17. XI 2020
Woke up this morning, decided to kill my ego, it ain’t ever done me no good, no how. (Sturgill Simpson, opening verse of Just Let Go)
Triple negatives exist only in country music and in statistics. While their deployment in everyday language is considered excessive and, outside of the bible belt, rejected as stylistically undesirable, musical lyrics sometime require them for rhythm or rhyme. In statistics, on the other hand, triple negatives play a specific role and are encountered regularly. They are there to prevent us from overstating a statistically significant result as certainty.
Statistical inference is based on drawing probabilistic conclusions based on the analysis of a finite sample of data. The sample that we work with is our universe, but although it can be representative, it never contains all information. Because of that, there is always residual skepticism that we didn’t see it all and have not taken everything into account. As a consequence, based on statistical analysis, it is unacceptable to say that something is true, but instead we have to settle for a slightly weaker statement that something is not false. In statistics, one cannot simply accept, but can only fail to reject. And that can be done only with a certain confidence level without which the statistical results are incomplete.
When testing a hypothesis in a statistical context, statements like “This will happen with certainty” have no place; the strongest statement one can make is “This will happen almost surely”. For example, if all air in a room is evacuated and we release a small amount in one corner, after some time the air molecules will be evenly distributed throughout the entire room. Their distribution will stay uniform almost surely. This does not mean that the air molecules cannot find themselves all in one corner again – there is nothing in physical laws that prohibits that — it is just that probability of that happening is extraordinarily small.
Results of current Presidential elections, as they are an outcome of statistical measurements and inferences, thus, need to be expressed accurately. First the outcome needs to be framed properly. Although the bottom line is that after Jan-20-2021, there will be a new president in the White House, if one is to be precise, Democrats didn’t win – Trump failed in not losing.
And to be perfectly clear, despite all his maneuvers and PR stunts, Trump doesn’t really believe that he won – he just fails to reject that he didn’t lose. Of course, the subtlety of this difference completely eludes him. In his overdramatized “revolt”, which is really a money raising scam, he is being less delusional and more ignorant and confused.
While the claim of victory represents a statement of certainty and, as such, can be easily invalidated, the failure to reject is incomplete without giving the confidence interval about that statement and, in its incompleteness, it is deprived of its probabilistic context. The official results of the Elections reject the fact that Trump did not lose with probability greater than 99.999%; Trump’s failure to accept the results is the complement of that – it has a probability, which is less than 1/1000th percent.
The consequence of the underlying incompleteness is that in the eyes of Trump’s innumerate followers, it opens an ill-conceived possibility of hope. The reality of the failure to reject is a very low confidence statement, one that has an infinitesimal probability of realization, at best less than a fraction of a 1/1000th of a percent.
While this is a probability that has no value for any practical purposes, when put in the context of the general Republican narrative of the last four years, the core of which had been based largely on conspiracy theories, with roughly the same order of magnitude probability of realization, it could have equal merit as everything else they have stood for. After all, more than 40% of Trump supporters are evangelical Christians who believe in the magic of thoughts and prayers and reality of miracles, which have even a lower probability of realization than Trump’s chance of successfully contesting the Election’s outcome.
 On Friday, 6-Nov, one day before the Elections would be called by the media, it was down to three states that remained to be counted: AZ, PA, and GA. Trump had to win all three in order to win. The probabilities of Trump’s victory in each state, which were at the time still called by different outlets were as follows: PA 8%, AZ 23%, and GA 12.4%. The probability of winning all three states (= product of the three probabilities) is 0.2%. Since then, with more votes counted, increased margins, dismissed disputes and further verification, the probabilities for Trump’s victory in each of the three states have declined precipitously and the joint probability of winning all three, which he would still need if his “arguments” are to hold, has dropped by several orders of magnitude.