The central argument in this article is that the probability of very large natural pandemics is more uncertain than either previous analyses or the historical record suggest. In public health and health security analyses, global catastrophic biological risks (GCBRs) have the potential to cause "sudden, extraordinary, widespread disaster," with "tens to hundreds of millions of fatalities." Recent analyses focusing on extreme events presume that the most extreme natural events are less likely than artificial sources of GCBRs and should receive proportionately less attention. These earlier analyses relied on an informal Bayesian analysis of naturally occurring GCBRs in the historical record and conclude that the near absence of such events demonstrates that they are rare. This ignores key uncertainties about both selection biases inherent in historical data and underlying causes of the nonstationary risk. The uncertainty is addressed here by first reconsidering the assumptions in earlier Bayesian analyses, then outlining a more complete analysis accounting for several previously omitted factors. Finally, relationships are suggested between available evidence and the uncertain question at hand, allowing more rigorous future estimates.