![]() ![]() ![]() We test the proposed model on data from both Monte-Carlo simulations and find that the proposed approach produces consistent parameter estimates while it significantly reduces the complexity of the problem. We propose a menu choice model that circumvents these two problems in a feasible and flexible, but parsimonious way. Second, the number of interactions among menu items also grows disproportionately to the number of items in the menu. ![]() First, the choice set (all possible menu selections) grows geometrically with the number of items in the menu. We show that modeling choices out of the typical menu leads to the “curse of dimensionality,” which transpires in two ways. This study focuses on the menus typically found in the marketplace (e.g., restaurants and Internet vendors), where the consumer may choose one or more from dozens of options or menu items, each at a posted price or fee. The numerical results indicate that the new algorithm is particularly useful for the problems for which the underlying system is strongly dependent. The new algorithm is tested on the spatial autologistic and autonormal models. Compared to the approximate exchange algorithms, such as the double Metropolis-Hastings sampler (Liang, 2010), the new algorithm overcomes their theoretical difficulty in convergence. ![]() Compared to the exchange algorithm, the new algorithm removes the requirement that the auxiliary variables must be drawn using a perfect sampler, and thus can be applied to many models for which the perfect sampler is not available or very expensive. The convergence of the algorithm is established under mild conditions. The new algorithm can be viewed as a MCMC extension of the exchange algorithm (Murray, Ghahramani and MacKay, 2006), which generates auxiliary variables via an importance sampling procedure from a Markov chain running in parallel. We propose a new algorithm, adaptive auxiliary variable exchange algorithm, or in short, adaptive exchange (AEX) algorithm, to tackle this problem. Sampling from the posterior distribution for a model whose normalizing constant is intractable is a long-standing problem in statistical research. The posterior predictions show that the model is able to repli-cate the homophily levels and the aggregate clustering of the observed network, in contrast with standard exponential family network models. The estimates detect high levels of racial homophily, and heterogeneity in both costs of links and payoffs from common friends. The posterior distribution of structural parameters and unobserved heterogeneity is estimated with school friendship network data from Add Health, using a Bayesian exchange algorithm. As a consequence the equilibrium networks are sparse and the sufficient statistics of the network concentrate around their mean. The model converges to a hierarchical exponential family random graph, with weak dependence among links. The probability of meeting people in different communities is smaller than the probability of meeting people in the same community, and it decreases with the size of the network. Players meet sequentially and decide whether to form bilateral links, after receiving a random matching shock. Players' payoffs vary by community and depend on the composition of direct links and common neighbors, allowing preferences to have a bias for similar people. Each player belongs to a community unobserved by the econometrician. I develop and estimate a structural model of strategic network formation with heterogeneous players and latent community structure, whose equilibrium networks are sparse and exhibit homophily and clustering. Social networks display homophily and clustering, and are usually sparse. ![]()
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