Bayesian Computation for High-Dimensional Continuous & Sparse Count Data
Probabilistic modeling of multidimensional data is a common problem in practice. When the data is continuous, one common approach is to suppose that the observed data are close to a lower-dimensional smooth manifold. There are a rich variety of manifold learning methods available, which allow mapping of data points to the manifold. However, there is a clear lack of probabilistic methods that allow learning of the manifold along with the generative distribution of the observed data. The best attempt is the Gaussian process latent variable model (GP-LVM), but identifiability issues lead to poor performance. We solve these issues by proposing a novel Coulomb repulsive process (Corp) for locations of points on the manifold, inspired by physical models of electrostatic interactions among particles. Combining this process with a GP prior for the mapping function yields a novel electrostatic GP (electroGP) process.
Another popular approach is to suppose that the observed data are closed to one or a union of lower-dimensional linear subspaces. However, popular methods such as probabilistic principal component analysis scale poorly computationally. We introduce a novel empirical Bayesian method that we term geometric density estimation (GEODE), which assumes the data is centered near a low-dimensional linear subspace. We show that, with mild assumptions on the prior, the subspace spanned by the principal axes of the data maximizes the posterior mode. Hence, leveraged on the geometric information of the data, GEODE easily scales to massive dimensional problems. It is also capable of learning the intrinsic dimension via a novel shrinkage prior. Finally we mix GEODE across a dyadic clustering tree to account for nonlinear cases.
When data is discrete, a common strategy is to define a generalized linear model (GLM) for each variable, with dependence in the different variables induced through including multivariate latent variables in the GLMs. The Bayesian inference for these models usually
rely on data augmented Markov chain Monte Carlo (DA-MCMC) method, which has a provable slow mixing rate when the data is imbalanced. For more scalable inference, we proposes Bayesian mosaic, a parallelizable composite posterior, for scalable Bayesian inference on a subclass of the multivariate discrete data models. Sampling is embarrassingly parallel since Bayesian mosaic is a multiplication of component posteriors that can be independently sampled from. Analogous to composite likelihood methods, these component posteriors are based on univariate or bivariate marginal densities. Utilizing the fact that the score functions of these densities are unbiased, we have shown that Bayesian mosaic is consistent and asymptotically normal under mild conditions. Since the evaluation of univariate or bivariate marginal densities could be done via numerical integration, sampling from Bayesian mosaic completely bypasses the traditional data augmented Markov chain Monte Carlo (DA-MCMC) method. Moreover, we have shown that sampling from Bayesian mosaic also has better scalability to large sample size than DA-MCMC.
The performance of the proposed methods and models will be demonstrated via both simulation studies and real world applications.
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