Exact Bayesian inference on state-space models~(SSM) is in general untractable, and unfortunately, basic Sequential Monte Carlo~(SMC) methods do not yield correct approximations for complex models. In this paper, we propose a mixed inference algorithm that computes closed-form solutions using belief propagation as much as possible, and falls back to sampling-based SMC methods when exact computations fail. This algorithm thus implements automatic Rao-Blackwellization and is even exact for Gaussian tree models.

Latent Factor Analysis of Gaussian Distributions under Graphical Constraints Md Mahmudul Hasan, Shuangqing Wei, Ali Moharrer Abstract --We explore the algebraic structure of the solution space of convex optimization problem Constrained Minimum Trace Factor Analysis (CMTF A), when the population covariance matrix Σ x has an additional latent graphical constraint, namely, a latent star topology. In particular, we have shown that CMTF A can have either a rank 1 or a rank n 1 solution and nothing in between. We found explicit conditions for both rank 1 and rank n 1 solutions for CMTF A solution of Σ x. As a basic attempt towards building a more general Gaussian tree, we have found a necessary and a sufficient condition for multiple clusters, each having rank 1 CMTF A solution, to satisfy a minimum probability to combine together to build a Gaussian tree. T o support our analytical findings we have presented some numerical demonstrating the usefulness of the contributions of our work. Index T erms --Factor Analysis, MTF A, CMTF A, CMDF A I. INTRODUCTION A. Motivation Factor Analysis (FA) is a commonly used tool in multivariate statistics to represent the correlation structure of a set of observables in terms of significantly smaller number of variables called "latent factors". With the growing use in data mining, high dimensional data analytics, factor analysis has already become a prolific area of research [1] [2]. Classical factor analysis models seek to decompose the correlation matrix of an n -dimensional random vector X R n, Σ x, as the sum of a diagonal matrix D and a Gramian matrix Σ x D .

The goal of any inference algorithm is to recover the hidden parameters related to those k hidden nodes (k may be unknown). Consider a special subset of graphical models, known as latent Gaussian trees, in which the underlying structure is a tree and the joint density of the variables is captured by a Gaussian density. The Gaussian graphical models are widely studied in the literature because of a direct correspondence between conditional independence relations occurring in the model with zeros in the inverse of covariance matrix, known as the concentration matrix. There are several works such as [1,2] that have proposed efficient algorithms to infer the latent Gaussian tree parameters. In fact, Choi et al., proposed a new recursive grouping (RG) algorithm along with its improved version, i.e., Chow-Liu RG (CLRG) algorithm to recover a latent Gaussian tree that is both structural and risk consistent [1], hence it recovers the correct value for the latent parameters. They introduced a tree metric as the negative log of the absolute value of pairwise correlations to perform the algorithm. Also, Shiers et al., in [3], characterized the correlation space of latent Gaussian trees and showed the necessary and sufficient conditions under which the correlation space represents a particular latent Gaussian tree. Note that the RG algorithm can be directly related to correlation space of latent Gaussian trees in a sense that it recursively checks certain constraints on correlations to converge to a latent tree with true parameters.