  In experimental design, we are given $n$ vectors in $d$ dimensions, and our
goal is to select $k\ll n$ of them to perform expensive measurements, e.g., to
obtain labels/responses, for a linear regression task. Many statistical
criteria have been proposed for choosing the optimal design, with popular
choices including A- and D-optimality. If prior knowledge is given, typically
in the form of a $d\times d$ precision matrix $\mathbf A$, then all of the
criteria can be extended to incorporate that information via a Bayesian
framework. In this paper, we demonstrate a new fundamental connection between
Bayesian experimental design and determinantal point processes, the latter
being widely used for sampling diverse subsets of data. We use this connection
to develop new efficient algorithms for finding $(1+\epsilon)$-approximations
of optimal designs under four optimality criteria: A, C, D and V. Our
algorithms can achieve this when the desired subset size $k$ is
$\Omega(\frac{d_{\mathbf A}}{\epsilon} + \frac{\log 1/\epsilon}{\epsilon^2})$,
where $d_{\mathbf A}\leq d$ is the $\mathbf A$-effective dimension, which can
often be much smaller than $d$. Our results offer direct improvements over a
number of prior works, for both Bayesian and classical experimental design, in
terms of algorithm efficiency, approximation quality, and range of applicable
criteria.
