Topics in inference and decision-making with partial knowledge

Two essential elements needed in the process of inference and decision-making are prior probabilities and likelihood functions. When both of these components are known accurately and precisely, the Bayesian approach provides a consistent and coherent solution to the problems of inference and decision-making. In many situations, however, either one or both of the above components may not be known, or at least may not be known precisely. This problem of partial knowledge about prior probabilities and likelihood functions is addressed. There are at least two ways to cope with this lack of precise knowledge: robust methods, and interval-valued methods. First, ways of modeling imprecision and indeterminacies in prior probabilities and likelihood functions are examined; then how imprecision in the above components carries over to the posterior probabilities is examined. Finally, the problem of decision making with imprecise posterior probabilities and the consequences of such actions are addressed. Application areas where the above problems may occur are in statistical pattern recognition problems, for example, the problem of classification of high-dimensional multispectral remote sensing image data

Maximum Entropy Vector Kernels for MIMO system identification

Recent contributions have framed linear system identification as a nonparametric regularized inverse problem. Relying on $\ell_2$-type regularization which accounts for the stability and smoothness of the impulse response to be estimated, these approaches have been shown to be competitive w.r.t classical parametric methods. In this paper, adopting Maximum Entropy arguments, we derive a new $\ell_2$ penalty deriving from a vector-valued kernel; to do so we exploit the structure of the Hankel matrix, thus controlling at the same time complexity, measured by the McMillan degree, stability and smoothness of the identified models. As a special case we recover the nuclear norm penalty on the squared block Hankel matrix. In contrast with previous literature on reweighted nuclear norm penalties, our kernel is described by a small number of hyper-parameters, which are iteratively updated through marginal likelihood maximization; constraining the structure of the kernel acts as a (hyper)regularizer which helps controlling the effective degrees of freedom of our estimator. To optimize the marginal likelihood we adapt a Scaled Gradient Projection (SGP) algorithm which is proved to be significantly computationally cheaper than other first and second order off-the-shelf optimization methods. The paper also contains an extensive comparison with many state-of-the-art methods on several Monte-Carlo studies, which confirms the effectiveness of our procedure

Markovian Dynamics on Complex Reaction Networks

Complex networks, comprised of individual elements that interact with each other through reaction channels, are ubiquitous across many scientific and engineering disciplines. Examples include biochemical, pharmacokinetic, epidemiological, ecological, social, neural, and multi-agent networks. A common approach to modeling such networks is by a master equation that governs the dynamic evolution of the joint probability mass function of the underling population process and naturally leads to Markovian dynamics for such process. Due however to the nonlinear nature of most reactions, the computation and analysis of the resulting stochastic population dynamics is a difficult task. This review article provides a coherent and comprehensive coverage of recently developed approaches and methods to tackle this problem. After reviewing a general framework for modeling Markovian reaction networks and giving specific examples, the authors present numerical and computational techniques capable of evaluating or approximating the solution of the master equation, discuss a recently developed approach for studying the stationary behavior of Markovian reaction networks using a potential energy landscape perspective, and provide an introduction to the emerging theory of thermodynamic analysis of such networks. Three representative problems of opinion formation, transcription regulation, and neural network dynamics are used as illustrative examples.Comment: 52 pages, 11 figures, for freely available MATLAB software, see http://www.cis.jhu.edu/~goutsias/CSS%20lab/software.htm