diff options
Diffstat (limited to 'problem.tex')
| -rw-r--r-- | problem.tex | 3 |
1 files changed, 2 insertions, 1 deletions
diff --git a/problem.tex b/problem.tex index 5fd8906..423e6fa 100644 --- a/problem.tex +++ b/problem.tex @@ -17,12 +17,13 @@ Note that the estimator $\hat{\beta}$ is a linear map of $y_S$; as $y_S$ is a mu Let $V:2^\mathcal{N}\to\reals$ be a value function, quantifying how informative a set of experiments $S$ is in estimating $\beta$. The standard optimal experimental design problem amounts to finding a set $S$ that maximizes $V(S)$ subject to the constraint $|S|\leq k$. -There is a variety of different value functions used in experimental design~\cite{pukelsheim2006optimal}; almost all are related to the covariance $(\T{X_S}X_S)^{-1}$ of the estimator $\hat{\beta}$. Due to its relationship to entropy, the \emph{$D$-optimality criterion} is commonly used: %which yields the following optimization problem +A variety of different value functions are used in experimental design~\cite{pukelsheim2006optimal}; almost all are related to the covariance $(\T{X_S}X_S)^{-1}$ of the estimator $\hat{\beta}$. A commonly used choice is the so-called \emph{$D$-optimality criterion} \cite{pukelsheim2006optimal,chaloner1995bayesian}: %which yields the following optimization problem \begin{align} V(S) &= \frac{1}{2}\log\det \T{X_S}X_S \label{dcrit} %\\ \end{align} As $\hat{\beta}$ is a multidimensional normal random variable, the $D$-optimality criterion is equal (up to a costant) to the negative of the entropy of $\hat{\beta}$. Hence, maximizing \eqref{dcrit} amounts to finding the set of experiments that minimizes the uncertainty on $\beta$, as captured by the entropy of its estimator. +Value function \eqref{dcrit} has several appealing properties. To begin with, it is a submodular set function (see Lemma~\ref{...} and Thm.~\ref{...}). In addition, the maximization of convex relaxations of this function is a well-studied problem \cite{boyd}. Note that \eqref{dcrit} is undefined when $\mathrm{rank}(\T{X_S}X_S)<d$; in this case, we take $V(S)=-\infty$ (so that $V$ takes values in the extended reals). %As discussed in the next section, in this paper, we work with a modified measure of information function, namely %\begin{align} %V(S) & = \frac{1}{2} \log\det \left(I + \T{X_S}X_S\right) |