Let us start by examining a part of the article that everyone can see is horrendous. When I supply proofs, in a future post, that other parts of the article are wrong, few of you will follow the details. But even the mathematically uninclined should understand, after reading what follows, that
- the authors of a grotesque mangling of lower-level mathematics are unlikely to get higher-level mathematics correct, and
- the reviewers and editors who approved the mangling are unlikely to have given the rest of the article adequate scrutiny.
Before proceeding, you must resolve not to panic at the sight of Greek letters and squiggly marks in the passage displayed below. Note that although the authors attempt to define a term, you need not know the meaning of the term to see that their “definition” is fubar. First, look over the passage (reading as “zeta” and as “epsilon”), and get what you can from it. Then move on to my explanation of the errors. Finally, review the passage.
This is just the beginning of a tour de farce. I encourage readers who have studied random variables to read Section 4.1, and post comments on the errors that they find.
Everyone should see easily enough that the authors have not managed to decide whether the “countable set of events” is a discrete probability distribution or a domain :
A discrete probability distribution is a countable set of events. The set of events [which the authors have just called a discrete probability distribution, and denoted ] is called the domain and signified by
However, puzzling over the undefined term event may keep you from fully appreciating the bizarreness of what you are seeing. The authors in fact use the term to refer to elements of the domain. Here is an emendation of the passage, with event defined in terms of domain, instead of the reverse:
A discrete probability distribution is a countable set of events. The set of events [which the authors have just called a discrete probability distribution, and denoted ] is called the domain and signified by [The authors refer to individual elements of the domain as events, though an event is ordinarily taken to be a subset of the domain.]
Next, the authors attempt (and fail) to define as a set:
Each event has a probability assigned
In plain language, Equation (30) says that is defined as the set of all elements of the set for which is a number between and The appearance of on both sides of the equation indicates that the authors have lapsed into circular definition — a fallacious attempt to define in terms of Furthermore, the authors use inconsistently. With the expression they indicate that is defined as the set of all events with a given property. However, their expression of the property, indicates that is not a set of events, but instead a function associating events with numbers. Equation (31) similarly treats as a function:
Each event has a probability assigned
such that the sum of all values is equal to one,
In plain language, Equation (31) says that the numbers associated with events in the domain sum to 1. Now, to gain an appreciation of how horribly confused the authors are, you should ponder the question of how they could use as a function in (31), associating events with numbers and yet believe that they were correct in defining as a set of events in (30). In what immediately follows, the authors refer to the probability distribution as though it were both a set of events and a function associating events with numbers:
For example, the distribution for the roll of a fair die has six events, each with probability and the sum of all event probabilities is 1.
What should the authors have written instead? The answer is not easy for general readers to understand, because I’ve written it as I would for mathematically literate readers. But here you have it:
A discrete probability distribution is a function from a countable set to the closed interval of real numbers, with For example, a roll of a fair die is modeled by distribution when for all in
the set of all possible outcomes of the roll.
This says that for each of the elements of the set the function associates with exactly one of the real numbers between 0 and 1. The number in the interval that is associated with is denoted I have omitted the term domain because the authors never use it. However, to plumb the depths of their incoherence, you need to know that the set is the domain of the function Here, to facilitate comparison with what I have written, is a repeat of the annotated passage:
Amusingly, the authors go on to provide examples of discrete probability distributions, in Equations (33) and (38), but misidentify the distributions as discrete random variables.
About one in ten of my sophomore students in computer science made errors like those on display above, so it would be unfair to sophomores to characterize the errors as sophomoric. I have never seen anything so ridiculous in the work of a graduate student, and have never before seen anything so ridiculous in a journal article. It bears mention that Nemati and Holloway were contemporaneous advisees of Robert Marks, the editor-in-chief of Bio-Complexity, as graduate students at Baylor University. Indeed, most of the articles published in Bio-Complexity have former advisees of Marks as authors.
[12/11/2019: Replaced a reference to (31) with a reference to (30).]