Bronk Ramsey 2009 - Other Group Priors Multiphase Models and Cross-Referencing

Summary

The uniform phase generalises to any within-phase density defined by two characteristic boundaries (ramped, exponential, normal), and multiphase models multiply group priors: overlapping phases are independent, contiguous phases share a boundary, and sequential phases with a hiatus add the constraint $t_{b} < t_{c}$ . Refinements keep the whole-model span prior neutral. Cross-referenced boundaries tie sub-models together.

Overview

If the uniform assumption is wrong (activity grows through the phase, material is residual, activity rises and tails off), only Eq. 15 changes; the “prior independent of the overall span” property is preserved.

Main Content

Within-phase distributions (Eqs. 17-22)

Given boundaries $t_{a} < t_{b}$ , events $t_{i}$ have conditional density:

Ramped (events grow in number): $p (t_{i} ∣ t_{a}, t_{b}) = 2 p_{H} (t_{i}, t_{b}) \frac{t _{i} - t _{a}}{( t _{b} - t _{a} ) ^{2}}$ (Eq. 17); overall prior $\propto p_{H} (t_{a}, t_{b}) \prod_{i} 2 p_{H} (t_{i}, t_{b}) (t_{i} - t_{a}) / (t_{b} - t_{a})^{2}$ (Eq. 18).

Exponential (definite end $t_{b}$ , earlier events with time constant $τ = t_{b} - t_{a}$ ): $p (t_{i} ∣ t_{a}, t_{b}) = p_{H} (t_{i}, t_{b}) \frac{exp { - ( t _{b} - t _{i} ) / ( t _{b} - t _{a} )}}{t _{b} - t _{a}}$ (Eq. 19), prior Eq. 20. Used for residual material; Figure 5 applies it to Stonehenge ditch antlers (Tau_Boundary), estimating $τ$ .

Normal (gradual onset and tail, $t_{a}, t_{b}$ are the $1 σ$ limits): $p (t_{i} ∣ t_{a}, t_{b}) = \frac{1}{( t _{b} - t _{a} ) π /2} exp {- \frac{( 2 t _{i} - t _{b} - t _{a} ) ^{2}}{2 ( t _{b} - t _{a} ) ^{2}}}$ (Eq. 21), prior Eq. 22. The latter two let events spill beyond the boundaries. OxCal keywords: Boundary, Zero_Boundary (ramp), Tau_Boundary (exponential), Sigma_Boundary (normal). The 4-parameter trapezium model (Karlsberg 2006) is mentioned but not implemented.

Multiphase priors (Eqs. 23-25)

With events $t_{i}$ in phase 1 (boundaries $t_{a}, t_{b}$ ) and $t_{j}$ in phase 2:

Overlapping (independent): $p (t) \propto \prod_{i} \frac{p _{H} ( t _{a} , t _{i} , t _{b} )}{t _{b} - t _{a}} \prod_{j} \frac{p _{H} ( t _{c} , t _{j} , t _{d} )}{t _{d} - t _{c}}$ (Eq. 23).

Contiguous (shared boundary $t_{b}$ ): $\prod_{i} \frac{p _{H} ( t _{a} , t _{i} , t _{b} )}{t _{b} - t _{a}} \prod_{j} \frac{p _{H} ( t _{b} , t _{j} , t _{c} )}{t _{c} - t _{b}}$ (Eq. 24).

Sequential with hiatus: $p_{H} (t_{b}, t_{c}) \prod_{i} \frac{p _{H} ( t _{a} , t _{i} , t _{b} )}{t _{b} - t _{a}} \prod_{j} \frac{p _{H} ( t _{c} , t _{j} , t _{d} )}{t _{d} - t _{c}}$ (Eq. 25).

Refinements keeping the span prior neutral (Eqs. 26-30)

With more than two boundaries the boundary events themselves are no longer uniform in span. Treat the inner boundaries $t_{k}$ as a uniform phase between the first $t_{a}$ and last $t_{m}$ : $p (t_{k} ∣ t_{a}, t_{m}) = p_{H} (t_{a}, t_{k}, t_{m}) / (t_{m} - t_{a})$ (Eq. 26), e.g. contiguous 2-phase: extra factor $1/ (t_{c} - t_{a})$ (Eq. 27), hiatus: $1/ (t_{d} - t_{a})^{2}$ (Eq. 28; Nicholls and Jones 2001). If $t_{a}, t_{m}$ also have limits (lower/upper $ll i m_{a}, u l i m_{a}, ll i m_{m}, u l i m_{m}$ ), multiply by $1/ g$ with
$g = min {(u l i m_{m} - ll i m_{a}) - (t_{m} - t_{a}), (t_{m} - t_{a}) - (ll i m_{m} - u l i m_{a}), u l i m_{a} - ll i m_{a}, u l i m_{m} - ll i m_{m}} (Eqs. 29-30)$
to maintain a uniform effective prior for $t_{m} - t_{a}$ (Bronk Ramsey 2008). OxCal 4 option UniformSpanPrior (default on). The authors stress these refinements “will only make very slight differences in practice”.

Beyond simple phases (Figure 7, Eq. 31). Tie sub-models by cross-referencing an event: in the Iron Age Israel model (Mazar and Bronk Ramsey 2008) a main two-phase sequence (start $t_{a}$ , transition $t_{b}$ , end $t_{c}$ ) is combined with a “Late Iron I” phase from $t_{d}$ to the same $t_{b}$ (Boundary("=Transition I/II")), giving $p \propto \prod_{i} \frac{p _{H} ( t _{a} , t _{i} , t _{b} )}{t _{b} - t _{a}} \prod_{j} \frac{p _{H} ( t _{b} , t _{j} , t _{c} )}{t _{c} - t _{b}} \frac{1}{t _{c} - t _{a}} \prod_{k} \frac{p _{H} ( t _{d} , t _{k} , t _{b} )}{t _{d} - t _{b}}$ .

Cautions. Use sparingly: cross-references slow MCMC; avoid putting an event in two versions of the same phase (duplicating prior); ask whether “synchronous” is reasonable (fine for a tephra horizon, doubtful for vegetation or cultural change). Framework hypotheses (Aegean Late Bronze Age, British Bronze Age) are working assumptions and results depend on them.

Examples

Stonehenge antlers with an exponential group (Figure 5)

Setup: nine antler 14C dates ( $4330 \pm 18$ to $4432 \pm 22$ BP and two $\pm 60$ ) from the main ditch, assumed to predate the construction event $E$ with exponentially distributed lag (Tau_Boundary("T") … Boundary("E"), $τ = E - T$ ). Result: the posterior for $τ$ (acquisition time scale) has 95.4% range 0-109 years. Interpretation: antlers may be somewhat older than the construction but within about a century; illustrative only, not a full site analysis.

Connections

Bronk Ramsey 2009 - Order Constraints and the Uniform Phase Prior, Bronk Ramsey 2009 - Deposition Models and Sequences, Hierarchical Models (boundaries as hyperparameters), Survival Analysis (exponential lag).

Second Brain

Explorer

Bronk Ramsey 2009 - Other Group Priors Multiphase Models and Cross-Referencing

Bronk Ramsey 2009 - Other Group Priors Multiphase Models and Cross-Referencing

Overview

Main Content

Examples

Connections

See Also

Graph View

Table of Contents

Backlinks