From a998a6545606eeb449b5b02886c8e562d0da754b Mon Sep 17 00:00:00 2001
From: Stratis Ioannidis <stratis@stratis-Latitude-E6320.(none)>
Date: Wed, 31 Oct 2012 11:56:10 -0700
Subject: result

---
 general.tex | 8 +++++---
 1 file changed, 5 insertions(+), 3 deletions(-)

(limited to 'general.tex')

diff --git a/general.tex b/general.tex
index e7a955f..23145ad 100644
--- a/general.tex
+++ b/general.tex
@@ -1,13 +1,15 @@
-\subsection{Bayesian Experimental Design}
-In this section, we extend our results to Bayesian experimental design \cite{chaloner1995bayesian}. In particular, we show that our choice of objective function \eqref{...} has a natural interpration in this context, further motivating its selection, and Theorem~\ref{...} has a natural generalization to this context.
+\subsection{Bayesian Experimental Design}\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 interpration in this context, further motivating its selection as our objective.  Moreover, we extend Theorem~\ref{thm:main} to a more general Bayesian setting.
 
-In the Bayesian setting, it is assumed that the experimenter has a prior distribution on $\beta$: in particular, $\beta$ is assumed to be sampled from a multivariate normal distribution with zero mean  and covariance $\sigma^2R\in \reals^{d^2}$ (where $\sigma^2$ is the noise variance). 
+In the Bayesian setting, it is assumed that the experimenter has a prior distribution on $\beta$: in particular, $\beta$ has a multivariate normal prior with zero mean  and covariance $\sigma^2R\in \reals^{d^2}$ (where $\sigma^2$ is the noise variance). 
 The experimenter estimates $\beta$ through \emph{maximum a posteriori estimation}: \emph{i.e.}, finding the parameter which maximizes the posterior distribution of $\beta$ given the observations $y_S$. Under the linearity assumption \eqref{model} and the gaussian prior on $\beta$, maximum a posteriori estimation leads to the following maximization \cite{hastie}: FIX!
 \begin{displaymath}
     \hat{\beta} = \argmin_{\beta\in\reals^d} \sum_i (y_i - \T{\beta}x_i)^2
     + \sum_i \norm{R\beta}_2^2
 \end{displaymath}
 This optimization, commonly known as \emph{ridge regression}, includes an additional penalty term compared to the least squares estimation \eqref{leastsquares}.
+
+
 Let $\entropy(\beta)$ be the entropy of $\beta$ under this distribution, and $\entropy(\beta\mid y_S)$ the  entropy of $\beta$ conditioned on the experiment outcomes $Y_S$, for some $S\subseteq \mathcal{N}$. In this setting, a natural objective to select a set of experiments $S$ that maximizes her \emph{information gain}:
 $$ I(\beta;y_S) = \entropy(\beta)-\entropy(\beta\mid y_S). $$
 
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