's balanced loss function (BLF). Kullback–Leibler (K-L) distance between gamma and some K-priors is tabulated to examine the quantitative prior robustness. Some numerical investigations are also conducted to illustrate the effects of a change in skewness and/or kurtosis on the Bayes predictor and the corresponding minimal Bayes predictive expected loss (MBPEL). Loss robustness with respect to the class of BLFs is also examined in terms of relative savings loss (RSL).

Much of science is (rightly or wrongly) driven by hypothesis testing. Even in situations where the hypothesis testing paradigm is correct, the common practice of basing inferences solely on p-values has been under intense criticism for over 50 years. We propose, as an alternative, the use of the odds of a correct rejection of the null hypothesis to incorrect rejection. Both pre-experimental versions (involving the power and Type I error) and post-experimental versions (depending on the actual data) are considered. Implementations are provided that range from depending only on the p-value to consideration of full Bayesian analysis. A surprise is that all implementations -- even the full Bayesian analysis -- have complete frequentist justification. Versions of our proposal can be implemented that require only minor modifications to existing practices yet overcome some of their most severe shortcomings., Comment: 62-00, 62-01, 62A01

The Bayesian estimation of unknown parameters of shifted Gompertz mixture model under type-I censorship, assuming both the informative and noninformative priors is investigated using different loss functions to obtain the Bayes estimates. It is seen that the closed-form expressions for the Bayes estimators cannot be obtained under shape and scale parameters. Some properties of the model with some graphs of the mixture density are discussed. These properties include Bayes estimators, posterior risks and reliability function under simulation scheme. The prior belief of the mixture model is represented by the uniform, and gamma prior. Bayes estimates are obtained considering two cases: (a) when the shape parameter is known and (b) when all parameters are unknown. A detailed simulation study is carried out to investigate the performance of the proposed set of estimators of the mixture model parameters. Posterior risks of the Bayes estimators are evaluated and compared to explore the effect of prior be...

Formal probabilistic models for common cognitive problems. How can an incomplete and uncertain model of the environment be used to perceive, infer, decide and act efficiently ? This is the challenge that both living and artificial cognitive systems have to face. Symbolic logic is, by its nature, unable to deal with this question. The subjectivist approach to probability is an extension to logic that is designed specifically to face this challenge. In this paper, we review a number of frequently encountered cognitive issues and cast them into a common Bayesian formalism. The concepts we review are ambiguities, fusion, multimodality, conflicts, modularity, hierarchies and loops. First, each of these concepts is introduced briefly using some examples from the neuroscience, psychophysics or robotics literature. Then, the concept is formalized using a template Bayesian model. The assumptions and common features of these models, as well as their major differences, are outlined and discussed., Comment un modèle incomplet et incertain de l’environnement peut-il être utilisé pour décider, agir, apprendre, raisonner et percevoir efficacement ? Voici le défi central que les systèmes cognitifs tant naturels qu’artificiels doivent résoudre. La logique, de par sa nature même, faite de certitudes et ne laissant aucune place au doute, est incapable de répondre à cette question. L’approche subjectiviste des probabilités est une extension de la logique conçue pour pallier ce manque. Dans cet article, nous passons en revue un ensemble de problèmes cognitifs usuels et nous montrons comment les formuler et les résoudre avec un formalisme probabiliste unique. Les concepts abordés sont : l’ambigüité, la fusion, la multi-modalité, les conflits, la modularité, les hiérarchies et les boucles. Chacune de ces questions est tout d’abord brièvement présentée à partir d’exemples venant des neurosciences, de la psychophysique ou de la robotique. Ensuite, le concept est formalisé en utilisant un modèle générique bayésien. Enfin, les hypothèses, les points communs et les différences de chacun de ces modèles sont analysés et discutés., Bessière Pierre, Diard Julien, Colas Francis. Modèles probabilistes formels pour problèmes cognitifs usuels. In: Intellectica. Revue de l'Association pour la Recherche Cognitive, n°65, 2016/1. Nouvelles approches en Robotique Cognitive. pp. 111-141.

