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Suppose we have a random variable $X \sim f(x|\theta)$. If $\theta_0$ were the true parameter, the the likelihood function should be maximized and the derivative equal to zero. This is the basic principle behind the maximum likelihood estimator.

As I understand it, Fisher information is defined as

$$I(\theta) = \Bbb E \Bigg[\left(\frac{\partial}{\partial \theta}f(X|\theta)\right)^2\Bigg ]$$

Thus, if $\theta_0$ is the true parameter, $I(\theta) = 0$. But if it $\theta_0$ is not the true parameter, then we will have a larger amount of Fisher information.

my questions

1. Does Fisher information measure the "error" of a given MLE? In other words, doesn't the existence of positive Fisher information imply my MLE can't be ideal?
2. How does this definition of "information" differ from that used by Shannon? Why do we call it information?

Suppose we have a random variable $X \sim f(x|\theta)$. If $\theta_0$ were the true parameter, the the likelihood function should be maximized and the derivative equal to zero. This is the basic principle behind the maximum likelihood estimator.

As I understand it, Fisher information is defined as

$$I(\theta) = \Bbb E \Bigg[\left(\frac{\partial}{\partial \theta}f(X|\theta)\right)^2\Bigg ]$$

Thus, if $\theta_0$ is the true parameter, $I(\theta) = 0$. But if it $\theta_0$ is not the true parameter, then we will have a larger amount of Fisher information.

my questions

1. Does Fisher information measure the "error" of a given MLE? In other words, doesn't the existence of positive Fisher information imply my MLE can't be ideal?
2. How does this definition of "information" differ from that used by Shannon? Why do we call it information?