Here we review the role of the Dirichlet process and related prior distribtions in nonparametric Bayesian inference. We discuss construction and various properties of the Dirichlet process. We then review the asymptotic properties of posterior distributions. Starting with the definition of posterior consistency and examples of inconsistency, we discuss general theorems which lead to consistency...

This paper develops a new representational model of similarity data that combines continuous dimensions with discrete features. An algorithm capable of learning these representations is described, and a Bayesian model selection approach for choosing the appropriate number of dimensions and features is developed. The approach is demonstrated on a classic data set that considers the similarities ...

We compare a number of widely used trend-cycle decompositions of output in a formal Bayesian model comparison exercise. This is motivated by the often markedly different results from these decompositions—different decompositions have broad implications for the relative importance of real versus nominal shocks in explaining variations in output. Using US quarterly real GDP, we find that the over...

This paper provides a brief review of the more popular methods for comparing models in a Bayesian framework. Personal experience in implementing these methods in problems requiring mixture models is also referenced.

We consider situations in Bayesian analysis where the prior is indexed by a hyperparameter taking on a continuum of values. We distinguish some arbitrary value of the hyperparameter, and consider the problem of estimating the Bayes factor for the model indexed by the hyperparameter vs. the model specified by the distinguished point, as the hyperparameter varies. We assume that we have Markov ch...

The choice of the summary statistics in Bayesian inference and in particular in ABC algorithms is paramount to produce a valid outcome. We derive necessary and sufficient conditions on those statistics for the corresponding Bayes factor to be convergent, namely to asymptotically select the true model. Those conditions, which amount to the expectations of the summary statistics to asymptotically...

The classical chi-square test of goodness-of-fit compares the hypothesis that data arise from some parametric family of distributions, against the nonparametric alternative that they arise from some other distribution. However, the chi-square test requires continuous data to be grouped into arbitrary categories. Furthermore, as the test is based upon an approximation, it can only be used if the...

No scientific conclusion follows automatically from a statistically non-significant result, yet people routinely use non-significant results to guide conclusions about the status of theories (or the effectiveness of practices). To know whether a non-significant result counts against a theory, or if it just indicates data insensitivity, researchers must use one of: power, intervals (such as conf...

The main objective of this paper is to find the relation between the adaptive significance level presented here and the sample size. We statisticians know of the inconsistency, or paradox, in the current classical tests of significance that are based on p-value statistics that are compared to the canonical significance levels (10%, 5%, and 1%): “Raise the sample to reject the null hypothesis” i...

Inference using significance testing and Bayes factors is compared and contrasted in five case studies based on real research. The first study illustrates that the methods will often agree, both in motivating researchers to conclude that H1 is supported better than H0 and the other way round, that H0 is better supported than H1. The next four however, show that the methods will also often disag...