In science and engineering, empirical observations of a physical system are frequently used to infer its underlying mathematical model structure and estimate its relevant parameters. These inverse problems-where the consequences are known but not the causes-are the basis of inductive learning and discovery.

The physical sciences have seen much synthesis of theoretical constructions, enabling highly effective predictions using detailed physically-based models. However, the biological sciences remain in the model-building knowledge-synthesis phase. Frequently, this process relies heavily on inductive learning from sparse and noisy empirical observations. Inverse problems play a central role in the process of inferring predictive models from biological data.

The solution to such problems can be very challenging. A range of models and parameter values frequently can reproduce observed measurements, particularly in the presence of measurement noise and uncertainty. From a range of possible approaches, Bayesian inference techniques are highly promising for solving inverse problems amid noise and uncertainty.

Sandia researchers are investigating the use of Bayesian inference techniques for a range of inverse problems, including in particular the reconstruction of gene regulatory networks (GRNs) from noisy microarray data. This work is funded through Sandia's President Harry S. Truman Fellowship in National Security Science and Technology, a new program that provides the opportunity for recipients to pursue independent research of their choosing that supports Sandia's national security mission.

While modern gene sequencing technologies have elucidated the genomic structure of many organisms they have also exposed the need for understanding the functions of these genes and their associated proteins-possibly an even more complex challenge. GRNs are characterized by feedback loops in which genes encode proteins that then regulate the expression of other genes, producing complex patterns of expression and regulation that not only differentiate cell types but vary in developmental time and response to external stimuli. Characterizing these networks with accurate models is essential to understanding the myriad complexity of living cells, elucidating differences and pathways between normal and disease states, and clarifying and manipulating cellular response to pathogens and toxins.

Observational data in general, and gene microarray data in particular, are invariably affected by noise and uncertainty, which translates to uncertainties in inferred model structure and parameters. A sound inference strategy not only arrives at specific parameter values and model structure, but also assigns uncertainties, confidence intervals, or error bars, to the inferred quantities. What is needed is a statement that a parameter or an element of model structure is known, within a stated probability, to be within given bounds. The Bayesian framework is ideally suited for these inference problems. The Bayesian approach provides a link between observed data and the parameters or models that may account for it. It allows inference from noisy, disparate sources of data and provides a quantitative description of the uncertainty of inferred results.

Bayesian ideas underlie much of estimation and decision theory and have found frequent and successful application in problems of model selection, parameter estimation, and pattern recognition in a variety of contexts. Moreover, Bayesian inference has particular advantages for model inversion and structural learning problems.