Fisher Information

Definition

Fisher Information is a measure of the amount of information that an observable random variable carries about an unknown parameter upon which the probability depends. It plays a critical role in estimating parameters, particularly through maximum likelihood estimation, where it quantifies the sensitivity of the likelihood function to changes in the parameter. Additionally, it is fundamental in deriving the Cramer-Rao lower bound, which sets a lower limit on the variance of unbiased estimators.

5 Must Know Facts For Your Next Test

1. Fisher Information can be calculated as the expected value of the squared derivative of the log-likelihood function with respect to the parameter.
2. Higher Fisher Information indicates that the data provides more information about the parameter, leading to more precise estimates.
3. In general, Fisher Information is additive for independent observations, meaning that if you have multiple independent samples, you can sum their Fisher Information.
4. The inverse of Fisher Information gives an idea of how much variability you can expect in your estimator for the parameter, which is crucial for designing experiments.
5. Fisher Information is often used to assess the efficiency of an estimator; if an estimator achieves the Cramer-Rao lower bound, it is considered efficient.

Review Questions

• How does Fisher Information relate to maximum likelihood estimation and why is it important in that context?
  • Fisher Information is essential in maximum likelihood estimation as it quantifies how much information the observed data provides about an unknown parameter. It determines how sensitive the likelihood function is to changes in this parameter. A high Fisher Information value indicates that small changes in the parameter will significantly affect the likelihood, allowing for more accurate parameter estimation.
• Discuss how Fisher Information contributes to establishing the Cramer-Rao lower bound and its implications for statistical estimation.
  • Fisher Information plays a critical role in deriving the Cramer-Rao lower bound, which states that no unbiased estimator can have a variance lower than the inverse of Fisher Information. This means that understanding Fisher Information allows us to evaluate how efficient our estimators are. If an estimator achieves this lower bound, it implies that we have reached optimal efficiency and cannot improve precision without introducing bias.
• Evaluate how increasing sample size impacts Fisher Information and its effect on estimation accuracy.
  • Increasing sample size directly increases Fisher Information because it typically leads to more accurate estimates of parameters. As more data points are collected, the variability in estimators decreases, allowing for tighter confidence intervals around estimated parameters. In essence, larger sample sizes provide more information about the underlying distribution, which improves precision and helps reach or approach the Cramer-Rao lower bound.

Related terms

Likelihood Function: A function that represents the probability of the observed data given specific values of the parameters in a statistical model.

Cramer-Rao Lower Bound: A theoretical lower bound on the variance of unbiased estimators, indicating the best precision that can be achieved for estimating a parameter.

Maximum Likelihood Estimation: A statistical method for estimating the parameters of a model by maximizing the likelihood function.