Disruptive Writing Style

In the pursuit of science, under whose umbrella I consider all intellectually rigorous activity to fall, the formulation and communication of ideas are critical. Here, I'll outline aspects of my own attitude to scientific communication.

In an earlier post on jargon, I advocated against over reliance on familiar terminology, as this can often give a false impression of understanding. I recommended occasionally throwing out unusual pieces of vocabulary, in the hope of ensuring that one's audience is engaged in the concepts, and not just semi-consciously following the signposts. This technique is a significant part of a communication strategy we might call 'disruptive writing style.'

When I say disruptive, I'm not talking about the content of an essay, article, speech, or whatever. I don't mean that the matter under discussion is disruptive, the way the birth of digital electronics represented a disruptive technology. Instead, I'm talking about the vehicle by which one conveys one's ideas to a wider appreciation. I'm talking about a style that occasionally strives to prevent the smooth progress of the reader (or listener) from beginning to end of your piece, in order to ensure that your thesis is being taken in.

Sums and Differences of Random Numbers

Here's a problem that cropped up in the course of some calculations I have been working on recently. In stating the problem, I'll include the physical context, thought this context isn't important for the rest of the discussion, which applies generally to certain manipulations on probability distributions:

A high-energy massive particle (e.g. a proton or α-particle), whose initial kinetic energy is governed by a certain probability distribution, passes through a slab of material. As it travels through the material, it scatters some of the electrons inside the slab, dissipating a small fraction of its energy. The amount of its energy that it loses also has some probability distribution (the so-called 'straggling function'). What is the probability distribution over the energy of the particle as it exits the slab of material? 

I'm sure all of us wrestle with exactly this question several times each day. The problem concerns the probability distribution over the difference between two independent random variables¹ (in this case the particle's initial energy and the energy it deposits in the slab of material).

The route to solving this problem involves utilizing the solution to a related problem, namely the probability distribution over the sum of two random variables, so first let's look at that.

Standard Error

In the quantification of uncertainty, there is an important distinction that's often overlooked. This is the distinction between the dispersion of a distribution, and the dispersion of the mean of the distribution. 

By 'dispersion of a distribution,' I mean how poorly is the mass of that probability distribution localized in hypothesis space. If half the employees in Company A are aged between 30 and 40, and half the employees in Company B are aged between 25 and 50, then (all else equal) the probability distribution over the age of a randomly sampled employee from Company B has a wider dispersion then the corresponding distribution for Company A.

A common measure of dispersion is the standard deviation, which is the average of the distance between all the parts of the distribution and the mean of that distribution.

Multi-level modeling

In a post last year, I went through some inference problems concerning a hypothetical medical test. For example, using the known rate of occurrence of some disease, and the known characteristics of a diagnostic test (false-positive and false-negative rates), we were able to obtain the probability that a subject has the disease, based on the test result.

In this post, I'll demonstrate some hierarchical modeling, in a similar context of medical diagnosis. Suppose we know the characteristics of the diagnostic test, but not the frequency of occurrence of the disease, can we figure this out from a set of test results?

A medical screening test has a false-positive rate of 0.15 and a false-negative rate of 0.1. One thousand randomly sampled subjects were tested, resulting in 213 positive test results. What is the posterior distribution over the background prevalence of the disease in this population?

Mean vs median - a careful balancing act

Two common measures of the location of a probability distribution are the mean and the median. While generally, they are quite different things, some familiar distributions have their mean and median at the same point (all such distributions are symmetric, (see comment, below) and vice versa).

The mean of a distribution, as we all know, is its average, while the median is, roughly speaking, the point at which the amount of probability mass to one side is the same as the amount on the other side. Upon hasty consideration, these definitions can appear to denote the same thing, and so confusion between the two concepts is common. Annoyingly, my own PhD thesis contains a sentence¹ that explicitly confuses the mean for the median (and furthermore, none of the half dozen eminent scientists whose job it was to assess my thesis (who otherwise all did an excellent job!) reported noticing this blunder).

Confusion between the mean and the median is highly analogous to a difficulty experienced by many young children when they try to balance asymmetric blocks on top of one another, as has been reported by cognitive scientist Annette Karmiloff-Smith².

The Fundamental Confidence Fallacy

The title of this post comes from an excellent recent paper (as far as I can tell, still in draft form) on misunderstandings of confidence intervals. The paper, 'The fallacy of placing confidence in confidence intervals', by R. D. Morey et al.¹ is by almost exactly the same set of authors whose earlier paper on a very similar topic I criticized, before, but the current paper does a far better job of explaining the authors' position, and arguing for it.

The authors identify the fundamental confidence fallacy (FCF) as believing automatically that,

If the probability that a random interval contains the true value is X%, then the plausibility (or probability) that a particular observed interval contains the true value is also X%.

Science is for Everyone

In the previous post, I explained that science is suitable for investigating all matters. Pursuing a similar theme, I want now to discuss how science is for all people, not just bearded academics with white lab coats. (Pardon the stereotype, and let me emphasize that there is no good reason why 50% of all scientists should not be women.)

I mentioned something in that last post that is also central to this discussion: scientific method is a graded affair - not black or white. Whatever we can learn by implementing a low level of scientific rigour, we can learn a little more, in a little more detail, and with a little more confidence, by applying a slightly more systematic procedure.