SUMMARYMany models for the study of point-referenced data explicitly introduce spatial random effects to capture residual spatial association. These spatial effects are customarily modelled as a zeromean stationary Gaussian process. The spatial Dirichlet process introduced by Gelfand et al. (2005) produces a random spatial process which is neither Gaussian nor stationary. Rather, it varies about a process that is assumed to be stationary and Gaussian. The spatial Dirichlet process arises as a probability-weighted collection of random surfaces. This can be limiting for modelling and inferential purposes since it insists that a process realization must be one of these surfaces. We introduce a random distribution for the spatial effects that allows different surface selection at different sites. Moreover, we can specify the model so that the marginal distribution of the effect at each site still comes from a Dirichlet process. The development is offered constructively, providing a multivariate extension of the stick-breaking representation of the weights. We then introduce mixing using this generalized spatial Dirichlet process. We illustrate with a simulated dataset of independent replications and note that we can embed the generalized process within a dynamic model specification to eliminate the independence assumption.

T his paper studies the effects of various types of online advertisements on purchase conversion by capturing the dynamic interactions among advertisement clicks themselves. It is motivated by the observation that certain advertisement clicks may not result in immediate purchases, but they stimulate subsequent clicks on other advertisements, which then lead to purchases. We develop a novel model based on mutually exciting point processes, which consider advertisement clicks and purchases as dependent random events in continuous time. We incorporate individual random effects to account for consumer heterogeneity and cast the model in the Bayesian hierarchical framework. We construct conversion probability to properly evaluate the conversion effects of online advertisements. We develop simulation algorithms for mutually exciting point processes to compute the conversion probability and for out-of-sample prediction. Model comparison results show the proposed model outperforms the benchmark models that ignore exciting effects among advertisement clicks. Using a proprietary data set, we find that display advertisements have relatively low direct effect on purchase conversion, but they are more likely to stimulate subsequent visits through other advertisement formats. We show that the commonly used measure of conversion rate is biased in favor of search advertisements and underestimates the conversion effect of display advertisements the most. Our model also furnishes a useful tool to predict future purchases and advertisement clicks for the purpose of targeted marketing and customer relationship management.

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Disease incidence or mortality data are typically available as rates or counts for specified regions, collected over time. We propose Bayesian nonparametric spatial modeling approaches to analyze such data. We develop a hierarchical specification using spatial random effects modeled with a Dirichlet process prior. The Dirichlet process is centered around a multivariate normal distribution. This latter distribution arises from a log-Gaussian process model that provides a latent incidence rate surface, followed by block averaging to the areal units determined by the regions in the study. With regard to the resulting posterior predictive inference, the modeling approach is shown to be equivalent to an approach based on block averaging of a spatial Dirichlet process to obtain a prior probability model for the finite dimensional distribution of the spatial random effects. We introduce a dynamic formulation for the spatial random effects to extend the model to spatio-temporal settings. Posterior inference is implemented through Gibbs sampling. We illustrate the methodology with simulated data as well as with a data set on lung cancer incidences for all 88 counties in the state of Ohio over an observation period of 21 years.

C rowdfunding, a peer-to-peer fundraising mechanism, solicits capital from individual backers to support entrepreneurial projects. Entrepreneurs set a funding target and deadline; the project will be funded only if it reaches this funding target by the deadline. Backers individually decide whether to contribute, but their total contributions collectively determine whether the project will be successfully funded. This study models the dynamics of backers' contributions in the presence of success uncertainty and analyzes managerial promotion strategies to maximize the likelihood of funding success. Two opposing forces affect backer decisions: backers are more likely to back a project that has already reached a greater fraction of its funding goal (positive externalities), but the backing propensity declines over time (negative deadline effects). These two competing forces give rise to a time-dependent critical threshold of funding that a project must attain to achieve successful funding. We evaluate actionable promotion strategies (when to promote the project and how much promotion effort to spend) for entrepreneurs to dynamically manage their crowdfunding campaigns.