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Diffstat (limited to 'problem.tex')
| -rw-r--r-- | problem.tex | 11 |
1 files changed, 10 insertions, 1 deletions
diff --git a/problem.tex b/problem.tex index 3ad3270..9d3fb9f 100644 --- a/problem.tex +++ b/problem.tex @@ -55,6 +55,9 @@ Under the linear model \eqref{model}, and the Gaussian prior, the information ga \begin{align} V(S) &= \frac{1}{2}\log\det(R+ \T{X_S}X_S) \label{dcrit} %\\ \end{align} +This value function is known in the experimental design literature as the +$D$-optimality criterion +\cite{pukelsheim2006optimal,atkinson2007optimum,chaloner1995bayesian}. %which is indeed a function of the covariance matrix $(R+\T{X_S}X_S)^{-1}$. %defined as $-\infty$ when $\mathrm{rank}(\T{X_S}X_S)<d$. %As $\hat{\beta}$ is a multidimensional normal random variable, the @@ -69,7 +72,13 @@ Under the linear model \eqref{model}, and the Gaussian prior, the information ga %\end{align} %There are several reasons %In addition, the maximization of convex relaxations of this function is a well-studied problem \cite{boyd}. -Our analysis will focus on the case of a \emph{homotropic} prior, in which the prior covariance is the identity matrix, \emph{i.e.}, $R=I_d\in \reals^{d\times d}.$ Intuitively, this corresponds to the simplest prior, in which no direction of $\reals^d$ is a priori favored; equivalently, it also corresponds to the case where ridge regression estimation \eqref{ridge} performed by $\E$ has a penalty term $\norm{\beta}_2^2$. In Section 5, we will address other $R$'s. +Our analysis will focus on the case of a \emph{homotropic} prior, in which the +prior covariance is the identity matrix, \emph{i.e.}, $R=I_d\in \reals^{d\times +d}.$ Intuitively, this corresponds to the simplest prior, in which no direction +of $\reals^d$ is a priori favored; equivalently, it also corresponds to the +case where ridge regression estimation \eqref{ridge} performed by $\E$ has +a penalty term $\norm{\beta}_2^2$. A generalization of our results to general +matrices $R$ can be found in Section~\ref{sec:ext}. %Note that \eqref{dcrit} is a submodular set function, \emph{i.e.}, %$V(S)+V(T)\geq V(S\cup T)+V(S\cap T)$ for all $S,T\subseteq \mathcal{N}$; it is also monotone, \emph{i.e.}, $V(S)\leq V(T)$ for all $S\subset T$. |