A method is presented to estimate the probability distributions of climate system properties based on a hierarchical Bayesian model. At the base of the model, we use simulations of a climate model in which the outputs depend on the climate system properties and can also be compared with observations. The degree to which the model outputs are "consistent" with the observations is used to obtain the likelihood for the climate system properties. We define the climate system properties as those properties of the climate model that control the large-scale response of the climate system to external forcings. In this paper, we use the MIT 2D climate model (MIT2DCM) to provide simulations of ocean, surface and upper atmospheric temperature behavior over zones defined by latitude bands. In the MIT2DCM, the climate system properties can be set via three parameters: Climate sensitivity (the equilibrium surface temperature change in response to a doubling of CO2 concentrations), the rate of deep-ocean heat uptake (as set by the diffusion of temperature anomalies into the deep-ocean below the climatological mixed layer), and net strength of the anthropogenic aerosol forcings. In this work, we use output from MIT2DCM coupled with historical temperature records to make inference about these climate system properties. Even though the MIT2DCM is far less computationally demanding than a full 3D climate model, the task of running the model for each combination of the climate parameters and processing its output is computationally demanding. Thus, a statistical model is required to approximate the model output. We obtain results that are critical for understanding uncertainty in future climate change and provide an independent check that the information contained in recent climate change is robust to statistical treatment.

Sansó, Forest and Zantedeschi (SFZ) have presented an excellent analysis that combines limited computer simulations with historical observations to provide inference about key climate system parameters. The Bayesian formulation used in this analysis allows the incorporation of a number of different sources of uncertainty in the final inference. We appreciate such a collaboration requires a substantial amount of effort from all involved. There is not much to criticize here. Most of our comments, which are motivated by the case study, apply generally to analyses involving the calibration of a physics-based simulation model using physical observations.

This paper analyzes the daily incidence of violence during the Second Intifada. We compare several alternative statistical models with different dynamic and structural stability characteristics while keeping modelling complexity to a minimum by only maintaining the assumption that the process under consideration is at most a second order discrete Markov process. For the pooled data, the best model is one with asymmetric dynamics, where one Israeli and two Palestinian lags determine the conditional probability of violence. However, when we allow for structural change, the evidence strongly favors the hypothesis of structural instability across political regime sub-periods, within which dynamics are generally weak.

The Full Bayesian Significance Test, FBST, is extensively reviewed. Its test statistic, a genuine Bayesian measure of evidence, is discussed in detail. Its behavior in some problems of statistical inference like testing for independence in contingency tables is discussed.

The title poses a deceptively simple question that must be addressed by any statistical model or computational algorithm for the clustering of points. Two distinct interpretations are possible, one connected with the number of clusters in the sample and one with the number in the population. Under suitable conditions, these questions may have essentially the same answer, but it is logically possible for one answer to be finite and the other infinite. This paper reformulates the standard Dirichlet allocation model as a cluster process in such a way that these and related questions can be addressed directly. Our conclusion is that the data are sometimes informative for clustering points in the sample, but they seldom contain much information about parameters such as the number of clusters in the population.

The Bayesian point-null testing problem is studied asymptotically under a high-dimensional normal-means model. A noninformative prior structure is proposed for general problems, and then refined for the specialized contexts of goodness-of-fit testing and functional data analysis. The associated tests are demonstrated on existing data sets and shown to provide a cornerstone for a toolbox of detailed analysis tools. The conceptual approach is to allow the prior null probability to vary with dimension and with prior dispersion parameters, then to guide its parametrization so that the posterior null probability behaves in accordance with Bayesian asymptotic-consistency concepts. Among the theoretical issues studied are the objectivity of setting the prior null probability to one-half, the Jeffreys-Lindley paradox, and the influence of smoothness constraints.

Ronald Fisher blieved that "The theory of inverse probability is founded upon an error, and must be wholy rejected." This note describes how Fisher divided responsibility for the error between Bayes and Laplace. Bayes he admired for formulating the problem, producing a solution and then withholding it; Laplace he blamed for promulgating the theory and for distorting the concept of probability to accommodate the theory. At the end of his life Fisher added a refinement: in the Essay Bayes had anticipated one of Fisher's own fiducial arguments.

For many classification and regression problems, a large number of features are available for possible use --- this is typical of DNA microarray data on gene expression, for example. Often, for computational or other reasons, only a small subset of these features are selected for use in a model, based on some simple measure such as correlation with the response variable. This procedure may introduce an optimistic bias, however, in which the response variable appears to be more predictable than it actually is, because the high correlation of the selected features with the response may be partly or wholly due to chance. We show how this bias can be avoided when using a Bayesian model for the joint distribution of features and response. The crucial insight is that even if we forget the exact values of the unselected features, we should retain, and condition on, the knowledge that their correlation with the response was too small for them to be selected. In this paper we describe how this idea can be implemented for "naive Bayes" models of binary data. Experiments with simulated data confirm that this method avoids bias due to feature selection. We also apply the naive Bayes model to subsets of data relating gene expression to colon cancer, and find that correcting for bias from feature selection does improve predictive performance.

Motivated by a large multilevel survey conducted by the US Veterans Health Administration (VHA), we propose a structural equations model which involves a set of latent variables to capture dependence between different responses, a set of facility level random effects to capture facility heterogeneity and dependence between individuals within the same facility, and a set of covariates to account for individual heterogeneity. Identifiability associated with structural equations modeling is addressed and properties of the proposed model are carefully examined. An effective and practically useful modeling strategy is developed to deal with missing responses and to model missing covariates in the structural equations framework. Markov chain Monte Carlo sampling is used to carry out Bayesian posterior computation. Several variations of the proposed model are considered and compared via the deviance information criterion. A detailed analysis of the VHA all employee survey data is presented to illustrate the proposed methodology.