This is achieved by first generating an SPD for each phase and normalizing

This is achieved by first generating an SPD for each phase and normalizing

Theoretically, a calibrated date should be a continuous probability density function (PDF); however, in practice a date is represented as a discrete vector of probabilities corresponding to each calendar year, and is therefore a probability mass function (PMF). This discretization (of both a proposed model probability distribution and a calibrated date probability distribution) provides the advantage that numerical methods can be used to calculate likelihoods.

Hypothetically, if a calibrated date was available with such precision that it could be attributed with certainty to just a single calendar year the model likelihood would trivially be the model probability at that date. Similarly, if the data comprised just two such point estimates (at calendar time points A and B), the model’s relative likelihood would trivially be the model probability at date A multiplied by the model probability at date B.

Therefore, the probability of a single calibrated date given the model can be calculated as the model probability at year A, or the model probability at year B etc., for all possible years, weighted by how probable the calibrated 14 C date is at each of those years. This can be calculated using the scalar product between model probabilities and calibrated date probabilities, and gives the probability of a single calibrated date under the model. This is repeated for every calibrated date, and the overall product gives the relative likelihood of the model, given the whole dataset.

This approach assumes each date is a fair and random sample, but where many dates are available from a single site-phase, it is sensible to first bin dates into phases. This is an important step in modelling population dynamics to adjust for the data ascertainment bias of some archaeological finds having more dates by virtue of a larger research interest/budget. These phase-SPDs are then combined and normalized to create a final SPD. …