\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, originally proposed by Lindley \cite{lindley1956measure}, is 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 (up to 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 $R$:

\thibaut{Discussion about the value function below}
\stratis{The text below is unpolished/not written for external consumption. Rather than trying to motivate $R>I$, it is probably better to figure out a lower bound.}

When there is an $R$ in the value function, it seems to make more sense to
study the modified value function:
\begin{displaymath}
    \tilde{V}(S) = \frac{1}{2}\log\det(R + \T{X_S}X_S) - \frac{1}{2}\log\det R
\end{displaymath}
For two reasons:
\begin{itemize}
    \item $\tilde{V}(\emptyset) = 0$: the value function is normalized, I think
    this assumption is needed somewhere in mechanism design.
    \item $\tilde{V}(S) = \frac{1}{2}\log\det(I_d + R^{-1}\T{X_S}X_S)$, so we
    can apply our result to get an $\alpha$ approximation ratio (see the value
    of $\alpha$ below). If we take $V$ instead of $\tilde{V}$ then one can
    write:
    \begin{displaymath}
        V(S) = \frac{1}{2}\log\det R + \tilde{V}(S)
    \end{displaymath}
    thus:
    \begin{displaymath}
        OPT(V) = \frac{1}{2}\log\det R + OPT(\tilde{V})
    \end{displaymath}
    we can find $S^*$ such that $OPT(\tilde{V}) \leq \alpha \tilde{V}(S)$, so:
    \begin{displaymath}
        OPT(V) = \frac{1}{2}\log\det R + \alpha\tilde{V}(S)
    \end{displaymath}
    But this does not give an $\alpha$ approximation ratio for $V$, because
    $\log\det R$ can be negative. This is only an \emph{asymptotic}
    approximation ratio\ldots.
\end{itemize}

\begin{theorem}
 For the function $\tilde{V}$ defined above, there is a truthful, budget
 feasible mechanism which achieves an approximation ratio of:
 \begin{displaymath}
    \frac{5e-1}{e-1}\frac{2\mu}{\log(1+\mu)} + 5.1
 \end{displaymath}
where $\mu$ is the smallest eigenvalue of $R$.
\end{theorem}

Note that everything becomes nice when $R \geq I_d$. In this case, the smallest
eigenvalue is larger than 1. Hence $\log\det R\geq 0$ and an approximation on
$\tilde{V}$ gives an approximation ration on $V$ (see discussion above).

\subsection{Beyond Linear Models}
Selecting experiments that maximize the information gain in the Bayesian setup leads to a natural generalization to other learning examples beyond linear regression. In particular, in the more general PAC learning setup \cite{...}, the features $x_i$, $i\in \mathcal{N}$ take values in some general set $\Omega$, called the \emph{query space}, and measurements $y_i\in\reals$ are given by 
 \begin{equation}\label{eq:hypothesis-model}
    y_i = h(x_i) + \varepsilon_i
\end{equation}
where $h\in \mathcal{H}$ for some subset $\mathcal{H}$ of all possible mappings $h:\Omega\to\reals$, called the \emph{hypothesis space}, and $\varepsilon_i$ are random variables in $\reals$, not necessarily identically distributed, that are independent \emph{conditioned on $h$}. This model is quite broad, and captures many learning tasks, such as:
\begin{enumerate}
\item\textbf{Generalized Linear Regression.} $\Omega=\reals^d$,  $\mathcal{H}$ is the set of linear maps $\{h(x) = \T{\beta}x \text{ s.t. } \beta\in \reals^d\}$, and $\varepsilon_i$ are independent zero-mean normal variables, where  $\expt{\varepsilon_i^2}=\sigma_i$. 
\item\textbf{Logistic Regression.} $\Omega=\reals^d$,  $\mathcal{H}$ is the set of maps $\{h(x) = \frac{e^{\T{\beta} x}}{1+e^{\T{\beta} x}} \text{ s.t. } \beta\in\reals^d\}$, and $\varepsilon_i$ are independent conditioned on $h$ such that $$\varepsilon_i=\begin{cases} 1- h(x_i),& \text{w.~prob.}~h(x_i)\\-h(x_i),&\text{w.~prob.}~1-h(x_i)\end{cases}$$
\item\textbf{Learning Binary Functions with Bernoulli Noise.} $\Omega = \{0,1\}^d$, and $\mathcal{H}$ is some subset of $2^{\Omega\times\{0,1\}}$, and $$\varepsilon_i =\begin{cases}0, &\text{w.~prob.}~p\\\bar{h}(x_i)-h(x_i), \text{w.~prob.}~1-p\end{cases}$$
\end{enumerate}

In this setup, assume that the experimenter has a prior distribution on the hypothesis $h\in \mathcal{H}$. Then, the information gain objective can be written again as the mutual information between $\beta$ and $Y_S$.
\begin{align}V(S) = \entropy(\beta) -\entropy(\beta\mid Y_S),\quad S\subseteq\mathcal{N} \label{general} \end{align}
This is a monotone set function, and it clearly satisfies $V(\emptyset)=0$. Though, in general, mutual information is not a submodular function, this specific setup leads indeed to a submodular formulation.
\begin{theorem}The value function given by the information gain \eqref{general} is submodular.
\end{theorem}
\begin{proof}
The theorem is proved in a slightly different context in \cite{guestrin}; we repeat the proof here for the sake of completeness: \stratis{add one-line proof} 
\end{proof}

This theorem that implies that learning an \emph{arbitrary hypothesis, under an arbitrary prior} when noise is conditionally independent leads to a submodular value function. Hence the universally truthful 7.91-approximate mechanism by Chen \emph{et al.} \cite{chen} immediately applies in the value query model. Moreover, in cases where maximizing \eqref{general} can be done in polynomial time in the full-information setup, the truthful $8.34$-appoximate mechanism of Chen \emph{et al.}~applies again. However that, in many scenarios covered by this model (inlcuding the last two examples above), even computing the entropy under a given set might be a hard task---\emph{i.e.}, the value query model may not apply. Hence, identifying learning tasks in the above class for which truthful or universally truthful constant approximation mechanisms exist, or studying these problems in the context of stronger query models such as the demand model \cite{dobz2011-mechanisms,bei2012budget} remains an interesting open question. 

%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). 
