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Diffstat (limited to 'problem.tex')
| -rw-r--r-- | problem.tex | 3 |
1 files changed, 1 insertions, 2 deletions
diff --git a/problem.tex b/problem.tex index 89ffa9d..1001d7b 100644 --- a/problem.tex +++ b/problem.tex @@ -54,8 +54,7 @@ Maximizing $I(\beta;y_S)$ is therefore equivalent to maximizing $\log\det(R+ \T{ $D$-optimality criterion \cite{pukelsheim2006optimal,atkinson2007optimum,chaloner1995bayesian}. -Note that the estimator $\hat{\beta}$ is a linear map of $y_S$. As $y_S$ is a multidimensional normal r.v., so is $\hat{\beta}$; in particular, $\hat{\beta}$ has -covariance $\sigma^2(R+\T{X_S}X_S)^{-1}$. As such, maximizing $I(\beta;y_S)$ can alternatively be seen as a means of reducing the uncertainty on estimator $\hat{\beta}$ by minimizing the product of the eigenvalues of its covariance (as the latter equals the determinant). +%Note that the estimator $\hat{\beta}$ is a linear map of $y_S$. As $y_S$ is a multidimensional normal r.v., so is $\hat{\beta}$; in particular, $\hat{\beta}$ has covariance $\sigma^2(R+\T{X_S}X_S)^{-1}$. As such, maximizing $I(\beta;y_S)$ can alternatively be seen as a means of reducing the uncertainty on estimator $\hat{\beta}$ by minimizing the product of the eigenvalues of its covariance (as the latter equals the determinant). %An alternative interpretation, given that $(R+ \T{X_S}X_S)^{-1}$ is the covariance of the estimator $\hat{\beta}$, is that it tries to minimize the %which is indeed a function of the covariance matrix $(R+\T{X_S}X_S)^{-1}$. |