Evidence, Temperature, and the Laws of Thermodynamics

Abstract

A primary purpose of statistical analysis in genetics is the measurement of the strength of evidence for or against hypotheses. As with any type of measurement, a properly calibrated measurement scale is necessary if we want to be able to meaningfully compare degrees of evidence across genetic data sets, across different types of genetic studies and/or across distinct experimental modalities. In previous papers in this journal and elsewhere, my colleagues and I have argued that geneticists ought to care about the scale on which statistical evidence is measured, and we have proposed the Kelvin temperature scale as a template for a context-independent measurement scale for statistical evidence. Moreover, we have claimed that, mathematically speaking, evidence and temperature may be one and the same thing. On first blush, this might seem absurd. Temperature is a property of systems following certain laws of nature (in particular, the 1st and 2nd Law of Thermodynamics) involving very physical quantities (e.g., energy) and processes (e.g., mechanical work). But what do the laws of thermodynamics have to do with statistical systems? Here I address that question.

Statistical Evidence Measured on a Properly Calibrated Scale across Nested and Non-nested Hypothesis Comparisons

“…Consider also the two hypotheses H1: "coin is biased towards tails" (θ ≤ ½), vs. H2: "coin is fair" (θ = ½). We articulate four BBPs up front (we have described the thought experiments used to motivate the BBPs in detail elsewhere; see, e.g., [5]; here we simply summarize the BBPs themselves). Note that the BBPs described in what follows are specific to this particular set up, which is the one we considered in previous work [4,5].…”

“…We articulate four BBPs up front (we have described the thought experiments used to motivate the BBPs in detail elsewhere; see, e.g., [5]; here we simply summarize the BBPs themselves). Note that the BBPs described in what follows are specific to this particular set up, which is the one we considered in previous work [4,5]. Extensions to other forms of hypotheses are considered in subsequent sections, albeit still in the context of the binomial model.…”

“…For example, the number of observations n on the statistical side does not function in our EqS as the analogue of the number of particles in physics; rather, the number of observations n appears to be part of the description of the statistical system's information "energy", rather than its size. (See [5] for discussion of this issue.) Therefore, but perhaps quite counter-intuitively, we do not expect to see a simple alignment of plerodynamics with statistical mechanical (microscopic) descriptions of physical systems, even in the event that we are able ultimately to consolidate the theory under the umbrella family of Boltzmann distributions.…”

“…Towards this end, we proposed adapting some of the mathematics of thermodynamics as the basis for an absolute (context-independent) measurement scale for evidence [4]. The result was a new theory of information-dynamics, in which different types of information are conserved and interconverted under principles that resemble the first two laws of thermodynamics, with evidence emerging as a relationship among information types under certain kinds of transformations [5]. As we argued previously, this provides us both with a formal definition of statistical evidence and with an absolute scale for its measurement, much as thermodynamics itself did for Kelvin's temperature T. But unless this new theory can produce something useful, it is purely speculative and not really a theory at all in the scientific sense, so much as an overgrown analogy.…”

“…While this issue was ameliorated by the introduction of wholly information-based versions of the 1st and 2nd laws of thermodynamics [5], it remained a concern, particularly in view of our inability to extend the theory beyond the one-sided binomial application.…”