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| -rw-r--r-- | problem.tex | 8 |
1 files changed, 4 insertions, 4 deletions
diff --git a/problem.tex b/problem.tex index 2e4db20..c3c7c09 100644 --- a/problem.tex +++ b/problem.tex @@ -56,7 +56,7 @@ $D$-optimality criterion Note that the estimator $\hat{\beta}$ is a linear map of $y_S$; as $y_S$ is a multidimensional normal r.v., so is $\hat{\beta}$ (the randomness coming from the noise terms $\varepsilon_i$ and the prior on $\beta$). 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 $\hat{\beta}$ my minimizing the product of the eigenvalues of its covariance. +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. %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 $(R+\T{X_S}X_S)^{-1}$. @@ -124,9 +124,9 @@ In particular, consider the greedy algorithm in which, for $S\subseteq\mathcal{N}$ the set constructed thus far, the next element $i$ included is the one which maximizes the \emph{marginal-value-per-cost}, \emph{i.e.}, -\begin{align} - i = \argmax_{j\in\mathcal{N}\setminus S}\frac{V(S\cup\{i\}) - V(S)}{c_i}\label{greedy} -\end{align} +%\begin{align} + $ i = \argmax_{j\in\mathcal{N}\setminus S}{(V(S\cup\{i\}) - V(S))}/{c_i}.$ %\label{greedy} +%\end{align} This is repeated until adding an element in $S$ exceeds the budget $B$. Denote by $S_G$ the set constructed by this heuristic and let $i^*=\argmax_{i\in\mathcal{N}} V(\{i\})$ be the element of maximum singleton value. Then, |