<p>Gaussian filtering provides a Bayesian approach to dynamic state estimation, but requires precise statistical information about observation noise. When this information is unavailable, it is necessary to estimate the measurement noise covariance based on the observation and/or innovation sequences. Common approaches are based on frequentist paradigms, such as maximum likelihood or least squares criteria, which are dependent on having a large amount of observations and do not guarantee the covariance positive semi-definiteness. We propose an alternative Bayesian approach to recursively estimate the innovation covariance concurrently with state estimation, guaranteeing the requisite properties of the covariance and eliminating the need for an explicit measurement noise statistics model. The proposed Bayesian statistics estimation (BSE) method is embedded within a sigma-point Kalman filter, and adapted to non-stationary noise processes using Gaussian filter consistency tests. The adaptive filter is validated through simulation, and compared to an established online approach based on frequentist covariance matching.</p>