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THE NATIONAL INTEREST JAN. 10, 2024

‘Biden Against Democracy,’ The Right’s Favorite Trump Rationale Why the anti-anti-Trump right loves Trump’s message.

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THE NATIONAL INTEREST JAN. 10, 2024

‘Biden Against Democracy,’ The Right’s Favorite Trump Rationale Why the anti-anti-Trump right loves Trump’s message.

ne of Donald Trump’s most consistent election messages is that Joe Biden, not he, is the threat to democracy. The New York Times has an excellent story explaining how this message, which Trump summarizes as “BAD” (Biden Against Democracy), is designed to neutralize Trump’s most important political weakness.

The article puts this strategy in the context of Trump’s lifelong habit of accusing his opponents of whatever Trump himself is doing in order to muddy the waters and foster cynicism. But there is another aspect of this argument the article does not consider: BAD is not only a Trumpian schoolyard taunt but also an argument that is being advanced by putatively serious conservative intellectuals.

The literal version of Trump’s argument — which casts Biden as an authoritarian tyrant who stole the election and is now hell-bent on imprisoning his opponent — is obviously promoted by his most enthusiastic supporters. But the main purpose of the claim is to turn the democracy question into a tie. Maybe Trump has been a bad boy (January 6 and all that), the argument will go, but Biden has also threatened democracy. Since both candidates are authoritarians, we might as well vote for the one who will support our favorite domestic policies.

Other Distribution Models

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Other Distribution Models

There are myriad distributions to account for all sorts of problems and uncertainties. We will explain a few of these here, but feel free to ignore this part if you find it too technical. Another famous one is the Poisson distribution—so-called after the 19th century French mathematician, Simeon Denis Poisson—which is used to model the occurrence of rare events, like earthquakes. See Figure 5-3.

Figure 5-3: The Poisson distribution for different values of the parameter lambda

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Another popular distribution is the exponential distribution. This is a distribution of waiting times. If we are modeling the arrival of ambulances, we might ask how long will it take until the next ambulance arrives? See Figure 5-4.

Figure 5-4: Exponential distribution for different parameter settings

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Then there is the geometric distribution, which can tell us how many failures we will get until we reach a success. For example, how many times do I need to roll a die, until I roll a 6?

The binomial distribution describes the number of successes in a number of trials—for example, if I am betting on getting a 6 on a roll of a die, how many successes will I get if I roll the die 100 times?

Chapter 5: What’s Math Got To Do With It? The Power of Probability Distributions

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Chapter 5: What’s Math Got To Do With It? The Power of Probability Distributions

Overview

“Probability theory is nothing but common sense reduced to calculation.”

—Pierre-Simon Laplace

A “probability distribution” is one of the most significant concepts ever devised in mathematics. In an uncertain context, we can’t predict the outcome 100 percent of the time. If we’re trying to predict the next roll of a die, or the outcome of a sports game, we will get it right sometimes, but not all the time. What if we had a way to express the probabilities of the different outcomes? From there, we can then calculate useful results, like the most likely outcome, or other useful quantities like the variance (which you’ll look at shortly).

Chapter 1 explained that the two kinds of uncertainty are epistemic and aleatoric. Epistemic uncertainty exists because of a lack of knowledge. For example, let’s say that we are using a medical instrument to take some blood measurements. We want to use these measurements to predict whether someone has contracted a serious virus. These measurements are not 100 percent accurate. In fact, we know that any measurement could be around 20 percent off. In ten years’ time we might have developed an instrument that is more precise—say only 10 percent off. This is epistemic uncertainty.

In a coin-tossing experiment, we face aleatoric uncertainty. Assuming the coin is fair, whether it lands on heads or tails is a random phenomenon. Likewise, we can’t know what the result will be before someone rolls the die. The word “epistemic” comes from the Greek for “knowledge,” episteme; “aleatoric” comes from the Latin word for a die, alea—or aleator, a dice player. It’s a useful way to remember the distinction.

After Laplace

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After Laplace

Before Bayes’ resurgence, the “central limit theorem” stole the show, and authorities started collecting data on everything, the opinion that probability could be subjective sounded specious and non-rigorous. John Stuart Mill argued that probability was ignorance, coined into science.