This paper derives Bayes shrinkage estimator of Rayleigh parameter and its associated risk based on conjugate prior under the assumption of general entropy loss function for progressive type-II censored data. Risk function of maximum likelihood estimate, Bayes estimate and Bayes shrinkage estimate have also been derived and compared. A procedure has been suggested to include a guess value in case of the Bayes shrinkage estimation. Risk function of empirical Bayes estimate and empirical Bayes shrinkage estimate have also been derived and compared. In conclusion, an illustrative example is presented to assess how the Rayleigh distribution fits a real data set.

This paper deals with the problem of testing on the equality of the scale parameters in the log-logistic distributions. We propose default Bayesian testing procedures for the scale parameters under the reference priors. The reference prior is usually improper which yields a calibration problem that makes the Bayes factor to be dened up to a multiplicative constant. Therefore, we propose the default Bayesian testing procedures based on the fractional Bayes factor and the intrinsic Bayes factor under the reference priors. To justify proposed procedures, a simulation study is provided and also, an example is given.

In this paper, we suggest the Bayes linear estimator (BLE) for randomized response model (RRM) to improve the efficiency of RR estimators, only using the first and second prior moments. The randomized response model is an indirect questioning technique used to protect the privacy of respondents in a survey regarding a sensitive characteristic. Meanwhile Bayes linear estimation is useful for parameter estimation compared to the typical Bayesian method because it only uses the first and second prior knowledge of the variable of interest. Also, it has an advantage of robustness with the distribution. We suggest the Bayes linear estimators for the two-stage and the stratified RRM and find the optimal sample size to minimize the Bayes risk for the stratified RRM. Also, we show the difference in efficiency between the Bayes linear estimators and the typical non-Bayesian RR estimators by simulation study.

I emphasize some common misuses of statistics that everyone, whether you do statistics or just read what others write, should be on the lookout for. I next discuss somewhat more complicated issues in archaeological method and theory and then conclude with a qualitative explanation of Bayesian methods and why they are often preferable to the frequentist methods advocated in many introductory statistics texts.

Bayes theorem is the normative method for revising probability forecasts using new information. However, for unaided forecasters its application can be difficult, effortful, opaque and even counter-intuitive. The study proposes two simple heuristics for approximating Bayes formula while yielding accurate decisions. Their performance was assessed where a decision is made on which of two events is most probable and where a choice is made between an option yielding an intermediate utility for something that is certain or for a gamble which will result in either a worse or better utility (“certainty or risk” decisions). For “most probable event” decisions the first heuristic always results in the correct decision when the reliability of the new information does not depend on which event will occur. In other cases, the second heuristic typically led to the correct decision for about 95% of “most probable event” decisions and 86% of “certainty or risk” decisions.

The Bayes factor for the integral priors in the one-way random effects model has recently been developed by Cano et al. (2007). We establish the consistency of this Bayes factor when the number of groups goes to infinity, when the number of observations per group goes to infinity, and when both go to infinity.

Summary This paper presents a comparison of three approaches to the teaching of probability to demonstrate how the truth table of elementary mathematical logic can be used to teach the calculations of conditional probabilities. Students are typically introduced to the topic of conditional probabilities—especially the ones that involve Bayes' rule—with the help of such traditional approaches as formula use or conversion to natural frequencies. The truth table approach is an alternative method for explaining the concept and calculation procedure of conditional probability and Bayes' rule.

A large literature suggests that many individuals do not apply Bayes’ Rule when making decisions that depend on them correctly pooling prior information and sample data. We replicate and extend a classic experimental study of Bayesian updating from psychology, employing the methods of experimental economics, with careful controls for the confounding effects of risk aversion. Our results show that risk aversion significantly alters inferences on deviations from Bayes’ Rule.

How much more likely is a positive test to be found in a patient with a pathological condition than in a patient without it? How much more likely is a negative test to be found in a patient without...

The problem of selecting a population according to “selection and ranking” is an important statistical problem. The ideas in selecting the best populations with some demands having optimal criterion have been suggested originally by Bechhofer (1954) and Gupta (1956, 1965). In the area of ranking and selection, the large part of literature is connected with a single criterion. However, this may not satisfy the experimenter’s demand. We follow methodology of Huang and Lai (1999) and the main focus of this article is to select a best population under Type-II progressively censored data for the case of right tail exponential distributions with a bounded and unbounded supports for μi. We formulate the problem and develop a Bayesian setup with two kinds of bounded and unbounded prior for μi. We introduce an empirical Bayes procedure and study the large sample behavior of the proposed rule. It is shown that the proposed empirical Bayes selection rule is asymptotically optimal.

