This page collects the prior art we found while designing the notation of the essay. Each entry states what the source says and why it matters here.
Hogg, D. W. (2012). “Probability calculus for inference.” arXiv:1205.4446. The closest cousin of this essay: a physics-flavoured teaching note that attaches units to every term of the Bayes rule and calls the overloaded \(p(\cdot)\) “a nomenclatural abomination”. All of Hogg’s units are reciprocal-argument units — \(p(a)\) carries \(a^{-1}\) so that \(\int p(a)\,da\) is dimensionless; forbidden expressions such as \(\int p(a|b)\,db\) fail the unit check. Our container reading of \([\Omega]\) (“per unit of data”) resolves to exactly Hogg’s scheme.
Shen, W. & Lin, D. K. J. (2019). “Statistical theories for dimensional analysis.” Statistica Sinica 29:527–550. Proves (their Lemma 3) that the probability of an event is dimensionless, because probability is invariant under a change of measurement system — and a unit is precisely what changes under such a change. Their Lemma 6: the density of a continuous dimensional variable carries the reciprocal dimension of that variable. This is the formal ground under our rule “a density carries [what it spreads] per [the space it spreads over]”.
Lee, T., Zidek, J. V. & Heckman, N. (2020). “Dimensional analysis in statistical modelling.” arXiv:2002.11259. Examines dimensional consistency in likelihood analysis; notes that transcendental functions such as \(\ln(x)\) require dimensionless arguments. In our framing only the ratio \(\ell/n\) is ever logged, so the requirement is met by construction.
de Finetti, B. (1974). Theory of Probability, vol. 1. Opens with “PROBABILITY DOES NOT EXIST” (his capitals): probability is a degree of belief held by someone, not a physical quantity. The subjectivist pole of the debate our belief/probability distinction walks through.
Hájek, A. “Interpretations of Probability.” Stanford Encyclopedia of Philosophy. Survey of the interpretations (frequentist, propensity, subjectivist and more). Background for why “probability is a property of the world” is a stance, not a neutral fact.
Good, I. J. “Weight of evidence: a brief survey.” Reports Turing’s wartime invention of the ban and deciban as named units for the logarithm of the Bayes factor — the historical precedent for giving an invented, pedagogically motivated unit to a Bayesian quantity. Good: “a deciban is about the smallest change in weight of evidence directly perceptible to human intuition.”
Jaynes, E. T. (2003). Probability Theory: The Logic of Science, §4.2. Works evidence in decibels through the hypothesis-testing chapter; the practical face of Turing’s unit.
Kennedy, A. “Types for units-of-measure: theory and practice.” The theory behind F#’s units-of-measure types; justified by dimensional invariance — program behaviour must not change when units change. The standard for what makes something a unit rather than a type tag.
Stan Reference Manual: constrained types
(simplex). Stan encodes “non-negative and sums to
one” as a type, not a unit. The closest production-grade prior art for
our conservation law \(\sum b =
1\,\mathrm{R}\), expressed in a type system instead of a unit
system.
Bliss, C. I. (1934). “The method of probits.” Science 79(2037):38–39. “Probability unit” already has a contracted name — probit — owned by dose–response statistics since 1934. A reminder that inventing terminology requires checking the neighbours.
\(\Omega\) in measure theory. The letter \(\Omega\) conventionally names the sample space in the probability triple \((\Omega, \mathcal{F}, P)\). Our \([\Omega]\) is a unit tag, not a sample space; readers arriving from measure theory should not conflate the two.