Kappa prior

The kappa parameter of the HKY model (Hasegawa et al, 1985) represents how large transition rates are when all transversion rates are set to 1. Since this is a rate parameter, a distribution with a range from 0 upwards is appropriate.

Rosenberg et al, 2003 found that a lognormal(M=1,S=1.25) distribution fits well based on empirical observations, so this is the default prior in BEAST.

References

Rosenberg MS, Subramanian S, Kumar S. Patterns of transitional mutation biases within and among mammalian genomes. Molecular biology and evolution. 2003 Jun 1;20(6):988-93. doi:10.1093/molbev/msg113.

Hasegawa M, Kishino H, Yano TA. Dating of the human-ape splitting by a molecular clock of mitochondrial DNA. Journal of molecular evolution. 1985 Oct;22(2):160-74. doi:10.1007/BF02101694.