Specification of the Truncated Cauchy calibration density in MCMCTree

Gustavo A. Ballen and Sandra Reinales

2025-08-19

It is possible to fix p to a given value describing the location of the distribution, that is, how close to the minimum age is the mode in the truncated Cauchy, and then estimate c so that the area under the curve attains a certain maximum age, allowing for some probability to be allocated to the right of it. The function c_truncauchy can be used provided that we have both age minimum and maximum, and define some probability pr to be allocated to the right of the maximum age.

In this example, our minimum age is e.g. \(1\) in units of \(100 Ma\), and our maximum age is \(4.93\) also in units of \(100 Ma\). The minimum is a soft bound allowing \(0.025\) of the density to be allocated to the left of it, whereas \(0.975\) is the percentile at which we observe the maximum age. The quantity p=0.001 has been chosen so that the mode is closer to the minimum age.

# load the package
library(tbea)

# estimate the c parameter for the L distribution
cparam <- c_truncauchy(tl=0.4094, tr=0.4160, p=0.001, pr=0.975, al=0.025)
cparam

## [1] 0.0009289821

The function gives us an estimated of approximately \(2.0\) for c. Thus we can specify this distribution in MCMCTree as L(0.4094, 0.1, 0.0009289821, 0.025). We can use the packages mcmc3r ¹ to plot the L density. The following code should do the trick.

# load the package
library(mcmc3r)

# using the function dL to plot the L density
curve(dL(x, tL=0.4094, p=0.001, c=cparam), from=0.4094, to=0.4160)