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We propose a procedure to undertake Bayesian variable selection and model averaging for a series of regressions located on a lattice. For those regressors that are in common in the regressions, we consider using an Ising prior to smooth spatially the indicator variables representing whether or not the variable is zero or nonzero in each regression. This smooths spatially the probabilities that each independent variable is nonzero in each regression and indirectly smooths spatially the regression coefficients. We discuss how single-site sampling schemes can be used to evaluate the joint posterior distribution. The approach is applied to the problem of functional magnetic resonance imaging in medical statistics, where massive datasets arise that require prompt processing. Here the Ising prior with a three-dimensional neighborhood structure is used to smooth spatially activation maps from regression models of blood oxygenation. The Ising prior also has the advantage of allowing incorporation of anatomic prior information through the external field. Using a visual experiment, we show how a single-site sampling scheme can provide rapid evaluation of the posterior activation maps and activation amplitudes. The approach is shown to result in maps that are superior to those produced by a recent Bayesian approach using a continuous Markov random field for the activation amplitude.KEY WORDS: Binary Markov random field; Human brain mapping; Ising prior; Markov chain Monte Carlo; Model averaging.

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The use of coupons delivered by mobile phone, so called "m-coupons," is growing rapidly. In this study we analyze consumer response to m-coupons for a two-year trial at a large shopping mall. About 8,500 people were recruited to a panel and then received 3 text message m-coupons whenever they "swiped" their mobile phone at the mall entrances, with downstream redemption recorded. Almost 144,000 m-coupons were delivered during the trial, comprising 38 stores that supplied 134 different coupons. We find that an important feature of m-coupons is where and when they are delivered, with location and time of delivery significantly influencing redemption. Also important is how long they are valid (expiry length), because redemption times for m-coupons are much shorter than for traditional coupons. This suggests that their expiration length should be shortened to help signal time urgency. Nevertheless, traditional coupon features, like face value, still dominate m-coupon effectiveness, as does the product type, with snack food coupons being particularly effective.

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In this research we introduce a new class of multivariate probability models to the marketing literature. Known as "copula models," they have a number of attractive features. First, they permit the combination of any univariate marginal distributions that need not come from the same distributional family. Second, a particular class of copula models, called "elliptical copula," has the property that they increase in complexity at a much slower rate than existing multivariate probability models as the number of dimensions increase. Third, they are very general, encompassing a number of existing multivariate models and providing a framework for generating many more. These advantages give copula models a greater potential for use in empirical analysis than existing probability models used in marketing. We exploit and extend recent developments in Bayesian estimation to propose an approach that allows reliable estimation of elliptical copula models in high dimensions. Rather than focusing on a single marketing problem, we demonstrate the versatility and accuracy of copula models with four examples to show the flexibility of the method. In every case, the copula model either handles a situation that could not be modeled previously or gives improved accuracy compared with prior models.Bayesian estimation, discrete copula, Markov chain Monte Carlo, Gaussian copula, media modeling, probability models, website page views

No abstract

Almost all existing nonlinear multivariate time series models remain linear, conditional on a point in time or latent regime. Here, an alternative is proposed, where nonlinear serial and cross-sectional dependence is captured by a copula model. The copula defines a multivariate time series on the unit cube. A drawable vine copula is employed, along with a factorization which allows the marginal and transitional densities of the time series to be expressed analytically. The factorization also provides for simple conditions under which the series is stationary and/or Markov, as well as being parsimonious. A parallel algorithm for computing the likelihood is proposed, along with a Bayesian approach for computing inference based on model averages over parsimonious representations of the vine copula. The model average estimates are shown to be more accurate in a simulation study. Two five-dimensional time series from the Australian electricity market are examined. In both examples, the fitted copula captures substantial asymmetric tail dependence, both over time and between elements in the series.