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diff --git a/general.tex b/general.tex index 429299a..5f80c58 100644 --- a/general.tex +++ b/general.tex @@ -29,9 +29,9 @@ where $\mu$ is the smallest eigenvalue of $R$. \subsection{Non-Bayesian Setting} In the non-bayesian setting, \emph{i.e.} when the experimenter has no prior -distribution on the model, the covariance matrix $R$ is the zero matrix and -ridge regression \eqref{ridge} reduces to simple least squares. In this case, -the $D$-optimal criterion takes the following form: +distribution on the model, the covariance matrix $R$ is the zero matrix. In this case, +the ridge regression estimation proceedure \eqref{ridge} reduces to simple least squares (\emph{i.e.}, linear regression), +and the $D$-optimality reduces to the entropy of $\hat{\beta}$, given by: \begin{equation}\label{eq:d-optimal} V(S) = \log\det(X_S^TX_S) \end{equation} @@ -46,16 +46,14 @@ and individual rationality. \begin{lemma} For any $M>1$, there is no $M$-approximate, truthful, budget feasible, individually rational mechanism for a budget feasible reverse auction with -value function $V(S) = \det{\T{X_S}X_S}$. For any $M>1$, there is no -$M$-approximate, truthful, budget feasible, individually rational mechanism for -a budget feasible reverse auction with $V(S) = \det{\T{X_S}X_S}$. +value function $V(S) = \det{\T{X_S}X_S}$. \end{lemma} \begin{proof} \input{proof_of_lower_bound1} \end{proof} -Beyond $D$-optimality, several other objectives such as $E$-optimality (maximizing the smallest eigenvalue of $\T{X_S}X_S$) or $T$-optimality (maximizing $\mathrm{trace}(\T{X_S}{X_S}))$ are encountered in the literature \cite{pukelsheim2006optimal}, though they do not relate to entropy as $D$-optimality. We leave the task of approaching the maximization of such objectives from a strategic point of view as an open problem. +%Beyond $D$-optimality, several other objectives such as $E$-optimality (maximizing the smallest eigenvalue of $\T{X_S}X_S$) or $T$-optimality (maximizing $\mathrm{trace}(\T{X_S}{X_S}))$ are encountered in the literature \cite{pukelsheim2006optimal}, though they do not relate to entropy as $D$-optimality. We leave the task of approaching the maximization of such objectives from a strategic point of view as an open problem. \subsection{Beyond Linear Models} Selecting experiments that maximize the information gain in the Bayesian setup |