Statistically consistent way to handle E(known outcomes) < deaths

[adamkucharski]

The current implementation in CFR is based on calculating E(known outcomes) to compare to totals deaths. However, in extreme examples, such as small outbreaks with a very high CFR (like Ebola in Yambuku in 1976), there can be occasionally situations where E(known outcomes) < deaths and hence the binomial likelihood calculation is not valid. In this situation the code currently returns NA to make the problem clear to the user.

In the longer-term, a more statistically consistent approach would be to integrate over the possible known outcomes, rather than just using the expectation. This would allow calculation on the plausible known outcomes < deaths and automatic omission of known outcomes > deaths. Something like the following:
$E(CFR) = \sum_i P(\text{i known outcomes so far | cases, deaths}) E(\text{CFR | i known outcomes so far}) $