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More Hard Math Does Not Necessarily Mean More Useful Solutions

It is sometimes tempting to overemphasize the math and underemphasize the relevance

Math is said to be the language of science in that most (but definitely not all) scientific models of the world involve mathematical equations. The Pythagorean theorem, the normal distribution, Einstein’s energy-mass equivalence, Newton’s second law of motion, Newton’s universal law of gravitation, Planck’s equation. How could any of these remarkable models be expressed without math?

Unfortunately, it is sometimes tempting to overemphasize the math and underemphasize the relevance. The brilliance of the models listed above lies not in mathematical pyrotechnics but, if anything, in their breath-taking simplicity. Useful models help us understand the world and make reliable predictions. Math for the sake of math does neither. Examples of mindless math are legion. I will give three very different examples.

The net present value (NPV) of an investment is calculated by discounting all of the costs and benefits by an interest rate that takes into account the time value of money. With a 10% interest rate, $10 received one year from now has a present value of $9.09 because $9.09 invested at 10% will be worth $10 a year from now. Projects with positive NPVs are financially attractive. Projects with negative NPVs are not.

The internal rate of return (IRR) is the interest rate that gives an NPV that is equal to 0. Many business managers prefer to work with IRRs because rates of return seem more intuitive than NPVs. However, there are many problems with IRRs; most fundamentally, they don’t tell us whether the NPV is positive or negative for other interest rates. For instance, a project with a 10% IRR has an NPV of 0 at a 10% interest rate but the NPVs for 5% or 20% interest rates could be positive (attractive) or negative (unattractive). We just don’t know without calculating the NPV. So, we should dispense with IRRs and focus on NPVs.

Believe it or don’t, a paper with the hopeful title, ”A Resolution to the NPV–IRR Debate?”, published in a respected finance journal, argued that the NPV and IRR approaches can be reconciled by considering IRRs that are complex numbers — those baffling square roots of negative one that puzzled us in high school. However, the existence of complex IRRs does not resolve any of the issues with IRRs. Specifically, complex IRRs tell us nothing about the NPVs for other interest rates. Even worse, complex IRRs are not economically meaningful. If the most important weakness of the NPV criterion is that it does not yield a rate of return that businesspeople can understand easily, this weakness is not solved by presenting them with meaningless complex-valued IRRs.

A very different example is a famous paper, published in the flagship journal of the American Psychological Association, which claimed that a 2.9013 ratio of positive to negative emotions separated individuals, marriages, and business teams that flourished from those that languished. One of the authors, Marcial Losada, immodestly named this magic ratio “the Losada line.”

It is absurd to think that a dividing line (to four decimal places!) could be applied to complex human emotions. Losada and his coauthor used physics and engineering equations that describe the movement of fluids to make their argument appear scientific but their use of the language of fluid dynamics might easily be mistaken for a parody: High performance teams operate “in a buoyant atmosphere created by the expansive emotional space;” low-performance teams are “stuck in a viscous atmosphere highly resistant to flow.”

A critique authored by Nick Brown (who describes himself as a “a self-appointed data police cadet”), Alan Sokal (a full-time physics and math professor and part-time BS debunker), and Harris Friedman (a distinguished psychology professor) concluded — no surprise — that,

We find no theoretical or empirical justification for the use of differential equations drawn from fluid dynamics, a subfield of physics, to describe changes in human emotions over time…The lack of relevance of these equations and their incorrect application lead us to conclude that Fredrickson and Losada’s claim to have demonstrated the existence of a critical minimum positivity ratio of 2.9013 is entirely unfounded.

Brown, N. J. L., Sokal, A. D., & Friedman, H. L. (2013). The complex dynamics of wishful thinking: The critical positivity ratio. American Psychologist, 68(9), 801–813. https://doi.org/10.1037/a0032850

My third example involves the regrettable fact that the current data deluge has supercharged the use of mindless math in data science and artificial intelligence. There are now data for so many variables that researchers seek ways to select a parsimonious number of explanatory variables to predict whatever it is that they are trying to predict. Many pruning procedures seem to be chosen simply because the math is impressively dense.

Big data concept with data mining analyst

For instance, principal components regression — or some variation of this procedure — is often used to reduce a large number of explanatory variables down to a small number of principal components. The mathematical procedure involves using a singular value decomposition (don’t ask) to create orthogonal eigenvectors and then dropping the eigenvectors with the smallest eigenvalues. Wow! What does that mouthful of mathese mean?

Obscured by the mathematical gymnastics, the procedure imposes constraints on the explanatory variables. Astonishingly, these constraints depend solely on the correlations among the explanatory variables with no regard whatsoever for the variable that the model is trying to predict. As a consequence, the final model may force a variable that has a positive effect on the dependent variable to have a negative effect (or vice versa) and force unrelated nuisance variables to be more important than the real explanatory variables.

For example, household spending might be positively related to both income and stock prices. A principal components regression that considers these two explanatory variables plus the price of tea in China might, because of the correlations among income, stock prices, and tea prices, make the estimated effect of income on spending negative and make the effect of tea prices larger than the effect of stock prices. Principal components regression not only distorts the estimated effects but hides them in a eigenvector blizzard so that the only thing we know is that some fancy math was used.

In all three of these examples, researchers do not consider the assumptions and implications of the mathematical tools they use. They evidently assume that impressive math yields impressive insights. This disconnect is acute in black-box artificial intelligence (AI) algorithms that few understand but many trust.

Just one of innumerable examples: The CEO and co-founder of the Lemonade insurance company noted that, “AI crushes humans at chess, for example, because it uses algorithms that no human could create, and none fully understand.” He argued that, in the same way, “Algorithms we can’t understand can make insurance fairer.” So far, Lemonade is a lemon, with its stock price down nearly 90%.

The invisibility of the mindless math inside black box algorithms is a flaw, not a feature. As I have said many times, the real danger today is not that computers are smarter than us but that we think computers are smarter than us and consequently trust them to make decisions they should not be trusted to make.

You may also wish to read: Step away from stepwise regression (and other data mining). Stepwise regression, which is making a comeback, is just another form of HARKing — Hypothesizing After the Results are Known. I ran a series of Monte Carlo simulations and found that stepwise regression was disappointing when applied to fresh data. (Gary Smith)

Gary N. Smith

Senior Fellow, Walter Bradley Center for Natural and Artificial Intelligence
Gary N. Smith is the Fletcher Jones Professor of Economics at Pomona College. His research on financial markets statistical reasoning, and artificial intelligence, often involves stock market anomalies, statistical fallacies, and the misuse of data have been widely cited. He is the author of dozens of research articles and 16 books, most recently, Distrust: Big Data, Data-Torturing, and the Assault on Science, Oxford University Press, 2023.

More Hard Math Does Not Necessarily Mean More Useful Solutions