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\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, $\beta$ 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}:
$$ I(\beta;y_S) = \entropy(\beta)-\entropy(\beta\mid y_S). $$
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).
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