The paper has two major themes. The first part of the paper establishes certain general results for infinite-dimensional optimization problems on Hilbert spaces. These results cover the classical representer theorem and many of its variants as special cases and offer a wider scope of applications. The second part of the paper then develops a systematic approach for learning the drift function of a stochastic differential equation by integrating the results of the first part with Bayesian hierarchical framework. Importantly, our Bayesian approach incorporates low-cost sparse learning through proper use of shrinkage priors while allowing proper quantification of uncertainty through posterior distributions. Several examples at the end illustrate the accuracy of our learning scheme.

Base calling is a critical step in the Solexa next-generation sequencing procedure. It compares the position-specific intensity measurements that reflect the signal strength of four possible bases (A, C, G, T) at each genomic position, and outputs estimates of the true sequences for short reads of DNA or RNA. We present a Bayesian method of base calling, BM-BC, for Solexa-GA sequencing data. The Bayesian method builds on a hierarchical model that accounts for three sources of noise in the data, which are known to affect the accuracy of the base calls: fading, phasing, and cross-talk between channels. We show that the new method improves the precision of base calling compared with currently leading methods. Furthermore, the proposed method provides a probability score that measures the confidence of each base call. This probability score can be used to estimate the false discovery rate of the base calling or to rank the precision of the estimated DNA sequences, which in turn can be useful for downstream analysis such as sequence alignment.