Bayesian 14C-rationality, Heisenberg Uncertainty, and Fourier Transform

Abstract

Following some 30 years of radiocarbon research during which the mathematical principles of 14C-calibration have been on loan to Bayesian statistics, here they are returned to quantum physics. The return is based on recognition that 14C-calibration can be described as a Fourier transform. Following its introduction as such, there is need to reconceptualize the probabilistic 14C-analysis. The main change will be to replace the traditional (one-dimensional) concept of 14C-dating probability by a two-dimensional probability. This is entirely analogous to the definition of probability in quantum physics, where the squared amplitude of a wave function defined in Hilbert space provides a measurable probability of finding the corresponding particle at a certain point in time/space, the so-called Born rule. When adapted to the characteristics of 14C-calibration, as it turns out, the Fourier transform immediately accounts for practically all known so-called quantization properties of archaeological 14C-ages, such as clustering, age-shifting, and amplitude-distortion. This also applies to the frequently observed chronological lock-in properties of larger data sets, when analysed by Gaussian wiggle matching (on the 14C-scale) just as by Bayesian sequencing (on the calendar time-scale). Such domain-switching effects are typical for a Fourier transform. They can now be understood, and taken into account, by the application of concepts and interpretations that are central to quantum physics (e.g. wave diffraction, wave-particle duality, Heisenberg uncertainty, and the correspondence principle). What may sound complicated, at first glance, simplifies the construction of 14C-based chronologies. The new Fourier-based 14C-analysis supports chronological studies on previously unachievable geographic (continental) and temporal (Glacial-Holocene) scales; for example, by temporal sequencing of hundreds of archaeological sites, simultaneously, with minimal need for development of archaeological prior hypotheses, other than those based on the geo-archaeological law of stratigraphic superposition. As demonstrated in a variety of archaeological case studies, just one number, defined as a gauge-probability on a scale 0–100%, can be used to replace a stacked set of subjective Bayesian priors.

A Neolithic population model based on new radiocarbon dates from mining, funerary and population scaled activity in the Saint-Gond Marshes region of North East France

“…These extremely conservative test results are then illustrated in red or blue vertical bands, where real data radiocarbon results deviate from the exponential null of taphonomic loss and population growth (Shennan et al 2013). Here, radiocarbon data are not normalized after calibration (Weninger et al 1986(Weninger et al , 2015(Weninger et al , 2017(Weninger et al , 2020Bevan et al 2017). The resultant plots then tell us whether our model demonstrates a statistically significant positive (vertical red band) or negative (vertical blue band) population event, to a 95% confidence threshold (grey band) in our population model.…”

Approximate Bayesian Computation of radiocarbon and paleoenvironmental record shows population resilience on Rapa Nui (Easter Island)

“…The rise and fall (i.e., "collapse") observed in their study between 1430 and 1550 AD falls entirely within a significant positive deviation from their null model. Given the known impact of 14 C normalization at this steep portion of the Southern Hemisphere calibration curve 23,46,47,75,78,79 , the population only appears to rise and decline. Moreover, Lima et al's study does not provide a clear rationale why continuous linear population growth would be expected for the small island of Rapa Nui.…”

“…Furthermore, while spikes and troughs in SPD curves, or deviations from null models, may appear to be correlated with paleoenvironmental events, several non-trivial technical issues prevent simple conclusions to be drawn from 'eye-balling' SPDs. First, 14 C normalization 46,47 and back-calibration methods, used to generate simulation envelopes for model fitting 23 , can cause spurious spikes in the SPD at steep portions of the 14 C calibration curve. In the case of 14 C normalization, individually calibrated distributions are standardized so the posterior probability density sums to 1.…”

“…In the case of 14 C normalization, individually calibrated distributions are standardized so the posterior probability density sums to 1. In constructing SPDs, normalization inflates spikes caused by steep portions of the calibration curve, whereas nonnormalized dates lack these artifacts 23,46,47 . In addition, comparing empirical SPDs to candidate models requires one to simulate distributions of radiocarbon dates consistent with the model expectations.…”

“…1375-1500 AD is steep, followed by a plateau and several 'wiggles' 78,79 . Rather than reflecting demographic events, it is likely that this spike is a predictable consequence of normalization 23,46,47 . Critically, Lima et al 29 do not account for the calibration curve and sample size effects by fitting their models directly to the normalized SPD.…”