Table 1.

An illustration of the difference between the restricted and full conditional prior using a four-taxa example

		Multiplicative	Conditional	Restricted conditional
Prior “correction term”	((a,b),(c,d)) $T_{CD}<T_{AB}$	–	$3\lambda e^{-3\lambda h_2}$	$12e^{-3\lambda h_2}(1-e^{-\lambda h_2})$
	((a,b),(c,d)) $T_{CD}\ge T_{AB}$
	(((ab),c),d)	–	$3\lambda e^{-3\lambda h_3}$	$4\lambda e^{-4\lambda h_3}$
	(((a,b),d),c)
Marginal Topology probability	((a,b),(c,d))	93%	94.2%	50%
	(((a,b),c),d)	3.5%	2.9%	25%
	(((a,b),d),c)	3.5%	2.9%	25%
Marginal calibration prior		$\frac{3\lambda e^{-3\lambda x}}{e^{-12\lambda }-e^{-18\lambda }}$	$\frac{1}{6-4}$	$\frac{1}{6-4}$

The prior is a pure birth process with a birth rate of $\lambda =\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$ and a uniform calibration density between four and six is applied to the clade (a,b). The uncorrected (multiplicative) prior is $\frac{1}{6-4}\frac{4!}{4}{\lambda }^3{e}^{-\lambda (2h_1+h_2+h_3)}$, and the table gives the conditional prior “correction terms” for each ranked topology, together with the induced prior probability of each unranked topology and the marginal density for the calibrated clade.