Approximate Computation and Implicit Regularization for Very Large-scale Data Analysis

Database theory and database practice are typically the domain of computer scientists who adopt what may be termed an algorithmic perspective on their data. This perspective is very different than the more statistical perspective adopted by statisticians, scientific computers, machine learners, and other who work on what may be broadly termed statistical data analysis. In this article, I will address fundamental aspects of this algorithmic-statistical disconnect, with an eye to bridging the gap between these two very different approaches. A concept that lies at the heart of this disconnect is that of statistical regularization, a notion that has to do with how robust is the output of an algorithm to the noise properties of the input data. Although it is nearly completely absent from computer science, which historically has taken the input data as given and modeled algorithms discretely, regularization in one form or another is central to nearly every application domain that applies algorithms to noisy data. By using several case studies, I will illustrate, both theoretically and empirically, the nonobvious fact that approximate computation, in and of itself, can implicitly lead to statistical regularization. This and other recent work suggests that, by exploiting in a more principled way the statistical properties implicit in worst-case algorithms, one can in many cases satisfy the bicriteria of having algorithms that are scalable to very large-scale databases and that also have good inferential or predictive properties.Comment: To appear in the Proceedings of the 2012 ACM Symposium on Principles of Database Systems (PODS 2012

A variational approach to path estimation and parameter inference of hidden diffusion processes

We consider a hidden Markov model, where the signal process, given by a diffusion, is only indirectly observed through some noisy measurements. The article develops a variational method for approximating the hidden states of the signal process given the full set of observations. This, in particular, leads to systematic approximations of the smoothing densities of the signal process. The paper then demonstrates how an efficient inference scheme, based on this variational approach to the approximation of the hidden states, can be designed to estimate the unknown parameters of stochastic differential equations. Two examples at the end illustrate the efficacy and the accuracy of the presented method.Comment: 37 pages, 2 figures, revise