A Bayesian method for estimating stochastic parameters

This presentation will discuss a Bayesian method for estimating the variance of stochastic terms in a stochastic-dynamical model. It is shown that the standard method for estimating uncertain model parameters, based on augmentation of the state vector, fails when applied to the variance of stochastic terms. The source of this failure is explained. A Bayesian method is proposed that simultaneously estimates the state and the variance of the stochastic terms. The state estimation reduces to the familiar Kalman Filter equations, whereas the parameter estimation requires solving a nonlinear equation. A fundamentally new quantity required in the latter equation is the derivative of the forecast covariance with respect to the parameter. That this quantity appears is reasonable because the stochastic term directly influences the forecast variance and hence changes in the forecast variance provide a basis for tuning the parameter. The Bayesian method is illustrated with a stochastic version of the Lorenz three-variable model and shown to give reasonable estimated of the parameter.