  Independence testing is a classical statistical problem that has been
extensively studied in the batch setting when one fixes the sample size before
collecting data. However, practitioners often prefer procedures that adapt to
the complexity of a problem at hand instead of setting sample size in advance.
Ideally, such procedures should (a) stop earlier on easy tasks (and later on
harder tasks), hence making better use of available resources, and (b)
continuously monitor the data and efficiently incorporate statistical evidence
after collecting new data, while controlling the false alarm rate. Classical
batch tests are not tailored for streaming data: valid inference after data
peeking requires correcting for multiple testing which results in low power.
Following the principle of testing by betting, we design sequential kernelized
independence tests that overcome such shortcomings. We exemplify our broad
framework using bets inspired by kernelized dependence measures, e.g., the
Hilbert-Schmidt independence criterion. Our test is also valid under
non-i.i.d., time-varying settings. We demonstrate the power of our approaches
on both simulated and real data.
