Variational Bayes Inference in Digital Receivers

The digital telecommunications receiver is an important context for inference methodology, the key objective being to minimize the expected loss function in recovering the transmitted information. For that criterion, the optimal decision is the Bayesian minimum-risk estimator. However, the computational load of the Bayesian estimator is often prohibitive and, hence, efficient computational schemes are required. The design of novel schemes, striking new balances between accuracy and computational load, is the primary concern of this thesis. Two popular techniques, one exact and one approximate, will be studied. The exact scheme is a recursive one, namely the generalized distributive law (GDL), whose purpose is to distribute all operators across the conditionally independent (CI) factors of the joint model, so as to reduce the total number of operators required. In a novel theorem derived in this thesis, GDL, if applicable, will be shown to guarantee such a reduction in all cases. An associated lemma also quantifies this reduction. For practical use, two novel algorithms, namely the no-longer-needed (NLN) algorithm and the generalized form of the Markovian Forward-Backward (FB) algorithm, recursively factorizes and computes the CI factors of an arbitrary model, respectively. The approximate scheme is an iterative one, namely the Variational Bayes (VB) approximation, whose purpose is to find the independent (i.e. zero-order Markov) model closest to the true joint model in the minimum Kullback-Leibler divergence (KLD) sense. Despite being computationally efficient, this naive mean field approximation confers only modest performance for highly correlated models. A novel approximation, namely Transformed Variational Bayes (TVB), will be designed in the thesis in order to relax the zero-order constraint in the VB approximation, further reducing the KLD of the optimal approximation.

Bayesian inference for PCA and MUSIC algorithms with unknown number of sources

Principal component analysis (PCA) is a popular method for projecting data onto uncorrelated components in lower dimension, although the optimal number of components is not specified. Likewise, multiple signal classification (MUSIC) algorithm is a popular PCA-based method for estimating directions of arrival (DOAs) of sinusoidal sources, yet it requires the number of sources to be known a priori. The accurate estimation of the number of sources is hence a crucial issue for performance of these algorithms. In this paper, we will show that both PCA and MUSIC actually return the exact joint maximum-a-posteriori (MAP) estimate for uncorrelated steering vectors, although they can only compute this MAP estimate approximately in correlated case. We then use Bayesian method to, for the first time, compute the MAP estimate for the number of sources in PCA and MUSIC algorithms. Intuitively, this MAP estimate corresponds to the highest probability that signal-plus-noise's variance still dominates projected noise's variance on signal subspace. In simulations of overlapping multi-tone sources for linear sensor array, our exact MAP estimate is far superior to the asymptotic Akaike information criterion (AIC), which is a popular method for estimating the number of components in PCA and MUSIC algorithms.

Copula Variational Bayes inference via information geometry

Variational Bayes (VB), also known as independent mean-field approximation, has become a popular method for Bayesian network inference in recent years. Its application is vast, e.g. in neural network, compressed sensing, clustering, etc. to name just a few. In this paper, the independence constraint in VB will be relaxed to a conditional constraint class, called copula in statistics. Since a joint probability distribution always belongs to a copula class, the novel copula VB (CVB) approximation is a generalized form of VB. Via information geometry, we will see that CVB algorithm iteratively projects the original joint distribution to a copula constraint space until it reaches a local minimum Kullback-Leibler (KL) divergence. By this way, all mean-field approximations, e.g. iterative VB, Expectation-Maximization (EM), Iterated Conditional Mode (ICM) and k-means algorithms, are special cases of CVB approximation. For a generic Bayesian network, an augmented hierarchy form of CVB will also be designed. While mean-field algorithms can only return a locally optimal approximation for a correlated network, the augmented CVB network, which is an optimally weighted average of a mixture of simpler network structures, can potentially achieve the globally optimal approximation for the first time. Via simulations of Gaussian mixture clustering, the classification's accuracy of CVB will be shown to be far superior to that of state-of-the-art VB, EM and k-means algorithms.