The article suggests an approach to determine structural elements technical condition, based on the mathematical probabilistic apparatus of technical diagnostics. Diagnostics are performed using probabilistic methods of complex technical systems conditions recognition. Probabilistic parameters are calculated according to Bayes’s rule. The paper shows a diagnostics example of intermediate floor elements and systems in the old urban development building. Both the suggested method and information theory methods are used during diagnostics.

This study considers the estimation problem for the parameter and reliability function of Generalized Half logistic distribution under record Data. We use the maximum likelihood and Bayesian procedures to obtain the estimators of parameter and reliability function of Generalized Half logistic distribution. We also obtain the Empirical Bayesian Estimators for the parameter and reliability function of Generalized Half logistic distribution and considered the problem of predicting future record in a Bayesian and Empirical Bayesian approaches. Comparisons are made between the different estimators based on a simulation study.

In this paper, we develop Bayesian inference of the finite population mean with the assumption of posterior linearity rather than normality of the superpopulation in the presence of auxiliary information under the balanced loss function. We compare the performance of the optimal Bayes estimator under the balanced loss function with ones of the classical ratio estimator and the usual Bayes estimator in terms of the posterior expected losses, risks and Bayes risks.

In this article, we study the problem of selecting the best population from among several exponential populations based on interval censored samples using a Bayesian approach. A Bayes selection procedure and a curtailed Bayes selection procedure are derived. We show that these two Bayes selection procedures are equivalent. A numerical example is provided to illustrate the application of the two selection procedure. We also use Monte Carlo simulation to study performance of the two selection procedures. The numerical results of the simulation study demonstrate that the curtailed Bayes selection procedure has good performance because it can substantially reduce the duration time of life test experiment.

In this paper, we consider the Bayesian approach to the model selection problem for nested linear regression models. Common Bayesian procedures to this problem are based on Zellner's g-prior with a hyper-prior for the scaling factor g. Maruyama and George (2011) recently adopted this procedure with the beta-prime distribution for g and derived an explicit closed-form Bayes factor without integral representation which is thus easy to compute. In addition, they have studied its corresponding model selection consistency for fixed number of parameters. Over recent years, linear regression models with a growing number of unknown parameters have gained increased popularity in practical applications, such as the clustering problem. This observation motivates us to further investigate the consistency of Bayes factor with the beta-prime distribution for g under a scenario in which the number of parameters increases with the sample size. Finally, the results presented here are compared with the ones for the Bayes factor under intrinsic priors in relevant literature.

A Bayesian network (or a belief network) is a probabilistic graphical model that represents a set of variables and their probabilistic independencies. Some researches often involve continuous random variables. In order to apply these continuous variables to BN models, these variables should convert into discrete variables with limited states, often two. During the discretization process, one problem that researchers faced is to decide the number of states for discretization. Does the number of states chosen for discretization impact models’ power? In this study, this issue is examined empirically. The study examines this issue in the financial distress prediction field. The sample consists of 144 firms listed in Tehran stock exchange from 1997 to 2007. In order to develop Naive Bayes models, two methods for choosing variables were used. The first method is based upon conditional correlation between variables and the second method is based upon conditional likelihood. The accuracy in predicting financial distress of the first naive Bayes model's performance that is based upon conditional correlation is 90% and the accuracy of the second naive Bayes model is 93%. Collectively, the results showed that the performance of the second naive Bayes model that based upon conditional likelihood is better than the first one. Further analyses also showed that the number of states chosen for discretization has effect on models’ performance. In comparing the model's performance when continuous variables are discretized into two, three, four and five states, the results showed that the naive Bayes model's performance increases when the number of states for discretization increases from two to three, and from three to four but when the number of states increases from four to five the model's performance decreased.

