The Connection Between Bayesian Estimation of a Gaussian Random Field and RKHS.

Reconstruction of a function from noisy data is key in machine learning and is often formulated as a regularized optimization problem over an infinite-dimensional reproducing kernel Hilbert space (RKHS). The solution suitably balances adherence to the observed data and the corresponding RKHS norm. When the data fit is measured using a quadratic loss, this estimator has a known statistical interpretation. Given the noisy measurements, the RKHS estimate represents the posterior mean (minimum variance estimate) of a Gaussian random field with covariance proportional to the kernel associated with the RKHS. In this brief, we provide a statistical interpretation when more general losses are used, such as absolute value, Vapnik or Huber. Specifically, for any finite set of sampling locations (that includes where the data were collected), the maximum a posteriori estimate for the signal samples is given by the RKHS estimate evaluated at the sampling locations. This connection establishes a firm statistical foundation for several stochastic approaches used to estimate unknown regularization parameters. To illustrate this, we develop a numerical scheme that implements a Bayesian estimator with an absolute value loss. This estimator is used to learn a function from measurements contaminated by outliers.

Estimating in vitro mitochondrial oxygen consumption during muscle contraction and recovery: a novel approach that accounts for diffusion.

A deconvolution algorithm, based on a Bayesian statistical framework and smoothing spline technique, is applied to reconstructing input functions from noisy measurements in biological systems. Deconvolution is usually ill-posed. However, placing a Bayesian prior distribution on the input function can make the problem well-posed. Using this algorithm and a computational model of diffusional oxygen transport in an approximately cylindrical muscle (about 0.5-mm diameter and 10-mm long mouse leg muscle), the time course of muscle oxygen uptake and mitochondrial oxygen consumption, both during isometric twitch contractions (at various frequencies) and the recovery period, is estimated from polarographic measurements of oxygen concentration on the muscle surface. An important feature of our experimental protocol is the availability of data for the apparatus characteristics. From these time courses, the actual mitochondrial consumption rates during resting and exercise states can be estimated. Mitochondrial oxygen consumption rate increased during stimulation to a maximum steady state value approximately five times of the resting value of 0.63 nmol/s/g wet weight for the stimulation conditions studied. Diffusion slowed the kinetic responses to the contraction but not the steady state fluxes during the stimulation interval.