TL;DR - the fundamental assumption of probability theory is one of ignorance. This assumption is too easy to break in most contexts and leads to unfounded confidence in conclusions.
There are many circumstances in which uncertainty is warranted.1 Gas temperature measurements, weather forecasts, horse races, coin flips, and clinical trials all have some uncertainty involved. Probability theory is the science that finds commonality among these seemingly disconnected phenomena. We can observe, for example, that the summation of many “repeatable” random events, properly normalized, begins to look like a gaussian distribution (aka the central limit theorem). We can notice common shapes in the histograms of these repeatable experiments, such as “fat-tailed” or “power law” distributions. And if the event is not repeatable, we can at least apply the rules of probability theory to avoid inconsistencies in our thinking (which would allow a savvy adversary to take advantage of us when gambling).
However, I believe there are tasteful and distasteful applications of probability theory. This is because the application of probability to a particular event requires a suspension of disbelief. To consider an event as repeatable and iid is to accept that the causal factors driving the outcome are (practically) unobservable and therefore ignorable. In effect, it means giving up on deeply understanding a causal explanation of the phenomena and instead sweeping the details under the rug of “the distribution.”
This makes probability theory the science of last resort. Only after truly exhausting your ability to investigate causal factors and processes should you indulge in probabilistic thinking. Doing otherwise is a cop-out, one that dangerously feels “scientific.”
Examples of Distaste
The tastefulness of a particular application of probability theory is a matter of context. Consider the humble coin flip. A lay-person may reasonably assume that this event is a repeatable, iid experiment with a uniform prior; they may be working with a variety of coins, fingers, and surfaces, and may not have equipment available to make precise measurements. To the physicist, however, this is obviously a cop-out. The physicist knows that the coin’s trajectory can be precisely captured by the laws of classical mechanics, and therefore predicted with almost certainty.2 Where the average person gives up and shrugs, the scientist continues searching for explanations.
Similar things can be said about drawing a card from a deck of cards. A casual observer may reasonably assign uncertainty to the event. But to the magician who controls the precise method of shuffling, this is obviously a cop-out.
Or consider a randomized clinical study in which a drug harms patients in 0.01% of cases. It’s easy to sweep the causal factors under the rug and assume the effects are “randomly distributed.” But we can imagine more information gathering revealing that all instances of harm occured in an ethnic minority. In practice, “randomness” is more often a cop-out than an unavoidable facet of the system under study.
My struggle with probability theory is that it lends itself to distaste. Humans are lazy, and when presented with the option of doing more investigative leg work or simply assuming data is randomly iid, they will often choose the latter, especially when the latter appears to be “scientifically and mathematically rigorous.” However, mathematics are only as correct as the assumptions made at the beginning, and by hiding the causal factors of an event behind the abstraction of a “probability distribution” we deprive ourselves of the ability to identify when those causal factors change and our assumptions no longer hold (i.e. the distribution shifts).3
And even when the assumption of iid is justified, the logic of probability theory is more often misapplied than not, despite supposedly being a “guide towards logical consistency.” In my experience, probability theory is more often used to prove a point in scientific papers than it is a self-check for correctness. Just look at the p-value crisis of the 2010’s.4 As they say, there are lies, damned lies, and statistics. Even Bayesians, who seem to think they’re always right, can occasionally get it wrong as evidenced by E.T. Jaynes’ humorous exploration of a paper that proved a woman had ESP.56
Given the prevalence of misapplication, I can only conclude that probability theory needs to be redesigned as a mental device. I don’t want to be a probability theory nazi, but all the evidence seems to indicate that probability theory is a science for people who have given up on science, rather than the rigorous system of analysis it purports to be.
Many events are unpredictable to us in practice, either because the laws governing the outcome are not known, or because the laws are known but the observations required are too arduous to make. Sometimes the required observations are too numerous to collect (as in statistical mechanics), and other times the non-linear, chaotic nature of the system necessitates observations that are too precise to be practical. Quantum experiments seem to be inherently unpredictable, although whether this is a fundamental facet of nature is a matter of debate. ↩
For example, I may experiment with a coin and decide that it is fair when tossing it onto a wooden surface, only to discover later that the coin is magnetized and slightly biased towards heads on metallic surfaces. ↩
I also highly recommend Chapter 10. ↩