Not really an answer ... more like an extended comment that is too long for the comment box. But I found the question interesting, for a number of reasons:

1. I did not know that Mma had a FisherInformation function hidden away where you found it - how DID you find it?

2. Your question actually highlights one of my pet dislikes - which is the naming of distributions as black boxes - which is exactly what you are trying to emulate. Even for something as well known as the Normal distribution, there are competing parameterisations. For other distributions, there is not even agreement on what functional form the distribution should take, or any number of competing common forms that co-exist. Even when there is such agreement as to functional form, your own example shows beautifully why using black box names is often just inappropriate. You enter:

$\rightarrow$

   FisherInformation[NormalDistribution[m, s]] // MatrixForm

There is no such thing as the Fisher Information of a Normal distribution. Fisher Information is carried out wrt parameters, not with respect to black boxes. Are the parameters of $N(\mu, \sigma^2)$:

• $\mu$ and $\sigma$, or
• $\mu$ and $\sigma^2$?

Well, it can be either. And we may want to consider both cases, or something entirely different. Each approach is a different problem, with a different solution.

Consider your example of a Normal random variable with pdf $f(x)$:

The Fisher Information on $(\mu, \sigma)$ is:

The Fisher Information on $(\mu, \sigma^2)$ is:

To avoid any confusion, I should note that I am using here the FisherInformation function in the mathStatica add-on to Mathematica, and as declaration, that I am one of the authors.

Not really an answer ... more like an extended comment that is too long for the comment box. But I found the question interesting, for a number of reasons:

1. I did not know that Mma had a FisherInformation function hidden away where you found it - how DID you find it?

2. Your question actually highlights one of my pet dislikes - which is the naming of distributions as black boxes - which is exactly what you are trying to emulate. Even for something as well known as the Normal distribution, there are competing parameterisations. For other distributions, there is not even agreement on what functional form the distribution should take, or any number of competing common forms that co-exist. Even when there is such agreement as to functional form, your own example shows beautifully why using black box names is often just inappropriate. You enter:

$\rightarrow$

   FisherInformation[NormalDistribution[m, s]] // MatrixForm

There is no such thing as the Fisher Information of a Normal distribution. Fisher Information is carried out wrt parameters, not with respect to black boxes. Are the parameters of $N(\mu, \sigma^2)$:

• $\mu$ and $\sigma$, or
• $\mu$ and $\sigma^2$?

Well, it can be either. And we may want to consider both cases, or something entirely different. Each approach is a different problem, with a different solution.

Consider your example of a Normal random variable with pdf $f(x)$:

_{(source: tri.org.au)}

The Fisher Information on $(\mu, \sigma)$ is:

_{(source: tri.org.au)}

The Fisher Information on $(\mu, \sigma^2)$ is:

_{(source: tri.org.au)}

To avoid any confusion, I should note that I am using here the FisherInformation function in the mathStatica add-on to Mathematica, and as declaration, that I am one of the authors.