"Parameter Estimates with Non-necessarily Gaussian Noise"

ABSTRACT

Parameter values in physical models are typically inferred from data observed in the field. In this work we make the following assumptions about this inference: (1) the discrete observation vector $d$ contains random variables which have non-necessarily Gaussian noise $\epsilon$ with known covariance matrix cov($\epsilon$)= $C_D$, (2) the physics are well understood and can be represented linearly by $G$, and (3) the true parameter values $m$ are not known but estimated by $m^*$ which is a vector of random variables from an unknown non-necessarily Gaussian prior distribution with cov($m^*$)=$C_M$. The parameter estimate $m^*$ could be from any method where the uncertainty in $m^*$ is unknown, and we need to determine the covariance matrix $C_M$. We will present an approach for estimating a range of viable matrices $C_M$ under the above stated assumptions. This approach is based on the fact that the minimum value of the cost function

\begin{displaymath}{\cal J}[m]= (d-Gm)^TC_D^{-1}(d-Gm) + (m-m^*)^TC_M^{-1}(m-m^*) \end{displaymath}

is a $\chi^2$ random variable. We use this fact to find values of $C_M$ that ensure the minimum value of ${\cal J}[m]$ lies within the specified confidence interval.