From cd2b03f18be9b110cbaa31dd0469139b4504baab Mon Sep 17 00:00:00 2001
From: Thibaut Horel <thibaut.horel@gmail.com>
Date: Wed, 31 Oct 2012 17:36:58 +0100
Subject: Some typo fixes in section 2

---
 problem.tex | 3 ++-
 1 file changed, 2 insertions(+), 1 deletion(-)

(limited to 'problem.tex')

diff --git a/problem.tex b/problem.tex
index 423e6fa..49a861c 100644
--- a/problem.tex
+++ b/problem.tex
@@ -8,11 +8,12 @@ More precisely, putting cost considerations aside, suppose that an experimenter
 \end{align}
 where $\beta$ a vector in $\reals^d$, commonly referred to as the \emph{model}, and $\varepsilon_i$ (the \emph{measurement noise}) are independent, normally distributed random  variables with zero mean and variance $\sigma^2$. 
 
-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
+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 \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 $|S|\leq k$. 
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