changes

1 files changed, 7 insertions, 7 deletions

diff --git a/problem.tex b/problem.tex
index 90203c6..5fd8906 100644
--- a/problem.tex
+++ b/problem.tex
@@ -10,18 +10,18 @@ where $\beta$ a vector in $\reals^d$, commonly referred to as the \emph{model},
 
 The purpose of these experiments is to allow the experimenter to estimate the model $\beta$. In particular, assuming gaussian noise, the maximum likelihood estimator of $\beta$ is the \emph{least squares} estimator: for $X_S=[x_i]_{i\in S}\in \reals^{|S|\times d}$ the matrix of experiment features and
 $y_S=[y_i]_{i\in S}\in\reals^{|S|}$ the observed measurements, 
-\begin{align*} \hat{\beta} &=\max_{\beta\in\reals^d}\prob(y_S;\beta) =\argmin_{\beta\in\reals^d } \sum_{i\in S}(\T{\beta}x_i-y_i)^2 \\
-& = (\T{X_S}X_S)^{-1}X_S^Ty_S\end{align*} 
+\begin{align} \hat{\beta} &=\max_{\beta\in\reals^d}\prob(y_S;\beta) =\argmin_{\beta\in\reals^d } \sum_{i\in S}(\T{\beta}x_i-y_i)^2 \nonumber\\
+& = (\T{X_S}X_S)^{-1}X_S^Ty_S \label{leastsquares}\end{align} 
 %The estimator $\hat{\beta}$ is unbiased, \emph{i.e.}, $\expt{\hat{\beta}} = \beta$ (where the expectation is over the noise variables $\varepsilon_i$). Furthermore, $\hat{\beta}$ is a multidimensional normal random variable with mean $\beta$ and covariance matrix $(X_S\T{X_S})^{-1}$. 
 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 $\varepsilon_i$). In particular, $\hat{\beta}$ has mean $\beta$ (\emph{i.e.}, it is an \emph{unbianced estimator}) and covariance $(\T{X_S}X_S)^{-1}$.
-
+ 
 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 capture this through some property the covariance $(\T{X_S}X_S)^{-1}$ of the estimator $\hat{\beta}$. Due to its relationship to entropy, a most commonly used is the \emph{$D$-optimality criterion}: %which yields the following optimization problem
+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
 \begin{align}
- V_D(S) &= \frac{1}{2}\log\det \T{X_S}X_S \label{dcrit} %\\
+ 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, selecting a set of experiments $S$ that maximizes $V_D(S)$ is equivalent to finding the set of experiments that minimizes the uncertainty on $\beta$, as captured by the entropy of its estimator.
+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.
 
 %As discussed in the next section, in this paper, we work with a modified measure of information function, namely 
 %\begin{align}
@@ -33,7 +33,7 @@ As $\hat{\beta}$ is a multidimensional normal random variable, the $D$-optimalit
 \subsection{Budget Feasible Mechanism Design}
 In this paper, we approach the problem of optimal experimental design from the perspective of \emph{a budget feasible reverse auction} \cite{singer-mechanisms}. In particular, we assume that each experiment $i\in \mathcal{N}$ is associated with a cost $c_i$, that the experimenter needs to pay in order to conduct the experiment.  The experimenter has a budget $B\in\reals_+$. In the \emph{full information case}, the costs are common knowledge; optimal design in this context amounts to selecting a set $S$ maximizing the value $V(S)$ subject to the constraint $\sum_{i\in S} c_i\leq B$.
 
-As in \cite{singer-mechanisms,chen}, we focus in a \emph{strategic scenario}: experiment $i$ corresponds to a \emph{strategic agent}, whose cost $c_i$ is private.  For example, each $i$ may correspond to a human participant; the feature vector $x_i$ may correspond to a normalized vector of her age, weight, gender, income, \emph{etc.}, and the measurement $y_i$ may capture some biometric information (\emph{e.g.}, her red cell blood count, a genetic marker, etc.). The cost $c_i$ is the amount the participant deems sufficient to incentivize her participation in the study. 
+As in \cite{singer-mechanisms,chen}, we focus on a \emph{strategic scenario}: experiment $i$ corresponds to a \emph{strategic agent}, whose cost $c_i$ is private.  For example, each $i$ may correspond to a human participant; the feature vector $x_i$ may correspond to a normalized vector of her age, weight, gender, income, \emph{etc.}, and the measurement $y_i$ may capture some biometric information (\emph{e.g.}, her red cell blood count, a genetic marker, etc.). The cost $c_i$ is the amount the participant deems sufficient to incentivize her participation in the study. 
 
 
 A mechanism $\mathcal{M} = (f,p)$ comprises (a) an \emph{allocation function} $f:\reals_+^n \to 2^\mathcal{N}$ and (b) a \emph{payment function} $p:\reals_+^n\to \reals_+^n$. The allocation function determines the set $S\subset \mathcal{N}$ of experiments to be conducted. The payment function returns a vector of payments $[p_i]_{i\in \mathcal{N}}$. As in \cite{singer-mechanisms, chen}, we study mechanisms that are normalized ($i\notin S$ implies $p_i=0$), individually rational ($p_i\geq c_i$) and have no positive transfers $p_i\geq 0$.