## 2020-10-19

### Book Review: "The Drunkard's Walk" by Leonard Mlodinow

There are other aspects of human behavior that this book fails to adequately capture; these may technically be beyond the scope of this book, but I think they are worth noting anyway, as they speak to larger problems with the ability of people (even those well-trained in STEM fields) to really understand probability theory. The second chapter goes over many examples of how, in the technical language of probability, given events $$A$$ and $$B$$, certain questions can be framed such that laypeople and professional specialists (particularly doctors & lawyers) fall into the trap of believing that $$\operatorname{Pr}(A \cap B) > \operatorname{Pr}(A)$$ even though the opposite is mathematically always true. However, I can already see that the phrasing of many of those questions, particularly the way that events $$A$$ & $$B$$ are juxtaposed (especially if $$B$$ is additional information that may be relevant to the assessment of $$A$$), may make people believe either that what should be interpreted as $$\operatorname{Pr}(A)$$ is actually $$\operatorname{Pr}(A \cap \neg B)$$, in which case $$\operatorname{Pr}(A \cap B) > \operatorname{Pr}(A \cap \neg B)$$ could in fact be true, or that what should be interpreted as $$\operatorname{Pr}(A \cap B)$$ is actually the conditional probability $$\operatorname{Pr}(A|B)$$, in which case $$\operatorname{Pr}(A|B) > \operatorname{Pr}(A)$$ could in fact be true. This speaks more to the way that natural human language is unsuited to the subtleties of the language of probability theory, yet rather than address these possibilities, the author again leaves the discussion there, implying disdain for people who are too stupid to know better. Another problem is that throughout the book, the author raises the question of how to determine whether a particular sequence of observations of outcomes for a process that may be random reflects a specific probability distribution model, but never clearly explains how to do this in practice, instead only giving hints about this through various examples. This is related to the question of why one may prefer an explanation based on probabilities than based on deterministic phenomena, particularly for small sample sizes. For this, I will give an example. Consider exactly 5 observations of an event, which has binary outcomes (either success or failure), for which no other observations are made, and for which in all of those 5 observations, success occurs every single time. Intuitively, laypeople might be inclined to believe that there is a deterministic cause of this, while if a probability theorist were to initially believe that this is consistent with a binomial distribution with $$(N, p) = (5, 0.6)$$ but then later revise this to $$(N, p) = (5, 0.99)$$, laypeople could reasonably wonder why this would be justified, and why the probability theorist refuses to believe in the possibility of some deterministic causal relationship. Of course, this is a contrived example, I understand why causation needs to be proved as an alternative to a null hypothesis, and I understand that probability distributions closer to uniform probabilities are favored as those that maximize entropy (which essentially means that subject to certain known constraints, the probability distribution that best reflects the state of ignorance about a system is closest to uniform), but the author does not properly explain these points. Finally, the broadest problem with this book is that the author only superficially acknowledges the issue that if every calculation in probability theory or statistics, whether of a certain event happening, a string of events being a true "hot streak", or a model fitting data correctly, is itself a probability, then the aforementioned disconnect of this language of probability from natural human language makes it difficult to translate probabilities into robust rules for deterministic (usually binary) decisions that laypeople must make; this is related to the idea that in game theory, a single person playing a single-shot game cannot play a mixed strategy, and the concept of a mixed strategy only makes sense in the context of observing a large ensemble of independent players, possibly playing repeatedly. Perhaps asking the author to address this problem is too much, but I still feel like such failures diminish the book in comparison to its hype.