The search for extraterrestrial (alien) life is one of the greatest scientific quests yet raises fundamental questions about just what we should be looking for and how. We approach alien hunting from the perspective of an experimenter engaging in binary classification with some true and confounding positive probability (TPP and CPP). We derive the Bayes factor in such a framework between two competing hypotheses, which we use to classify experiments as either impotent, imperfect or ideal. Similarly, the experimenter can be classified as dogmatic, biased or agnostic. We show how the unbounded explanatory and evasion capability of aliens poses fundamental problems to experiments directly seeking aliens. Instead, we advocate framing the experiments as looking for that outside of known processes, which means the hypotheses we test do not directly concern aliens per se. To connect back to aliens requires a second level of model selection, for which we derive the final odds ratio in a Bayesian framework. This reveals that it is fundamentally impossible to ever establish alien life at some threshold odds ratio, $\mathcal{O}_{\mathrm{crit}}$, unless we deem the prior probability that some as-yet-undiscovered natural process could explain the event is less than $(1+\mathcal{O}_{\mathrm{crit}})^{-1}$. This elucidates how alien hunters need to carefully consider the challenging problem of how probable unknown unknowns are, such as new physics or chemistry, and how it is arguably most fruitful to focus on experiments for which our domain knowledge is thought to be asymptotically complete.