Complication Is Different

In this section, some Bayesian insights will be brought to bear on the study of condensed-matter physics.

In addition to increasing and decreasing length scales (section 4.1,) Anderson [2] and Coleman [10] have pointed out that there is a third way, in which scientists, and particularly physicists, have recently begun to seek out evidence, to draw more robust inferences about theories, namely measuring processes of greater complication. While biological and social scientists have long studied very complicated systems, there is still a substantial niche, including [1] the study of solid state physics, for physical scientists to fill, before the three disciplines start competing directly.

There seem to be two conflicting approaches to the physics of complication. The first approach is that which has led to numerical band-structure calculations in solid-state physics: the ``first principles'' of quantum mechanics, or some well-understood approximation thereto, are given to a computer, along with a precise specification of the atomic-scale configuration of the material, and computational resources are used to make theoretical predictions about observable characteristics of the material. In solid-state physics, this approach is at the end of a process of implementing Schrödinger equations with ever more realistic approximations to the Hamiltonian. Such an approach has [51,47,50] been used, with a series of choices for the approximation to quantum mechanics, and the atomic configuration, to predict properties of layered structures of manganese, cobalt, and copper.

Anderson [1] points out a difficulty with this approach: any system, more complicated than a molecule of four atoms or so, is pretty well never in an eigen-state of its Hamiltonian, so the Schrödinger equation doesn't describe the state of the system. This is because the tunnelling-like processes, which would otherwise collapse the system into an eigen-state of its Hamiltonian, are very slow for complicated systems: often very slow compared with the age of the universe, and certainly very slow compared with the rate of occurrence of measurement-like interactions with the outside world, which collapse the system into eigen-states of operators other than the Hamiltonian: for example, the transition, with increasing temperature, from Schrödinger-controlled evolution to interaction-controlled evolution, was [13] recently observed experimentally, in Buckminster Fullerene molecules. Therefore, it can't be guaranteed that the model which implements a Schrödinger equation with the most realistic Hamiltonian will always be the most useful in describing the real behaviour of the system. Anderson [1] intended this principle to apply generally to the relationship between any discipline and a successful theory in a slightly simpler discipline, not just to that between solid state physics and quantum mechanics.

The second approach is that of emergent phenomena [2,10], in which the theories used have postulates other than the ``first principles'' of quantum mechanics, which can make testable predictions, without stretching computing capacity to its limits in the way that band-structure calculations do, perhaps succeeding in encoding the various atomic configurations in continuous, analytical parameters, such as the roughness parameters in ``Correlation between dynamic magnetic hysteresis loops and nanoscale roughness of ultrathin $Co$ films'' [37], or the classical electric potentials and magnetic flux densities in ``Classical-Field Theory of Electron Waves as a Polarized Radiation Probe of Magnetic Surfaces'' [31].

These postulates may either be derived rigorously from statistical properties of quantum mechanics, in which case the postulates are not really new, just a re-encoding of those of quantum mechanics¹⁶, or may be introduced on an ad hoc basis, usually with the use of some clues from quantum mechanics, or some approximation thereto, such as classical mechanics.

Here, it is argued that there is a second difficulty with the ``first principles'' approach: in band-structure calculation, there are many plausible choices of the precise atomic configuration, i.e. many possible theories of similar form, for which it is [42,43] therefore reasonable to assign similar prior probabilities; when many theories have similar prior probabilities, those prior probabilities will [42,43] be small. Therefore, although the greater use of the clues available from quantum mechanics, in ``first principles'' theories, renders it reasonable to assign a higher prior probability to the disjunction of all ``first principles'' theories, than to a similarly rich disjunction of ``emergent phenomena'' theories, each individual atomic configuration in the ``first principles'' theories will have a tiny prior probability.

This matters because the introduction of a numerical-computational method prevents the use of a likelihood for the observed data, expressed as an analytical function of a continuous parameter, which represents which of the many possible choices is considered, and forces the likelihood to be evaluated individually for each precise configuration. Given realistic computing resources, one cannot do this for all the possible configurations¹⁷, and many will therefore end up in the situation outlined in section 5.1, i.e. all of the available evidence will be in the position, which appears to be identified with the experiences of the marginalized in standpoint epistemology (section 3.1.) It is reasonable to suppose that the disjunction of the small number of atomic configurations, for which the band-structure calculation can be completed, will not have a very large prior probability compared with an ``emergent phenomena'' theory, the microscopic variability of which is encoded in continuous, adjustable parameters, in terms of which an analytical likelihood formula can be obtained.

There are four possible responses to this situation. The first is to use the encoding of evidence, for which likelihoods are not accurately known, in a prior probability distribution, described in section 5.1. However, this kind of guesswork about the inferences which can be drawn from evidence is only intended as a short-term, interim solution, while deductions from theories are allowed to catch up with experiment. The application of band-structure calculations, even to the most relevant, relatively few, atomic configurations, seems set to be a very long-term project indeed.

The second is to attempt to model the predictions for atomic configurations, for which the numerical band-structure calculations have not been performed, by some distribution about the predictions resulting from those for which the band-structure calculations have been performed, this distribution representing an added postulate in the ``first principles'' theory. However, the predictions for one configuration cannot easily be used as a guide to the predictions for similar configurations; certainly, in the study of magnetic films, small changes in atomic arrangement and in approximation method can lead to substantial changes in observed properties, both experimentally [37,15], and in the calculations [51,47,50]. Therefore, the selection of the distribution about the known results, which is to be added to the ``first principles'' theory is a non-trivial task, and might in itself require a process of inference from experimental evidence, to choose between distributions. Also, this will leave the likelihood distribution $P(E\vert T)$, for those theories $T$ that are treated in this way, with a substantial Shannon entropy, rendering the greedy top-down approximation to Bayesian experiment selection (section 5.2) useless.

The third is to undertake inference with the theories for which predictions are available, effectively assigning a zero prior probability to those atomic configurations for which the band-structure calculations have not yet been performed, and leaving the specific ``first principles'' theories for which the band-structure calculations have been performed with their small prior probabilities, and the adjustable ``emergent phenomena'' theories with their large prior probabilities. However, it is not (section 2.2.2) entirely satisfactory to assign zero prior probabilities to theories, the existence of which, if not their details, is known.

The fourth is to ignore the evidence, for which some theories have not predicted likelihoods, until such time as the theoretical deductions are complete. However, it is not entirely satisfactory to ignore such a large body of evidence, for the length of time that would be involved in this mammoth theoretical project.