Glucose and willpower

In The Physiology of Willpower: Linking Blood Glucose to Self-Control (Gailliot and Baumeister) (local copy: PDF) the authors look at the effect that exerting one’s willpower has on blood glucose levels. There are some pretty strong connections between glucose tolerance and antisocial behaviour:

Several other studies have shown that improving the diet of incarcerated adolescents or adult prisoners reduces the incidence of violence in prison (e.g., “New Studies Show Strong Link,” 2004). It is plausible that the reduction in prison violence by improved diet is partially attributable to improved glucose levels or glucose tolerance.

A remarkable longitudinal field study of criminal recidivism by Virkkunen, DeJong, Bartko, Goodwin, and Linnoila (1989) sought to predict violent criminal acts over several years after release from prison. It found that glucose tolerance correctly predicted further violence for 84% of criminals. In view of the many extraneous variables and error variance in measuring criminal behavior, it would be hard to ask for a stronger result. Prisoners who exhibited poor glucose tolerance were much more likely to commit violent acts years later as compared to prisoners with better glucose tolerance. This relationship did not appear to be attributable to any other relevant physiological or demographic factors. That glucose tolerance at one point predicted violent acts at a later point is consistent with the interpretation that poor glucose tolerance partially caused the violent acts. Individuals who exhibit poor glucose tolerance as a stable trait seem predisposed to behave violently when opportunities beckon.

Triangle dissections

Some of my research is on dissections of triangles into equilateral triangles. Here’s an example:

So the outer equilateral triangle is cut up into smaller equilateral triangles, and no triangles overlap except along a common edge or point.

One of the earliest references on triangle dissections is this paper: The Dissection of Equilateral Triangles into Equilateral Triangles, W.T.Tutte, Proceedings of the Cambridge Philosophical Society, Vol. 44, pages 464-82, 1948. There, Tutte showed a connection between equilateral triangle dissections and electrical networks. See also http://www.squaring.net/tri/twt.html and the paper The dissection of rectangles into squares – R. L. Brooks, C. A. B. Smith, A. H. Stone and W. T. Tutte; 312-340.

A perfect dissection has no two triangles of the same size in the same orientation (up or down). Tutte conjectured that the smallest perfect dissection has size 15, and some recent enumeration work shows this to be the case (a paper will come out soon with these results). Here are the two perfect dissections of size 15, and perfect dissections of size 16 and 17:

Those graphics are produced using PyX and Sage. Full PDFs are available here.

Honesty in science

In my day job I work on mathematics (discrete mathematics, mainly to do with latin squares). My research output is modest. It is by no means ground-breaking stuff. I doubt that I’ll ever win the Clay maths prize for solving the Riemann hypothesis, nor will I ever win the Fields medal.

I can, however, take credit that I have always focussed on the mathematics itself. I’ve stated my claims as clearly as possible, provided source code to support computational claims, and made as much of my work freely available on reliable sites (mainly on the arXiv). What you see is what you get. On a daily basis I throw away a lot of what I do. That’s the nature of mathematics and science.

It then saddens me to read things like this: Dembski does it again. The paper in question is pushing another agenda through an otherwise normal scientific journal. This is an offence to the journal and the other people who have published there.

The paper itself is surreal. On p. 1054:

The simple idea of importance sampling is to query more frequently near to the target. As shown in Fig. 1, active information is introduced by variation of the distribution of the search space to one where more probability mass is concentrated about the target.

Our use of importance sampling is not conventional. The procedure is typically used to determine expected values using Monte Carlo simulation using fewer randomly generated queries by focusing attention on queries closer to the target. We are interested, rather, in locating a single point in T.

The term “importance sampling” is very well known in the statistics community. It has a very clear and agreed-upon meaning. To change that is silly at best, and dishonest at worst. In fact, misusing well known terms is a good sign that you’re a crank.

One standard application of importance sampling is to estimate an integral numerically. For a simple example, you might want to estimate the area under the curve in this picture:

The main benefit of importance sampling is that your estimates will be “close together” (mathematically, their standard deviation will be smaller than a uniform sampling Monte Carlo approach). The red dots in the following plot are the absolute errors from a uniform sampling estimator of the value of the integral, while the green dots are the absolute errors from an importance sampling estimator:

More precisely, the standard deviation of the errors of the uniform and importance sampling estimates is 0.0193 and 0.0001, respectively. The code is available as a Sage worksheet: Importance_sampling.sws.

For some reason, Dembski says that “We are interested, rather, in locating a single point in T”. In our picture, the only sensible thing that his statement could mean is that he wants to find the peak of the curve. Right? But we have algorithms specifically formulated for doing that sort of thing. Back in undergrad I remember learning about Newton’s method and other search/optimisation algorithms.

If you’re searching for a point in some space, then just say that.

For a thorough rebuke of the information theory rubbish in Dembski’s paper, see Mark C. Chu-Carroll’s post.