Is it possible to solve the following equation analytically for $\beta$:
$$y'(A+\beta B)^{-1}y = \alpha,$$
where $A$ and $B$ are both positive-semidefinite and symmetric matrices (essentially, some covariance matrices), $\alpha$ and $\beta$ - constants, and $y$ is some vector?
The problem arises from an attempt to automatically correct observation variance in an Update step of Kalman Filter to adjust for outliers. More specifically, if $y_k$ is a measurement error vector (multidimensional), $y_k$~$N(0,S_k)$, then $y_k'S_k^{-1}y_k$~$\chi^2(p)$. The problem is to find $\beta$, such that substituting $S_k=H_kP_{k|k-1}H_k'+\beta R_k$ (note that we've scaled covariance of observation noise $R_k$ with $\beta$), into statistics above would produce equality: $y_k'(H_kP_{k|k-1}H_k'+\beta R_k)^{-1}y_k=\chi^2_{\alpha}(p)$.
The alternative would be to actually scale the whole $S_k$, not just $R_k$. That's trivial to solve, but that doesn't have such an intuitive interpretation and it's going to result in another fit (hard to tell whether it would be materially worse).