From e73a4651379d2c34855f1bc6fe5c0abef039b1d5 Mon Sep 17 00:00:00 2001
From: Thibaut Horel <thibaut.horel@gmail.com>
Date: Mon, 11 Feb 2013 10:39:39 -0800
Subject: Mending section 3 and 5

---
 general.tex | 33 ++++++++++++++++++++++-----------
 1 file changed, 22 insertions(+), 11 deletions(-)

(limited to 'general.tex')

diff --git a/general.tex b/general.tex
index 3c7eddf..429299a 100644
--- a/general.tex
+++ b/general.tex
@@ -1,16 +1,16 @@
-\subsection{Strategic Experimental Design with non-homotropic prior}\label{sec:bed}
+\subsection{Strategic Experimental Design with Non-Homotropic Prior}\label{sec:bed}
 
 %In this section, we extend our results to Bayesian experimental design
 %\cite{chaloner1995bayesian}. We show that objective function \eqref{modified}
 %has a natural interpretation in this context, further motivating its selection
 %as our objective.  Moreover, 
 
-If the general case where the prior distribution of the experimenter on the
+In the general case where the prior distribution of the experimenter on the
 model $\beta$ in \eqref{model} is not homotropic and has a generic covariance
 matrix $R$, the value function takes the general form given by
 \eqref{dcrit}.
 
-Applying the mechanism described in algorithm~\ref{mechanism} and adapting the
+Applying the mechanism described in Algorithm~\ref{mechanism} and adapting the
 analysis of the approximation ratio, we get the following result which extends
 Theorem~\ref{thm:main}:
 
@@ -26,20 +26,31 @@ Theorem~\ref{thm:main}:
 where $\mu$ is the smallest eigenvalue of $R$.
 \end{theorem}
 
-\subsection{Other Experimental Design Criteria}
+\subsection{Non-Bayesian Setting}
 
-A value function which is frequently used in experimental design is the
-$D$-optimality criterion obtained by replacing $R$ by the zero matrix in
-\eqref{dcrit}:
+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:
 \begin{equation}\label{eq:d-optimal}
 V(S) = \log\det(X_S^TX_S)
 \end{equation}
-Since \eqref{eq:d-optimal} may take arbitrarily small negative values, to define a meaningful approximation one would consider  the (equivalent) maximization of $V(S) = \det\T{X_S}X_S$. %, for some strictly increasing, on-to function $f:\reals_+\to\reals_+$. 
-However, the following lower bound implies that such an optimization goal cannot be attained under the constraints of  truthfulness, budget feasibility, and individual rationality.
+A natural question which arises is whether it is possible to design
+a deterministic mechanism in this setting. Since \eqref{eq:d-optimal} may take
+arbitrarily small negative values, to define a meaningful approximation one
+would consider  the (equivalent) maximization of $V(S) = \det\T{X_S}X_S$.
+However, the following lower bound implies that such an optimization goal
+cannot be attained under the constraints of  truthfulness, budget feasibility,
+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 $V(S) =  \det{\T{X_S}X_S}$.
+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 $V(S) =  \det{\T{X_S}X_S}$.
 \end{lemma}
+
 \begin{proof}
 \input{proof_of_lower_bound1}
 \end{proof}
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