[smarnach]

> we should default to the calculation of 2-4x the rate.

No we should not. We should accept that we don't have any statistically meaningful number at all, since we only have a single incident.

Let's assume we roll a standard die once and it shows a six. Statistically, we only expect a six in one sixth of the cases. But we already got one on a single roll! Concluding Waymo vehicles hit 2 to 4 times as many children as human drivers is like concluding the die in the example is six times as likely to show a six as a fair die.

[akoboldfrying]

More data would certainly be better, but it's not as bad as you suggest -- the large number of miles driven till first incident does tell us something statistically meaningful about the incident rate per mile driven. If we view the data as a large sample of miles driven, each with some observed number of incidents, then what we have is "merely" an extremely skewed distribution. I can confidently say that, if you pick any sane family of distributions to model this, then after fitting just this "single" data point, the model will report that P(MTTF < one hundredth of the observed number of miles driven so far) is negligible. This would hold even if there were zero incidents so far.

[smarnach]

We get a statistically meaningful result about an upper bound of the incident rate. We get no statistically meaningful lower bound.

[NewJazz]

Uh, the miles driven is like rolling the die, not hitting kids.

[smarnach]

Sure, but we shouldn't stretch the analogy too far. Die rolls are discrete events, while miles driven are continuous. We expect the number of sixes we get to follow a binomial distribution, while we expect the number of accidents to follow a Poisson distribution. Either way, trying to guess the mean value of the distribution after a single incident of the event will never give you a statistically meaningful lower bound, only an upper bound.

[akoboldfrying]

The Poisson distribution is well approximated by the binomial distribution when n is high and p is low, which is exactly the case here. Despite the high variance in the sample mean, we can still make high-confidence statements about what range of incident rates are likely -- basically, dramatically higher rates are extremely unlikely. (Not sure, but I think it will turn out that confidence in statements about the true incident rate being lower than observed will be much lower.)