Donald Trump's re-election and Thomas Bayes' theory of conditional probability

Donald Trump was elected as the 47th president in the election held last week, securing a second term after a four-year hiatus with a significant margin in both votes and delegates; he will take office in January. In last week's issue of MIT's Download magazine, one of the United States' prominent technology universities, there was the following commentary regarding Trump: "Donald Trump's decisive victory is a stunning setback for the fight against climate change." Many other commentaries mention that he will end the Russia-Ukraine war, prioritize China as the primary adversary rather than Russia, will not exert effort to stop the inhumane war in Gaza, and will adopt a production-oriented economic model while supporting the widespread adoption of cryptocurrencies. 

These and similar commentaries emphasize that Trump's second term will be spent in chaos. There may be some truth to the chaos argument, as during his first term, we would give examples of his actions and statements in our university lectures when discussing the concepts of "post-truth" and "fake news." In fact, as will be remembered, Twitter founder Jack Dorsey suspended the account Trump used at the time, citing his post-truth statements as the reason. Later, Elon Musk, who bought Twitter and renamed it X, and who is currently Trump's most important supporter and the world's most significant opinion leader or "influencer," had the account reopened by conducting a poll on X in the name of "giving freedom to all thoughts." He also managed to be included in the family photo after Trump's election victory. When the Internet was opened to individual and commercial use worldwide in 1993, while the Democrats were in power, then-Vice President Al Gore also emphasized freedom with the slogan "building highways for the free flow of information." 

The reason I mention all of this is to suggest that past events can have a significant impact on events that may occur in the future. Of course, this idea is not new and was also theorized by the famous mathematician Thomas Bayes (1702-1761). As one of the important figures of the Enlightenment era, the ideas and theories he put forward are still used today in the fields of statistics and economics. The algorithms used by social media platforms—which some of us complain about and others are pleased with—that decide what we will like, what we will buy, and who we will vote for, are fundamentally based on his theories. Conveying this theory through a historical story from David Salsburg's book "The Lady Tasting Tea" (2001) might make it easier to both read and understand.

The Republic of Venice was a major power in the Mediterranean from the eighth century until the beginning of the eighteenth century. At the height of its empire, Venice controlled much of the Adriatic coast and the islands of Crete and Cyprus; it also held a monopoly on trade from the East to Europe. Venice was governed by noble families who established a form of democracy among themselves. The person holding the title of head of state was called the "doge." From the founding of the Republic in 697 until Venice was captured by Austria in 1797, more than one hundred and fifty people served as doge; while some served for less than a year, one remained in this office for thirty-four years. Upon the death of a doge, the republic would enter a complex election process. A small group determined by lottery from among the senior members of noble families was chosen as "electors." These electors would then determine new members to join them, and then a small number from this expanded group would be chosen by lottery. This process continued through several stages, and in the final stage, a final group was chosen from among the electors to determine the doge. Finally, this group would deliberate among themselves to elect the head of state.

In the early periods of the republic's history, negotiators (electors) were chosen by preparing a group of wax balls, some of which were empty and some of which contained pieces of paper with the word "elector" written on them, at each stage. By the seventeenth century, the final stages were carried out using gold and silver balls of the same size. 

When Doge Rainieri Zeno died in 1268, there were thirty negotiators in the second stage, and thirty wax balls had been prepared. Nine of these contained papers with the word "elector" written on them. A small child was brought in to pick the balls and give them to the candidates. The child would pick a ball from the basket and give them to the negotiators one by one. The negotiators would open the ball to see if they were a negotiator in the next stage, and the process continued like this.

Before the child picked the first ball, the probability of each member of the group being a negotiator in the next stage was 9/30. If the first ball was empty, the probability of each of the remaining members being chosen rose to 9/29. If the first ball contained a paper, the probability of each of the remaining members being chosen fell to 8/29. When the second ball was picked and opened, the probability of the next member being chosen as a negotiator would similarly decrease or increase depending on the result of that draw. This process continued until all nine marked balls were picked. At that point, the chance of the remaining members being a negotiator in the next stage dropped to zero.

This is a typical example of conditional probability. The probability of a specific member being a negotiator in the next stage depended on the selection of the balls before that member's selection. Similarly, we know today that the probability of a patient having small-cell lung cancer depends on that patient's smoking history. 

All the formulas developed in the eighteenth century to deal with conditional probabilities were based on the idea that the condition events occurred before the event being sought. Towards the end of the century, Thomas Bayes made a surprising discovery while playing with conditional probability formulas: The formulas had an internal symmetry.

Suppose there are two events that occur over a period of time, such as shuffling a deck of cards and then dealing a five-card poker hand. Let's call these events "before" and "after." It makes sense to talk about the probability of the "after" event depending on the "before" event. If we cannot shuffle the cards well, this affects the probability of finding two aces in a poker hand. Bayes discovered that we could also calculate the probability of the "before" event depending on the "after" event. But this seemed illogical. This was like determining the probability that a deck of cards contained four aces once a poker hand containing two aces had been dealt. Or like determining the probability that a patient had smoked based on the knowledge that they had lung cancer. Or like determining that a lottery draw was fair, solely by considering that a person named Ali Lucky (who actually won the lottery) was the winner.

Bayes did not continue working on these calculations, but his work was found among his papers after his death and was published posthumously. Bayes' theorem has since complicated the mathematics of statistical analysis. Bayes' inversion of conditional probability is often very logical. We call this a case-control study. For example, cases of a disease (we could call this the post-disease group) are brought together and compared with a group of people who do not have this disease but are similar to the disease-holders in other respects (we could call this the pre-disease group or control group). The effects of smoking on both heart disease and lung cancer were first discovered in this way. 

It is quite simple to say, and possible to explain with the theory of conditional probability: Trump did not get along well with China during his first term. He imposed sanctions. Therefore, it is highly probable that we will experience a similar situation in this second term. So, how can we make a prediction about Trump's second term using the internal symmetry of Bayes' theory of conditional probability? Because since Bayes discovered that we could also calculate the probability of the "before" event depending on the "after" event, why shouldn't we apply this to Trump's new term? With a simplified example, we can think that we can do it like this: We can compare the counter-moves China is likely to make in the new Trump term with China's counter-attitudes to the sanctions Trump imposed on China during his first term. Thus, we can see how probable the sanctions Trump applied in his previous term are against those moves China might make. In other words, we can calculate the probabilities of him re-imposing the sanctions he applied against China during his first term. Or we can just sit and wait to see what Trump will do.