However, some scientists still used it. Once such person was Joseph Louis François Bertrand (1822–1900), a French mathematician with contributions to many scientific fields, from economics to thermodynamics. Bertrand devised a method, based on Bayes, for artillery firings. The method dealt with uncertainties, such as the wind, or the enemy’s position, in order to optimize the firings.

The French polymath Henri Poincaré (1854–1912) intervened in the famous Dreyfus Affair, proffering Bayes’ theorem in court as evidence that Lieutenant Alfred Dreyfus wasn’t a traitor. It was one of the most famous trials in modern French history and Bayes’ theorem saved Dreyfus from life imprisonment in what was then French Guinea. Bayes’ theorem is arguably the only sensible way to treat evidence in court, and Poincaré recognized this, in spite of him being a frequentist.

It is deeply unfortunate that a court in England in 2011 ruled that Bayes’ theorem can no longer be used in trials. [3] It might seem odd for a mathematical theory (or lack thereof) to have such a bearing on—for example—cases of first-degree murder. But that’s exactly what happened to Sally Clark in 1999, when she was wrongfully convicted of smothering her two infant sons; the jurors and judges erroneously rested their decision on the statistical unlikelihood of two siblings dying of cot death. In fact, it was far more statistically rare for a mother to willfully kill both her children, and Clark’s sentence was overturned and she was finally freed in 2003. It might sometimes be difficult to grasp the intuition behind it, but Bayes theorem is very powerful.

The Formulation of Bayes’ Theorem

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The Formulation of Bayes’ Theorem

Don’t be put off by the esoteric appearance of the following equations. At first glance, you might feel like you’ve been sent back to an incoherent algebra class, but understanding Bayes’ theorem is likely to be easier than you think.

Bayes’ theorem is formulated as follows:

Recall that Bayes used conditional probability. This is the probability of an event given another event. The conditional probability of an event A taking place, given an event B, is written as P(A|B).

Bayes’ Theorem translates in plain language as follows:

“The probability of A taking place, given that B has taken place, is equal to the probability of B taking place, given A has taken place, times the probability of A, over the probability of B.”

Another way to break the denominator down (which might make things clearer for those with more mathematical affinity) is the following:

Frequentist or Bayesian?

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Frequentist or Bayesian?

The most popular definition of probability, and arguably the most intuitive, is the frequentist one (also known as frequentism). According to frequentists, an event’s probability is defined as the limit of the event’s frequency in a large number of trials.

What does this mean? Let’s go back to the example of flipping a fair coin. You said that the probability of rolling heads on a single roll is 50 percent. However, how do you know this to be true? What if you roll tails ten times in a row? Would this change the probability of rolling heads? Obviously not. Intuitively, this makes sense, but why?

I ran a coin-tossing experiment (simulated in the R programming language [1]); you can see the results in Figure 4-1. The proportion of heads very quickly converges to 50 percent.

Figure 4-1: Coin tossing experiment

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This is the definition of frequentist probability in practice. If you execute an experiment a large number of times, then the frequencies will converge to their true probabilities.

Frequentist statistics have been the orthodox branch of statistics for most of history. In his Rhetoric, Aristotle wrote that “the probable is that which for the most part happens.” The practice of statistics is based on the belief that you can extract a sample from a population, and then study properties of the population. If we treat each entity in this sample as an experiment, then the more samples we collect, the closer we will get to the truth.

Chapter 4: Probability: To Bayes or Not To Bayes?

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Chapter 4: Probability: To Bayes or Not To Bayes?

Overview

“Life is a school of probability.”

—William Bagehot

What is probability?

If you’re not a data scientist, you likely have an intuitive understanding of probability from school lessons or based on common sense. For many events, it is impossible to always correctly predict the outcome in advance. But we know that some outcomes are more likely than others. Put simply, probability is a way to express and study which outcomes are more, or less, likely to happen. The etymology of the word “probable” has an interesting bearing on how we use it today; it comes from the 14th century French word probable, meaning “provable or demonstrable,” originally derived from the Latin verb probare—“to try, to test.”

