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I'm looking for a parameter transformation of a probability distribution such that the resulting parameters are orthogonal. That is, the off-diagonal elements of the Fisher Information matrix of the distribution after parameter transformation should be zero.

Is there a general way to find such transformations or to express the orthogonality constraints in some form that is more compact than just setting the off-diagonal elements of the reparameterized matrix to zero?

My background is in statistics. It appears that fisher information relates to some well-studied concepts in math and physics, so perhaps the problem of finding orthogonal parameterization has been studied in these field. I would be thankful for any pointers to literature that considers how to obtain orthogonal parameterization. I'm interested in distributions with two or three parameters.

matus
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Since you tagged your question with information-geometry I'm assuming that you are working with a statistical model that is smooth enough to be (partially) treated as a manifold. Let $\mathcal{M}$ be your statistical model with Fisher metric $g$. Your question now reduces to the question whether there exists a a coordinate system in which the representation of $g$ is the diagonal metric, i.e. you are wondering whether the metric $g$ is locally diagonizable. Unfortunately, this does not need to hold in all cases (see Existence of orthogonal coordinates on a Riemannian manifold).

If you are looking for a reparametrization that only yields orthogonal coördinates around some fixed point of your model (e.g. the true data distribution, which may be all you need in some cases) you can find so called Riemannian normal coördinates. However, these only satisfy your desired property in a single point and not locally.

FHoogenboom
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  • Thanks for pointing out the terminology. I'm interested in locally diagonalizable manifolds of dimensionality 2 and 3 so the linked discussion suggests that the orthogonal parametrization does exist. However, in addition, I asked about how to obtain the orthogonal parametrization. It doesn't look like the proofs of the existence of orthogonal coordinates (in DeTurck & Yang, 1983 and Detruck and Kazdan, 1981) are constructive. But I could be wrong. Can you please enhance your answer to say how to obtain the orthogonal parametrization in cases where it exists? – matus Apr 08 '18 at 10:07