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The Nature of Solutions

April 12, 2010

Oh my, we seem to be heading into Changeux-Connes territory with this conversation turning to mind, matter, and mathematics.  So in some sense, with this Newtonian synthesis concept, you bring up the question of what constitutes a solution to a problem.  As your doctoral advisor Alan Willsky likes to point out with regard to the evolution of Bayesian estimation theory, the Wiener filter is a formula, the Kalman filter is an algorithm, and the particle filter is essentially a simulation.  Incidentally, as I think you know, I have some interest in the history of particle filters but have not yet found a description of why the British defense establishment was originally interested in them.

Pablo Parrilo, in a talk with Stephen Boyd at the Paths Ahead in the Science of Information and Decision Systems symposium, discussed the nature of solutions in an era with huge computational resources.  How do you think solutions specified as convex programming or Monte Carlo simulation (like a particle filter) fit into the combination of the logical and the statistical in the grand scheme of reasoning?

Anyway, as you had suggested, it is superior to be practical, so let me come back to the potentially practical question of learning a scientific law, here in the context of mind and matter.

The physical and biological basis for intelligence has been a question that has been around since the time of the ancients.  In particular, people have wondered why some individuals are smarter than others.  Is there something about the brains of the intelligent that is different from the brains of the unintelligent.  This article reviews many facts and theories, including the classical fact that total brain volume is correlated with intelligence.  A more recent finding is that intelligent brains are more efficient in the sense of information processing as measured through graph-theoretic quantities.  Some of my own work looks at the relationship between brain volume and memory capacity as well as the potential functional significance of structural properties of neuronal networks, so this is all very interesting to me.

The argument that brain volume is correlated with intelligence leads to the argument that brain volume is correlated with survival.  That is, larger brains give survival advantage.  But one might ask whether there is any direct way to measure survival of the fittest, a central underpinning of evolutionary theory.  A classic data set that has been analyzed several times (including work using logistic regression) is the Bumpus field sparrow data set (available from The Field Museum in Chicago, whose biological specimen collection I saw in 1999).  During a winter storm, some sparrows were killed and some survived.  It has been argued that the survivors were more fit than the others.  But was it because they had bigger or more efficient brains?

So now with all of that lead-up, finally to the practical question.  Given this data set with measurements in several phenotypic dimensions and binary survival labels, can an expert in  dimensionality reduction and supervised classification learn a scientific law to explain it?  Moreover, will the scientific law take the form of a formula, an algorithm, a simulation, a decision boundary, a story, or something else altogether?

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