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diff --git a/problem.tex b/problem.tex index 15f0c11..c21135b 100644 --- a/problem.tex +++ b/problem.tex @@ -3,7 +3,10 @@ The theory of experimental design \cite{pukelsheim2006optimal,atkinson2007optimum,chaloner1995bayesian} considers the following formal setting. % studies how an experimenter \E\ should select the parameters of a set of experiments she is about to conduct. In general, the optimality of a particular design depends on the purpose of the experiment, \emph{i.e.}, the quantity \E\ is trying to learn or the hypothesis she is trying to validate. Due to their ubiquity in statistical analysis, a large literature on the subject focuses on learning \emph{linear models}, where \E\ wishes to fit a linear function to the data she has collected. %Putting cost considerations aside -Suppose that an experimenter \E\ wishes to conduct $k$ among $n$ possible experiments. Each experiment $i\in\mathcal{N}\defeq \{1,\ldots,n\}$ is associated with a set of parameters (or features) $x_i\in \reals^d$, normalized so that $$b\leq \|x_i\|_2\leq 1,$$ for some $b>0$. Denote by $S\subseteq \mathcal{N}$, where $|S|=k$, the set of experiments selected; upon its execution, experiment $i\in S$ reveals an output variable (the ``measurement'') $y_i$, related to the experiment features $x_i$ through a linear function, \emph{i.e.}, +Suppose that an experimenter \E\ wishes to conduct $k$ among $n$ possible +experiments. Each experiment $i\in\mathcal{N}\defeq \{1,\ldots,n\}$ is +associated with a set of parameters (or features) $x_i\in \reals^d$, normalized +so that $$b\leq \|x_i\|^2_2\leq 1,$$ for some $b>0$. Denote by $S\subseteq \mathcal{N}$, where $|S|=k$, the set of experiments selected; upon its execution, experiment $i\in S$ reveals an output variable (the ``measurement'') $y_i$, related to the experiment features $x_i$ through a linear function, \emph{i.e.}, \begin{align}\label{model} \forall i\in\mathcal{N},\quad y_i = \T{\beta} x_i + \varepsilon_i \end{align} |