If I asked you what’s the probability of getting tails on a coin flip on a fair coin, you would probably say 50 percent. Intuitively it makes sense. However, you might not be able to explain why this should be true mathematically. Coin-flipping is a simple case for calculating probabilities. How would you define the probability of Manchester United winning the next Champions League? You might have your own personal beliefs about which football team is the best (and this belief could change every year), but the way you come up with this probability “feels” different from the coin-flipping case.

Occam’s Razor, Space Invaders, and Lizard People

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Occam’s Razor, Space Invaders, and Lizard People

“Reports that say that something hasn't happened are always interesting to me, because as we know, there are known knowns; there are things we know we know. We also know there are known unknowns; that is to say, we know there are some things we do not know. But there are also unknown unknowns—the ones we don't know we don't know. […] The absence of evidence is not evidence of absence, or vice versa.”

—Donald Rumsfeld

There may never be a better quote than this to capture the implications of decision-making under uncertainty. Donald Rumsfeld was the secretary of defense under George Bush Jr. from 2001 until 2006. Rumsfeld gave this answer when he was asked whether the United States had information about Saddam Hussein selling weapons of mass destruction to terrorist groups. We constantly face decisions that are shrouded in uncertainty, with bigger and smaller consequences. For Rumsfeld and Bush’s government, their decision was, of course, catastrophically massive.

When faced with uncertainty or a shortage of data, we might come up with all sorts of different hypotheses. But how can we choose among different ones? Is it more likely that someone is hiding weapons of mass destruction or not? Is your spouse cheating on you? Is there some secret conspiracy controlling a large part of humanity? We need a rule to help us figure this out.

Chapter 3: Swans and Space Invaders

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Chapter 3: Swans and Space Invaders

Overview

“Mediocristan is where we must endure the tyranny of the collective, the routine, the obvious, and the predicted; Extremistan is where we are subjected to the tyranny of the singular, the accidental, the unseen, and the unpredicted.”

—Nassim Nicholas Taleb, The Black Swan: The Impact of the Highly Improbable

One of the most famous current scholars of uncertainty is Nassim Nicholas Taleb. Now a scholar, essayist, and author of the Incerto collection of works (of which The Black Swan is the second title), Taleb used to work in finance—arguably the industry most exposed to uncertainty. Fortunes are made and lost in the blink of an eye.

Taleb’s view is that one of the main issues around finance and economics is that they belong to what he calls “Mediocristan” (think of it as a place); on the other hand, much of our world belongs to “Extremistan,” What Taleb refers to as Mediocristan has dominated Western thought to a large extent. Mediocristan describes statistical thought that is concentrated around averages and the “normal” distribution. As learned saw earlier, that method of reasoning has important consequences.

Much of our world is characterized by normality. You can see what a normal distribution looks like in Figure 3-1.

Figure 3-1: The standard normal distribution. The standard normal distribution has a mean of 0 and a standard deviation of 1. It is used in countless mathematical models

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The First Black Swan

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The First Black Swan

“The problem with experts is that they do not know what they do not know.”

―Nassim Nicholas Taleb, The Black Swan: The Impact of the Highly Improbable

Let’s say that there’s a new exotic creature that has appeared in the realm of zoology, called a “swan.” This is pretty interesting—no one has ever heard of swans before. The few people who have encountered swans say that all of them have been white.

You are a swan enthusiast and, charmed by the stories you’ve heard about these magnificent creatures, you decide to study them more carefully. You embark on a journey to take as many photos of swans as possible. After years of laborious efforts, you capture 1,000 pictures of swans, all of them white. Can you conclude that all swans on the planet are white? How certain are you? Can you express this certainty in a probability, p1, from 0 to 1?

How much would your certainty level change if you had taken 10,000 photographs, or a million photographs? Can you come up with a new probability, p2? Would p1 be smaller or larger than p2? If you’re like most people, you would—reasonably—assume that ten thousand or a million photographs give you more evidence and greater confidence than only 1,000 photographs. So, p2 > p1.

Let’s say that all swanologists in the world for ten years have recorded only pictures of white swans. So, the consensus is that swans are white.

But then someone shows up with a picture of a black swan. [3] What now? Now the probability of the statement “all swans are white” is 0. Pure zero. Absolute certainty about the negation of a fact.