# 费雪信息 (Fisher information) 的直观意义是什么？

MLAPP中Jeffreys priors 中提到，但是不知道fisher information的直观意义是是什么，其数学定义能够说明什么问题，为什么有用

632

38094

#### 9 个回答

What is an intuitive explanation of Fisher information?
Let's consider the one dimensional case with a log-likelihood function where is the parameter of interest. The observed fisher information is the curvature at the peak of this function, that is , which intuitively tells us how peaked the likelihood function is or how "well" we know the parameter after data has been collected. A log-likelihood which is not terribly peaked is somewhat spread out, and we don't really have much confidence in what is after having collected data and conversely, a very peaked likelihood implies we have a great deal of "confidence" of the precise value of .

The expected fisher information applies the same concept except we average out the data, and we treat as a constant: it's . So it tells us on average how curved or peaked the likelihood function will be after the data has been collected, for a prescribed value of .

In the multi-dimensional setting, we simply take the Hessian as opposed to the second derivative to measure curvature.

Conceptually, I find the idea of functionals of the likelihood as a statistic itself quite funny to wrap my head around: instead of a single number, we have an entire (random) data dependent function that encapsulates something about the parameter of interest.