Bayesian inference is one of the more controversial approaches to statistics. The fundamental objections to Bayesian methods are twofold: on one hand, Bayesian methods are presented as an automatic inference engine, and this raises suspicion in anyone with applied experience. The second objection to Bayes comes from the opposite direction and addresses the subjective strand of Bayesian inference. This article presents a series of objections to Bayesian inference, written in the voice of a hypothetical anti-Bayesian statistician. The article is intended to elicit elaborations and extensions of these and other arguments from non-Bayesians and responses from Bayesians who might have different perspectives on these issues.

A concept map is a data collection tool developed in psychology and education to obtain information about mental representations of concept associations. This methodology has recently been introduced to marketing to study consumers' brand perceptions (John et al. (2006); Joiner (1998)) and attitudes towards health risk (e.g., Huang (1997)). In conjunction with other more established methods (e.g., Multidimensional scaling), concept maps provide an additional valuable tool for researchers to understand consumers' structural knowledge about different important marketing concepts.

Building on the introduction by John et al. (2006), we propose a descriptive probability model of concept map formation, along with concept map analyses based on parameter estimates. In particular, we demonstrate how to test hypotheses about differences between two groups of maps, and how to aggregate across individual concept maps to form a "consensus map." To demonstrate our methodology, we apply our model to a dataset that uses concept maps to study college students' perceptions of Sexually Transmitted Diseases (STDs), an important topic of growing interest in health marketing (e.g., Hill (1988); LaTour and Pitts (1989); Raghubir and Menon (1998); Treise and Weigold (2001)). Though parsimonious in nature, our model adequately recovers map-level, concept-level, and link-level summary statistics commonly considered by other researchers, yet rarely modeled directly.

We introduce a class of shape mixtures of skewed distributions and study some of its main properties. We discuss a Bayesian interpretation and some invariance results of the proposed class. We develop a Bayesian analysis of the skew-normal, skew-generalized-normal, skew-normal-t and skew-t-normal linear regression models under some special prior specifications for the model parameters. In particular, we show that the full posterior of the skew-normal regression model parameters is proper under an arbitrary proper prior for the shape parameter and noninformative prior for the other parameters. We implement a convenient hierarchical representation in order to obtain the corresponding posterior analysis. We illustrate our approach with an application to a real dataset on characteristics of Australian male athletes.

Bayesian methods are presented for categorical sampling when some observations are censored (i.e., suffer missing distinctions between categories). Such problems have been researched over the years, as they can be important in applications. However, previous work has assumed strong restrictions, such as truthful reporting, noninformative censoring, etc.Here, we attempt to remove such restrictions. In particular, we remove two of the three restrictions imposed by Dickey, Jiang and Kanade (1987). We provide Bayesian methods for cases more general than those considered by Paulino and de B. Pereira (1992, 1995), and others. Thus, it will no longer be necessary to make unrealistic assumptions commonly employed regarding the censoring model. A theorem of Identifiability-by-Conditioning is provided, allowing familiar improper prior densities. By this theorem, we obtain identical Bayesian updating results by imposing constraints on either prior, likelihood, or posterior directly. Several computational procedures are suggested, and an example is used to illustrate methods.

This article considers the application of particle filtering to continuous-discrete optimal filtering problems, where the system model is a stochastic differential equation, and noisy measurements of the system are obtained at discrete instances of time. It is shown how the Girsanov theorem can be used for evaluating the likelihood ratios needed in importance sampling. It is also shown how the methodology can be applied to a class of models, where the driving noise process is lower in the dimensionality than the state and thus the laws of the state and the noise are not absolutely continuous. Rao-Blackwellization of conditionally Gaussian models and unknown static parameter models is also considered.

In this paper, we consider theoretical and computational connections between six popular methods for variable subset selection in generalized linear models (GLM's). Under the conjugate priors developed by Chen and Ibrahim (2003) for the generalized linear model, we obtain closed form analytic relationships between the Bayes factor (posterior model probability), the Conditional Predictive Ordinate (CPO), the L measure, the Deviance Information Criterion (DIC), the Aikiake Information Criterion (AIC), and the Bayesian Information Criterion (BIC) in the case of the linear model. Moreover, we examine computational relationships in the model space for these Bayesian methods for an arbitrary GLM under conjugate priors as well as examine the performance of the conjugate priors of Chen and Ibrahim (2003) in Bayesian variable selection. Specifically, we show that once Markov chain Monte Carlo (MCMC) samples are obtained from the full model, the four Bayesian criteria can be simultaneously computed for all possible subset models in the model space. We illustrate our new methodology with a simulation study and a real dataset.

Bayesian belief networks provide estimates of conditional probabilities, called query responses. The precision of these estimates is assessed using posterior distributions. This paper discusses two claims and a conjecture by Kleiter (1996) concerning the exact posterior distribution of queries. The two claims provide conditions where a query has an exact beta distribution. The first claim is clarified by the following generalization. Assuming a BDe prior and complete data, a query has the same distribution under equivalent network structures. If the query can be represented as a network parameter under an equivalent structure, it must then have a beta distribution. Kleiter's second claim is contradicted by a counter-example. His conjecture, concerning finite mixtures of beta distributions, is also disproved.

Map positional error refers to the difference between a feature's coordinate pair on a map and the corresponding true, unknown coordinate pair. In a geographic information system (GIS), this error is propagated through all operations that are functions of position, so that lengths, areas, etc., are uncertain. Often, a map's metadata provides a nominal statement on the positional error of a map, and such information has frequently been used to study the propagation of error through such operations. This article presents a statistical model for map positional error, incorporating positional error metadata as prior information, along with map coordinates, and, in particular, the information contained in the linearity of features. We demonstrate that information in the linearity of features can greatly improve the precision of true location predictions.