Postscript on Stormy Daniels for the Statistically Inclined

In the original draft we summed up our findings as follows:

Ultimately, our model suggests that the probability of a set of payments coincidentally coming so close to $130,000 is approximately .1%, or one out of one thousand. Therefore, the probability that the Trump campaign payments were somehow related to the Daniels payoff is about 99.9%.

As several people pointed out, this is a somewhat overbroad statement of our conclusions, because it implies the probability that the payments were Daniels-related is, quite literally, one minus the probability of the null.

This is not quite accurate. First, there is also some probability that another non-random factor caused these seemingly unrelated payments, such as the campaign laundering $130,000 for another purpose the same week Daniels was negotiating for her payout. The payments would not be caused by random chance, but they would not be related to Daniels, either.

But we believe that any reasonable observer would conclude that the Daniels story is, by a very substantial margin, the most probable non-random explanation — so much so that other non-random explanations can be largely ignored.

This is primarily because the revelation that five payments summed to $130,000 was not random discovery, but based on an initial theory derived from earlier reporting. That reporting included both the sum and timing of the Daniels payout. Simply put, to find another hidden payment of $130,000 within this period would be a mind-boggling coincidence.

The more robust critique of the original finding was Bayesian. If the prior probability of the Trump campaign using its funds to pay Daniels was extremely low, then even demonstrating that the observed outcome was very unlikely would not create a significant probability that the payments were related to Daniels. However, in order to significantly undermine our findings, the prior probability would have to be deemed almost infinitesimally small — nearly unthinkable. We trust that most observers do not find the prospect of Trump campaign payoffs so unlikely.

If we accept even a moderately low initial prior probability that Trump’s campaign funding was used to pay Daniels — say, a 1% chance — the extremely low probability of the observed outcome overwhelms that prior. As a rough estimate using the 1% figure, a simple application of Bayes’ theorem puts the probability of Trump’s campaign paying Daniels, given payments summing to $130,000, around 80%. If we use a more realistic prior probability of 5% or 10%, the likelihood that the payments were Daniels-related quickly climbs back to 95% or higher.

As this is a meant to be a practical analysis and not a conclusive determination of the “correct” probability that the payments were related to Daniels, we feel comfortable discussing the probabilities as essentially falling into two buckets: the null hypothesis, and the Daniels hypothesis. (We also feel that keeping the findings relatively comprehensible to lay readers is important, given the political nature of the issue.) Regardless of whether that correct probability is 80%, 95%, or 99.9%, we believe these payments are worth further investigation.