Two matrix Fisher distributions on SO(3)?

Question

After the uniform distribution (normalized Haar measure), the matrix Fisher distribution seems to be the most popular probability distribution on the Lie group SO(3). The density is proportional to exp tr(K M^T X), where M is the mode and K (symmetric, often positive-definite) is the concentration. There are two ways to fill in the details:

1. SO(3) is essentially the 3x2 matrices X such that X^TX = I₂, since the last column is determined by the other two. So M is 3x2 and K is 2x2. See e.g. Downs (1972).
2. O(3) is the 3x3 matrices X such that X^TX = I₃. Here, M is 3x3 and K is 3x3. Restrict this distribution to the SO(3) component, and renormalize. See e.g. Chang and Rivest (2001), Kent et al. (2013).

As far as I can tell, the second model is strictly larger than the first, due to the extra freedom in K. (I can explain if needed.) Some papers treat one version while others treat the other. Results proved for one are often not directly adaptable to the other. Is this a correct summary of the situation?

Answer 1

The question is discussed heavily by Sei, T., Shibata, H., Takemura, A., Ohara, K., & Takayama, N. (2013). Properties and applications of Fisher distribution on the rotation group. Journal of Multivariate Analysis, 116, 440-455. So I consider the question to be answered. But I will continue to check this page for a while, for other answers.