Which form of Jeffrey's prior can be used for a three-parameter distribution? Let X be a random variable which follows a distribution, say S with parameters a, b and c. Knowing that or Assuming that a, b and c are independent of one another, which one is reasonable to do?
a) Is it okay to find joint Jeffrey's prior as the product of three Jeffrey's prior (for a, b and c)?
b) Is it better to use the square root of determinant of a 3x3 Fisher's Infromation matrix?
Even for a normal distribution with two unknown parameters, joint Jeffrey's prior based on the product of two Jeffrey's prior (mu and sigma) yields a slighly different prior than the one based on the determinant of a 2x2 Fisher's Information matrix.

The Jeffreys prior for a three parameter model is the square root of the determinant of the Fisher information matrix.
If you take the product of individual Jeffreys priors, this is called the "independence Jeffreys prior".
Here is an example of an independence Jeffreys prior for 3-parameter models:
On the independence Jeffreys prior for skew-symmetric models
In many cases, this yields to a better prior as it reduces the dependence between parameters. See:
A catalogue of noninformative priors
