We consider regularized least-squares (RLS) with a Gaussian kernel. We prove that if we let the Gaussian bandwidth σ → ∞ while letting the regularization parameter λ → 0, the RLS solution tends to a polynomial whose order is controlled by the rielative rates of decay of 1 σ2 and λ: if λ = σ−(2k+1), then, as σ → ∞, the RLS solution tends to the kth order polynomial with minimal empirical error. We illustrate the result with an example.

We present a new statistically optimal approach to estimate transcript levels and ratios from one or more gene array experiments. The Bayesian Estimation of Array Measurements (BEAM) technique uses a model of measurement noise and prior information to estimate biological expression levels. It provides a principled method to deal with negative expression level measurements, combine multiple measurements, and identify changes in expression level. BEAM is more flexible than existing techniques, because it does not assume a specific functional form for noise and prior models. Rather, it uses a more accurate noise model developed from experimental data, a process we illustrate here using Affymetrix yeast chips.