ANOVA for Some Non-Normal Data by Inverting Reliability Estimators

Standard analysis of variance assumes observations are normally distributed within groups. This paper develops some analysis of variance tests for data which are Bernoulli, Poisson, exponential, or geometric distributed within groups. The tests are shown in Table 1. For natural exponential family data with conjugate priors for the distribution of means, reliability estimators directly estimate the posterior shrinkage. Using the linear posterior expectation induced by conjugate prior, a method is developed to construct an analysis of variance test by determining an appropriate transformation of a reliability estimator. The sampling distribution of the transformed reliability estimator under the assumption of group mean equality is derived to construct an appropriate test statistic. This method is used to invert the generalized KR21 estimators of Foster (2021) for some non-normal data, and it is also shown that the standard analysis of variance F-test statistic can be transformed into a consistent reliability estimator under the same assumptions. A limited simulation study shows that the inverted KR21 test has, in some scenarios, higher power than a standard analysis of variance or a generalized linear model analysis of variance.

Lucky or clever? From expectations to responsibility judgments

How do people hold others responsible for the consequences of their actions? We propose a computational model that attributes responsibility as a function of what the observed action reveals about the person, and the causal role that the person's action played in bringing about the outcome. The model first infers what type of person someone is from having observed their action. It then compares a prior expectation of how a person would behave with a posterior expectation after having observed the person's action. The model predicts that a person is blamed for negative outcomes to the extent that the posterior expectation is lower than the prior, and credited for positive outcomes if the posterior is greater than the prior. We model the causal role of a person's action by using a counterfactual model that considers how close the action was to having been pivotal for the outcome. The model captures participants' responsibility judgments to a high degree of quantitative accuracy across three experiments that cover a range of different situations. It also solves an existing puzzle in the literature on the relationship between action expectations and responsibility judgments. Whether an unexpected action yields more or less credit depends on whether the action was diagnostic for good or bad future performance.

Bayesian optimization of generalized data

Direct application of Bayes' theorem to generalized data yields a posterior probability distribution function (PDF) that is a product of a prior PDF of generalized data and a likelihood function, where generalized data consists of model parameters, measured data, and model defect data. The prior PDF of generalized data is defined by prior expectation values and a prior covariance matrix of generalized data that naturally includes covariance between any two components of generalized data. A set of constraints imposed on the posterior expectation values and covariances of generalized data via a given model is formally solved by the method of Lagrange multipliers. Posterior expectation values of the constraints and their covariance matrix are conventionally set to zero, leading to a likelihood function that is a Dirac delta function of the constraining equation. It is shown that setting constraints to values other than zero is analogous to introducing a model defect. Since posterior expectation values of any function of generalized data are integrals of that function over all generalized data weighted by the posterior PDF, all elements of generalized data may be viewed as nuisance parameters marginalized by this integration. One simple form of posterior PDF is obtained when the prior PDF and the likelihood function are normal PDFs. For linear models without a defect this PDF becomes equivalent to constrained least squares (CLS) method, that is, the χ2 minimization method.

Multi-sensor Information Fusion Method Based on BP Neural Network

This study aims to solve the problem of multi-sensor information fusion, which is a key issue in the multi-sensor system development. The main innovation of this study is to propose a novel multi-sensor information fusion algorithm based on back propagation neural network and Bayesian inference. In the proposed algorithm, a triple is defined to represent a probability space; thereafter, the Bayesian inference is used to estimate the posterior expectation. Finally, we construct a simulation environment to test the performance of the proposed algorithm. Experimental results demonstrate that the proposed algorithm can significantly enhance the accuracy of temperature detection after fusing the data obtained from different sensors.

An application of Bayesian QTL mapping to early development in double haploid lines of rainbow trout including environmental effects

A Bayesian model and variable dimensional parameter estimation based on Markov chain Monte Carlo was applied to map quantitative trait loci (QTLs) in a doubled haploid mapping population of rainbow trout. To increase power, the analysis was performed using the multiple-QTL model, which simultaneously accounted for all the environmental and genetic main effects that influence the expression of early development life history traits. By doing so we obtained the posterior estimated effects for the environmental factors as well as the number, positions, and the effects for the QTLs. The analyses revealed QTLs for time at hatching, embryonic length and weight at swim-up stage. The posterior expectation of the number of QTLs in different linkage groups shows that at least four QTLs are needed to explain the observed differences in early development between the clonal lines. The Bayesian method effectively combined all the information available to accurately position these QTLs in the rainbow trout genome.