A fundamental challenge for developing a cost-sensitive Naive Bayes method is how to effectively classify an instance based on the cost-sensitive threshold computed under the assumption of knowing the instance's true classification probabilities and the highly biased estimations of these probabilities by the Naive Bayes method. To address this challenge, we develop a cost-sensitive Naive Bayes method from a novel perspective of inferring the order relation (e.g., greater than or equal to, less than) between an instance's true classification probability of belonging to the class of interest and the cost-sensitive threshold. Our method learns and infers the order relation from the training data and classifies the instance based on the inferred order relation. We empirically show that our proposed method significantly outperforms major existing methods for turning Naive Bayes cost-sensitive through experiments with UCI data sets and a real-world case study.

We examine the large sample properties of Bayes procedures in a general framework, where data may be dependent and models may be misspecified and nonsmooth. The posterior distribution of parameters is shown to be asymptotically normal, centered at the quasi maximum likelihood estimator, under mild conditions. In this framework, the Bayes factor for the test problem of Davies (1997, 1987), where a parameter is unidentified under the null hypothesis, is analyzed. The probability that the Bayes factor leads to a correct conclusion about the hypotheses in Davies’ problem is shown to approach to one.

In this study, the theory of statistical kernel density estimation has been applied for deriving non-parametric kernel prior to the empirical Bayes which frees the Bayesian inference from subjectivity that has worried some statisticians. For comparing the empirical Bayes based on the kernel prior with the fully Bayes based on the informative prior, the mean square error and the mean percentage error for the Weibull model parameters are studied based on these approaches under both symmetric and asymmetric loss functions, via Monte Carlo simulations. The results are quite favorable to the empirical Bayes that provides better estimates and outperforms the fully Bayes for different sample sizes and several values of the true parameters. Finally, a numerical example is given to demonstrate the efficiency of the empirical Bayes.

We use Bayes’ estimator of a consumer-surplus probit model to study the relevance of the prior in a discrete choice model. We take random subsamples of varying sizes of stated preference data regarding ultra-low emission vehicle purchases in California and focus on the willingness-to-pay for improvements in driving range. Prior information is obtained from a meta-analysis of consumer valuation of driving range. We find the posterior distribution of the willingness-to-pay using a tight and a weakly informative prior, and also analyze the nonparametric estimates of the posterior compare these with the likelihood function of the problem. It is found that the weight of the prior is relevant for very small samples, but for standard sample sizes the prior vanishes. Thus, the Bayes estimator of a static discrete choice model is in general equivalent to the maximum likelihood estimator, although for some intermediate sample sizes the prior provides more realistic values.

Asymptotic cumulants of the Bayes and pseudo Bayes estimators of ability in item response theory are obtained up to the fourth order with the higher-order asymptotic variance under possible model misspecification. Typical estimators are treated as special cases of the (pseudo) Bayes estimator with the general weight. The asymptotic cumulants of the estimators after studentization are also derived. From the comparison of the mean square errors, the Bayes modal estimator with the standard normal prior is recommended for point estimation. For interval estimation, however, the maximum likelihood estimator is appropriate considering its small bias after studentization.

We show that under the null, the 2log(Bayesfactor) is asymptotically distributed as a weighted sum of chi-squared random variables with a shifted mean. This claim holds for Bayesian multi-linear regression with a family of conjugate priors, namely, the normal-inverse-gamma prior, the g-prior, and the normal prior. Our results have three immediate impacts. First, we can compute analytically a p-value associated with a Bayes factor without the need of permutation. We provide a software package that can evaluate the p-value associated with Bayes factor efficiently and accurately. Second, the null distribution is illuminating to some intrinsic properties of Bayes factor, namely, how Bayes factor quantitatively depends on prior and the genesis of Bartlett’s paradox. Third, enlightened by the null distribution of Bayes factor, we formulate a novel scaled Bayes factor that depends less on the prior and is immune to Bartlett’s paradox. When two tests have an identical p-value, the test with a larger power tends to have a larger scaled Bayes factor, a desirable property that is missing for the (unscaled) Bayes factor. Supplementary materials for this article are available online.

In this engaging article, Zhang and Zhang (2013) (henceforth ZZ) assembled examples from clinical trials to make a compelling case for medical science's need to measure the evidence for composite h...