The PERT distribution is a beta distribution, stretched out, ripped up and dumped down far from home. Poor thing. No actual beta distributions were harmed in preparing this article. All drawings by the author.

The trouble that lurks in the parameterization and the properties of the PERT distribution

For many, the PERT distribution is now the go-to distribution for encoding quantitative insights from subject matter experts.

But the PERT distribution is an ad hoc, belt-and-braces distribution determined far more by calculation convenience than any meaningful correspondence with the uncertainties we encounter. In this article, I will discuss the trouble that lurks both in the parameterization and the properties of the PERT distribution.

What is the PERT distribution?

For a comprehensive introduction to the PERT distribution as well as an insightful discussion of its origins, see Stephen Grey’s excellent article here.

The PERT distribution describes the uncertainty on a variable that takes values between…


Black Swans call for a radical reappraisal of the way we model, but we are just as deluded about their significance as we are about our ability to explain and predict them.

According to Nassim Nicholas Taleb, the author of the highly influential book “The Black Swan”, a Black Swan event is characterized as follows:

First, it is an outlier, as it lies outside the realm of regular expectations, because nothing in the past can convincingly point to its possibility. Second, it carries an extreme ‘impact’. Third, in spite of its outlier status, human nature makes us concoct explanations for its occurrence after the fact, making it explainable and predictable.

I stop and summarize the triplet: rarity, extreme ‘impact’, and retrospective (though not prospective) predictability. A small number of Black Swans explains…


Thank you and thank you for your thought-provoking comment. Much appreciated

You characterization of Popper is spot on, I think. And you're absolutely right that DS is inherently inductive. It doesn't get much more inductive than regression and a great deal of data science is basically regression on roller skates.

And yes, I'd say a lot of DS is pseudo-science, though not so much because its hypotheses are not falsifiable, but more because data scientists don't really try. In fact, the "Data Speak" perspective is so ground in, that they accept the "conclusions" of their data (analysis) more or less…


Archimedes (but maybe Euclid) democratizing mathematics. Detail from Raphael’s The School of Athens, 1509–1511, fresco at the Raphael Rooms, Apostolic Palace, Vatican City. Wikimedia Commons.

The demystification of mathematical modelling is the first step to raising the level of mathematical literacy in public debate

Under COVID, our lives are dictated by the mandate of a handful of mathematical models. The public understanding of those models has never been more essential. Yet models are often discussed as if they were the exclusive province of a mystical caste of mathematician priests, while at the same time, our news media are full of pundits (and politicians) pontificating about the prognostications of these models with no apparent appreciation either for what they can reasonably predict or what they can not.

In this article, I will argue that there is nothing mystical about mathematics, and that mathematical modelling is…


How high functioning in other intellectual disciplines can be an obstacle to learning mathematics

In a famous and controversial public lecture in 1959, the scientist and author C. P. Snow lamented the increasing polarization of western intellectual activity into two distinct “cultures”: science, engineering and mathematics on the one hand, and what he called literary humanities on the other. Snow argued that the phenomenon was particularly pernicious, because the ranks of the political class were increasingly drawn from the latter, leading to an under-representation of scientific insight in the ruling elite — a predicament cast into vivid contemporary relief in the light of diverse national responses to the unfolding of the COVID-19 pandemic.

Snow…


Photo by Nik Albert on Unsplash

A layman’s look at the foundations of Frequentism and Bayesianism and how you can have the best of both.

What is probability? We often hear that there are two schools of thought, Bayesian and Frequentist; that Frequentists believe in an objective measure of probability, which we can only access through large numbers of repeated experiments; and that Bayesians, on the other hand, hold that there is no objective measure of probability, but rather that probability is an inherently subjective measure of belief, which we can only refine and modify with data.

As I discuss in my article “Why probability is hard”, to the pupil of probability, this is all rather unsatisfactory. What about when you can’t repeat an experiment…


Photo by Tom Pumford on Unsplash

It’s not because you’re stupid or weren’t concentrating in school

In 1982, Kahnemann, Slovic and Tversky published “Judgement under uncertainty: Heuristics and biases” and shattered humanity’s collective self-delusion that we had any functional intuition for even the most rudimentary problems in probability theory. This work has seen a renaissance in popularity since the publication of Kahnemann’s rather more accessible “Thinking fast and slow”.

Kahnemann is broadly sympathetic to our struggles, but much of the follow-up literature and course material has a slightly disparaging, not to say patronizing odour, as if reliable probabilistic intuition is just a question of a little hard work and application.

I have taught probability theory to…


Only one of them is actually a book on leadership

Photo by j zamora on Unsplash

The five books that taught me most about leadership are

  1. On Grand Strategy by John Lewis Gaddis
  2. War and Peace by Leo Tolstoy
  3. The Open Society and its Enemies by Karl Popper
  4. Talleyrand by Duff Cooper
  5. Middlemarch by George Eliot

By leadership, I mean the act of forging and articulating a vision and then engaging and motivating individuals in its execution. …


Author: Farcaster at English Wikipedia (Wikimedia Commons)

The prevailing paradigm of data analytics is rooted in the philosophy of Logical Positivism. Institutions who do not primarily deal with data need a more flexible approach.

The prevailing paradigm of data analysis is buried deep in the arid soil of a philosophical school called Logical Positivism. The framework is characterized by three mythical tenets derived from Positivism

  • Data are the only natural point of departure for all analysis
  • We ensure objectivity in our models by gathering data impartially
  • Data “speak”. That is, they guide our minds in the construction of models.

As I discuss in my article “Myths of Modelling: Data Speak”, Positivism — and, by association, its mythical beliefs — had been pretty thoroughly discredited by the 1960s. Unfortunately, as if often the case…


Thank you so much for your thoughtful and thought-provoking commentary.

It was absolutely not my intention to dismiss attempts to refute theories, only that we shouldn't (and don't) *only* try to refute theories, that confirmation - and other criteria - can have a place in an open contest between rival theories, but also that refutation is harder than we might think.

I left Kuhn out, because the article was already long enough, but Kuhn is clearly essential in all this. I'd argue, though, that Kuhn, like Popper, is most persuasive in his rejection of the positivist view that science ought…

Graeme Keith

Mathematical modelling for business and the business of mathematical modelling. See stochastic.dk/articles for a categorized list of all my articles on medium.

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