From 166bcc95424910868c54b091eb94573ce3ffef0f Mon Sep 17 00:00:00 2001
From: Stratis Ioannidis <stratis@stratis-Latitude-E6320.(none)>
Date: Wed, 31 Oct 2012 10:26:41 -0700
Subject: changes

---
 general.tex | 22 +++++++++++++++++++++-
 1 file changed, 21 insertions(+), 1 deletion(-)

(limited to 'general.tex')

diff --git a/general.tex b/general.tex
index c90a439..e7a955f 100644
--- a/general.tex
+++ b/general.tex
@@ -1,5 +1,25 @@
 \subsection{Bayesian Experimental Design}
-TODO: Introduce prior with covariance $\sigma^2 R$. Change in entropy/ mutual information is then ... So our scheme can be seen as Baysian prior with $R=I_d$. Extension of our main theorem.
+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.
+
+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). 
+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}:
+$$ I(\beta;y_S) = \entropy(\beta)-\entropy(\beta\mid y_S). $$
+
+Assuming normal noise variables, the information gain is  equal (upto a constant) to the following value function \cite{chaloner1995bayesian}:
+\begin{align}
+V(S)  = \frac{1}{2}\log\det(R + \T{X_S}X_S)\label{bayesianobjective}
+\end{align}
+Our objective \eqref{,,,} clearly follows from \eqref{bayesianobjective} by setting $R=I_d$. Hence, our optimization can be interpreted as a maximization of the information gain when the prior distribution has a covariance $\sigma^2 I_d$, and the experimenter is solving a ridge regression problem with penalty term $\norm{x}_2^2$. 
+
+Moreover, our results can be extended to the general Bayesian case, by replacing $I_d$ with the positive semidefinite matrix $R$:
+
+TODO: state theorem, discuss dependence on $\det R$.
 
 \subsection{Beyond Linear Models}
 TODO: Independent noise model. Captures models such as logistic regression, classification, etc. Arbitrary prior. Show that change in the entropy is submodular (cite Krause, Guestrin). 
-- 
cgit v1.2.3-70-g09d2

