Conditional Density Estimation via Weighted Logistic Regressions

Compared to the conditional mean as a simple point estimator, the conditional density function is more informative to describe the distributions with multi-modality, asymmetry or heteroskedasticity. In this paper, we propose a novel parametric conditional density estimation method by showing the connection between the general density and the likelihood function of inhomogeneous Poisson process models. The maximum likelihood estimates can be obtained via weighted logistic regressions, and the computation can be significantly relaxed by combining a block-wise alternating maximization scheme and local case-control sampling. We also provide simulation studies for illustration.

On Robust Probabilistic Principal Component Analysis using Multivariate $t$-Distributions

Principal Component Analysis (PCA) is a common multivariate statistical analysis method, and Probabilistic Principal Component Analysis (PPCA) is its probabilistic reformulation under the framework of Gaussian latent variable model. To improve the robustness of PPCA, it has been proposed to change the underlying Gaussian distributions to multivariate $t$-distributions. Based on the representation of $t$-distribution as a scale mixture of Gaussians, a hierarchical model is used for implementation. However, although the robust PPCA methods work reasonably well for some simulation studies and real data, the hierarchical model implemented does not yield the equivalent interpretation. In this paper, we present a set of equivalent relationships between those models, and discuss the performance of robust PPCA methods using different multivariate $t$-distributed structures through several simulation studies. In doing so, we clarify a current misrepresentation in the literature, and make connections between a set of hierarchical models for robust PPCA.